research article numerical solution of fuzzy fractional...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 304739 17 pageshttpdxdoiorg1011552013304739

Research ArticleNumerical Solution of Fuzzy Fractional Pharmacokinetics ModelArising from Drug Assimilation into the Bloodstream

Ali Ahmadian12 Norazak Senu12 Farhad Larki3 Soheil Salahshour4

Mohamed Suleiman1 and Md Shabiul Islam3

1 Institute for Mathematical Research Universiti Putra Malaysia (UPM) 43400 Serdang Selangor Malaysia2Mathematics Department Science Faculty Universiti Putra Malaysia (UPM) 43400 Serdang Selangor Malaysia3 Institute ofMicroengineering andNanoelectronics (IMEN)Universiti KebangsaanMalaysia (UKM) 43600Bangi SelangorMalaysia4Department of Computer Engineering Mashhad Branch Islamic Azad University Mashhad Iran

Correspondence should be addressed to Ali Ahmadian ahmadianhosseinigmailcom

Received 22 August 2013 Revised 8 October 2013 Accepted 9 October 2013

Academic Editor Ali H Bhrawy

Copyright copy 2013 Ali Ahmadian et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose a Jacobi tau method for solving a fuzzy fractional pharmacokinetics This problem can model the concentration of thedrug in the blood as time increases The proposed approach is based on the Jacobi tau (JT) method To illustrate the reliability ofthe method some special cases of the equations are solved as test examples The method reduces the solution of the problem to thesolution of a system of algebraic equations Error analysis included the fractional derivative error estimation and the upper boundof the absolute errors is introduced for this method

1 Introduction

Pharmacokinetics is defined as the study of the time courseof drug absorption distribution metabolism and excretion[1] Pharmacokinetics determines bioavailability volume ofdistribution and clearance Bioavailability is the fraction ofa drug absorbed into the systemic circulation while volumeof distribution and clearance are measure of apparent spacein the body available to contain the drug and measure ofthe bodyrsquos ability to eliminate the drug respectively Inpharmacokinetics as a basic hypothesis always a relation-ship exists between the pharmacologic or toxic responseto a drug and the concentration of the drug in the blood(plasma) However for some drugs there is no straight-forward relationship between concentration in plasma andpharmacologic effects [2] Generally the concentration ofdrug in the systemic circulation is related to the concentrationof drug at its sites of action Absorption drug distribution(target tissues) drug concentration in circulation systemrate of eliminationplasma concentration and elimination(metabolism excretion) are some parameters that directlymodify the pharmacokinetics parameters

Mathematics is widely used for the quantitative descrip-tion of drug absorption distribution metabolism and excre-tion (ADME) Some parameters in pharmacokinetics aredefined by mathematic equations which can be obtaineddirectly by measurement or through calculation using exper-imental data based on developed mathematical equationsFor example the first and simplest model used for drugabsorption from the gastrointestinal (GI) tract assumesa ldquopseudosteady staterdquo and utilizes the physicochemicalproperties of the compound in conjunction with the ldquopH-partitionrdquo hypothesis to predict the fraction of dose absorbed[3ndash5] Tissue distribution is another important determinantof the pharmacokinetics profile of a drug Hence in drugdevelopment the prediction of tissue distribution would helppredict the in vivo pharmacokinetics of a compound priorto any experiments in animals or man Two methods areavailable to predict tissue distributionThesemethods predicteither tissue plasma ratios or the volume of distributionat steady state (V) From molecular descriptors propertiessuch as lipophilicityhydrophobicity are estimated [6] Insome approaches plasma protein binding is also takeninto consideration [7] Physiological information on tissue

2 Abstract and Applied Analysis

composition the blood composition and blood flow tothe tissue is utilized to develop a partitioning model [8ndash10] Mohler et al [11] proposed a model that describes cellgrowth and detailed investigations on the metabolism andthe kinetics of the influenza infection cycle which allowfor the optimization of influenza virus vaccine productionIn addition to these simple models more complex math-ematical manipulations called mathematical models havealso been used to describe pharmacokinetics [12] Thereare many classifications of pharmacokinetics models In aspecific classification for pharmacokinetics modeling twogeneral approaches can be considered compartment-basedmodeling [13] and noncompartment-based modeling [14]Both types of modeling take advantage of the quantitativestructure-pharmacokinetics relationships that are describedby empirical mathematical algorithms They can be used toestimate the activity of a compound based on its chemicalstructure in a numeric format [15]

In this study our focus is to find the approximate solutionof the fuzzy fractional model of the compartment model forthe flow of antihistamine in the blood To this end we usethe operational matrix of the Caputo fractional derivative ofthe Jacobi tau approximation based on the Jacobi polynomialsto derive the fuzzy fractional approximate solution underHukuhara differentiability (H-differentiability) To the bestof our knowledge there are no results on the JT method forsolving pharmacokinetics equations arising in mathematicalphysics This partially motivated our interest in such modeland solution method

11 Compartment Model The compartment model frame-work is an extremely natural and valuable means with whichto formulate models for processes which have inputs andoutputs over time [16] Compartment model itself can beclassified in one-compartment model and two-compartmentmodel In the former one the body is depicted as a kineticallyhomogeneous unit while in the latter one the body resolvesinto a central compartment and a peripheral compartment[17] According to the one-compartment model the druginstantaneously distributes throughout the body and at thesame time it equilibrates between tissues Thus the drugconcentration time profile shows a monophasic responseOn the other hand in two-compartment model the drugdoes not achieve instantaneous equilibration between the twocompartments Although in this model two compartmentshave no anatomical or physiological meaning it is usuallysurmised that the central compartment is composed of tissuesthat are highly perfused such as heart lungs kidneys liverand brain while the peripheral compartment comprises lesswell-perfused tissues such as muscle fat and skin [17]

12 Methods Have Been Proposed to Solve Fractional Pharma-cokinetics Models While an increasing number of fractionalorder integrals and differential equations applications havebeen reported in the physics [18 19] signal processing[20] engineering and bioengineering literatures [21 22]little attention has been paid to this class of models in thepharmacokinetics-pharmacodynamic (PKPD) literatureThereasons for the lack of application to pharmacodynamics are

mainly two first and foremost PKPD models incorporatingfractional calculus have not been proposed second is compu-tational while the analytical solution of fractional differentialequations is available in special cases it turns out that eventhe simplest PKPD models that can be constructed usingfractional calculus do not allow analytical solutions [23 24]

In this paper we propose new category of PKPD mod-els incorporating fuzzy fractional calculus and investigatetheir behavior using purposely written algorithm The mainpurpose of this paper is to attract the attention of thefield into these possibly interesting families of PKPD mod-els In particular we will focus on compartment modelsdirect concentration-response relationships describing mod-els relating drug concentration (in plasma or biophase) to apharmacodynamic effect

The concept of fractional or noninteger order derivationand integration can be traced back to the genesis of integerorder calculus itself [25ndash27] Due to its tremendous scope andapplications in several disciplines a considerable attentionhas been given to exact and numerical solutions of fractionaldifferential equations which is an extremely difficult taskMoreover the solution techniques and their reliability arerather more important aspects Several methods have beenproposed by many researchers to solve the fractional orderdifferential equations such as Taylor series [28] variationaliteration method [29ndash31] adomian decomposition method[32 33] fractional differential transform method (FDTM)[34 35] and homotopy analysis method [36 37]

As it is known the spectral method is one of the flexiblemethods of discretization for most types of differentialequations [38ndash40] Historically spectral method has beenrelegated to fractional calculus but in few years it has beensuccessfully applied for the fractional equation models basedon the different types of orthogonal polynomials such asBlock pulse functions [41 42] Legendre polynomials [43ndash46] Chebyshev polynomials [47ndash49] Laguerre polynomials[50ndash53] and Bernstein polynomials [54ndash56] Doha et al[57] introduced the shifted Jacobi operational matrix offractional derivative which is based on Jacobi tau method forsolving numerically linear multiterm fractional differentialequations with initial or boundary conditions SubsequentlyKazem [58] generalized Jacobi integral operational matrix tofractional calculus Afterwards Doha et al [59] proposed adirect solution techniques for solving the linear multiorderfractional initial value problems with constant and variablecoefficients using shifted Jacobi tau method and quadratureshifted Jacobi tau method respectively In this paper weintend to extend the application of the Jacobi polynomi-als to solve fuzzy fractional pharmacokinetics model oforder [0 1]

On the other hand a considerable attention has beenmade to the fractional differential equations in the senseof fuzzy setting theory Agarwal et al [60] is among thepioneerswhopresented the concept of solutions for fractionaldifferential equations with uncertainty Thereafter the exis-tence and uniqueness of the solution of the fuzzy fractionaldifferential equations (FFDEs) under the Riemann-Liouvilleand Caputorsquos fuzzy fractional differentiability were investi-gated in the literature [61ndash64] However the application of

Abstract and Applied Analysis 3

the numerical methods for solving FFDEs is unknown andtraceless in the literature of the fuzzy fractional calculus andonly a few number of researches have been reported for theapproximate solution of FFDEs [64ndash69]

In the present paper we intend to introduce new familiesof PKPD models based on the application of fractionalcalculus to PKPD models The aim of the paper is not toclaim the superiority of fractional dynamics models withrespect to standard ones but it is simply to define the newfamilies and provide some insights into their qualitativebehavior The main purpose is to apply the fuzzy logic indifferential equations of fractional order which has been usedas an effective tool for considering uncertainty in modelingthe processes FFDEs can also offer a more comprehensiveaccount of the process or phenomenon specifically foranalyzing the behavior of the PKPDmodels Furthermore wesuggest possible applications and stimulate further researchwhich might or might not demonstrate the applicability andimportance of spectral methods by using of the orthogonalpolynomials for finding the approximate solutions of thePKPD models based on the fuzzy fractional calculus

The structure of this paper is as follows In the nextsection we briefly recall the mathematical foundations offractional calculus required definitions of fuzzy setting the-ory and summarize the properties of Jacobi polynomialsIn Section 3 we then provide illustrations of the govern-ing fraction equation In Section 4 the proposed methodis explained for numerical solution of the derived FFDESection 5 is devoted to the numerical solution of the problemwith different dose of the drug and the error analysis isprovided to demonstrate the applicability and validity ofthe method A final discussion comments on the results arepresented in Section 6

2 Preliminaries and Notations

In this section we are going to state the definition andpreliminaries of fuzzy mathematics [63 70 71] fractionalcalculus [25 27] and some properties of shifted Jacobipolynomials

21 The Fuzzy Settings Definitions

Definition 1 Let 119906 be a fuzzy set in R 119906 is called a fuzzynumber if

(i) 119906 is normal there exists 1199090isin R such that 119906(119909

0) = 1

(ii) 119906 is convex for all 119909 119910 isin R and 0 le 120582 le 1 it holdsthat

119906 (120582119909 + (1 minus 120582) 119910 ge min 119906 (119909) 119906 (119910)) (1)

(iii) 119906 is upper semicontinuous for any 1199090isin R it holds

that

119906 (1199090) ge lim

119909rarr119909plusmn

0

119906 (119909) (2)

(iv) [119906]0 = supp(119906) is a compact subset of R

In this paper the set of all fuzzy numbers is denotedby RF

Definition 2 Let 119906 isin RF and 119903 isin [0 1] The 119903-cut of 119906 isthe crisp set [119906]119903 that contains all elements withmembershipdegree in 119906 greater than or equal to 119903 that is

[119906]119903= 119909 isin R | 119906 (119909) ge 119903 (3)

For a fuzzy number 119906 its 119903-cuts are closed intervals in R andwe denote them by

[119906]119903= [119906

119903

1 119906

119903

2] (4)

According to Zadehrsquos extension principle the operationof addition on RF is defined as follows

(119906 + V) (119909) = sup119910isinR

min 119906 (119910) V (119909 minus 119910) 119909 isin R (5)

and scalar multiplication of a fuzzy number is given by

(119896 ⊙ 119906) (119909) =

119906(119909

119896) 119896 gt 0

0 119896 = 0

(6)

where 0 isin RF

Definition 3 (see [70]) The distance 119863(119906 V) between twofuzzy numbers 119906 and V is defined as

119863 (119906 V) = sup119903isin[01]

119889119867([119906]

119903 [V]

119903) (7)

where

119889119867([119906]

119903 [V]

119903) = max 1003816100381610038161003816119906

119903

1minus V

119903

1

1003816100381610038161003816 1003816100381610038161003816119906

119903

2minus V

119903

2

1003816100381610038161003816 (8)

is the Hausdorff distance between [119906]119903 and [V]119903

It is easy to see that 119863 is a metric in RF and has thefollowing properties (see [70 72])

(i) 119863(119906 oplus 119908 V oplus 119908) = 119863(119906 V) forall119906 V 119908 isin RF(ii) 119863(119896 ⊙ 119906 119896 ⊙ V) = |119896|119863(119906 V) forall119896 isin R 119906 V isin RF(iii) 119863(119906 oplus V 119908 oplus 119890) le 119863(119906 119908) + 119863(V 119890) forall119906 V 119908 isin RF(iv) 119863(119906 + V 0) le 119863(119906 0) + 119863(V 0) forall119906 V isin RF(v) (RF 119863) is a complete metric space

Definition 4 (see [73]) Let 119891 and 119892 be the two fuzzy-number-valued functions on the interval [119886 119887] that is 119891 119892 [119886 119887] rarr RF The uniform distance between fuzzy-number-valued functions is defined by

119863lowast(119891 119892) = sup

119909isin[119886119887]

119863(119891 (119909) 119892 (119909)) (9)

Remark 5 (see [73]) Let 119891 [119886 119887] rarr RF be fuzzy contin-uous Then from property (iv) of Hausdorff distance we candefine

119863(119891 (119909) 0) = sup119903isin[01]

max 1003816100381610038161003816119891119903

1(119909)

1003816100381610038161003816 1003816100381610038161003816119891

119903

2(119909)

1003816100381610038161003816 forall119909 isin [119886 119887]

(10)


Definition 6 (see [74]) Let 119909 119910 isin RF If there exists 119911 isin

RF such that 119909 = 119910 oplus 119911 then 119911 is called the H-differenceof 119909 and 119910 and it is denoted by 119909 ⊖ 119910

In this paper the sign ldquo⊖rdquo always stands for H-differenceand note that 119909 oplus 119910 = 119909 + (minus119910) Also throughout the paperit is assumed that the Hukuhara difference and generalizedHukuhara differentiability exist

Theorem 7 (see [75]) Let 119865 (119886 119887) rarr RF be a functionand denote [119865(119905)]119903 = [119891

119903(119905) 119892

119903(119905)] for each 119903 isin [0 1] Then

(1) if 119865 is (1)-differentiable then 119891119903(119905) and 119892

119903(119905) are dif-

ferentiable functions and

[1198651015840(119905)]

119903

= [1198911015840

119903(119905) 119892

1015840

119903(119905)] (11)


119903(119905) are dif-


[1198651015840(119905)]

119903

= [1198921015840

119903(119905) 119891

1015840

119903(119905)] (12)

Definition 8 (see [76]) Consider the 119899 times 119899 linear system ofthe following equations

119886111199091+ 119886

121199092+ sdot sdot sdot + 119886

1119899119909119899= 119910

1

119886211199091+ 119886

221199092+ sdot sdot sdot + 119886

2119899119909119899= 119910

2

11988611989911199091+ 119886

11989921199092+ sdot sdot sdot + 119886

119899119899119909119899= 119910

119899

(13)

The matrix form of the above equations is

119860119883 = 119884 (14)

where the coefficient matrix 119860 = (119886119894119895) 1 le 119894 119895 le 119899 is a

crisp 119899 times 119899 matrix and 119910119894isin RF 1 le 119894 le 119899 This system is

called a fuzzy linear system (FLS)

Definition 9 (see [76]) A fuzzy number vector (1199091 119909

2

119909119899)119905 given by 119909

119894= (119909

119894

119903

minus 119909

119894

119903

+) 1 le 119894 le 119899 0 le 119903 le 1 is called a

solution of the fuzzy linear system (2) if

(

119899

sum119895=1

119886119894119895119909119895)

119903

minus

=

119899

sum119895=1

(119886119894119895119909119895)119903

minus= 119910

119903

119894minus

(

119899

sum119895=1

119886119894119895119909119895)

119903

+

=

119899

sum119895=1

(119886119894119895119909119895)119903

+= 119910

119903

119894+

(15)

If for a particular 119896 119886119896119895gt 0 1 le 119895 le 119899 we simply get

119899

sum119895=1

119886119896119895119909119903

119895minus

= 119910119903

119896minus

119899

sum119895=1

119886119896119895119909119903

119895+

= 119910119903

119896+

(16)

To solve fuzzy linear systems see [77]

In this part we firstly give some basic definitions andsome properties of fractional calculus [27] Afterwards theextension of the fractional differentiability in the sense offuzzy concept is provided and some relevant propertieswhich are used in the rest of the paper are given [61 63 64]

Let 119898 be the smallest integer that exceeds V thenCaputorsquos fractional derivative operator of order V gt 0 isdefined as

119888119863

V119891 (119909) =

119869119898minusV

119863119898119891 (119909) if 119898 minus 1 lt V lt 119898

119863119898119891 (119909) if V = 119898 119898 isin N

(17)

where

119869V119891 (119909) =

1

Γ (V)int

119909

0

(119909 minus 119905)Vminus1119891 (119905) 119889119905 V gt 0 119909 gt 0 (18)

For the Caputo derivative we have119888119863

V119862 = 0 (119862 is a constant)

119888119863

V119909120573

=

0 for 120573 isin N0 120573 lt lceilVrceil

Γ (120573 + 1)

Γ (120573 + 1 minus V)119909120573minusV

for 120573 isin N

0 120573 ge lceilVrceil

or120573 notin N 120573 gt lfloorVrfloor

(19)

Caputorsquos fractional differentiation is a linear operationnamely

119888119863

V(120582119891 (119909) + 120583119892 (119909)) = 120582

119888119863

V119891 (119909) + 120583

119888119863

V119892 (119909) (20)

where 120582 and 120583 are constants

Theorem 10 (generalized Taylor formula [78]) Supposethat 119863119896120572

119886119891(119905) isin 119862(119886 119887] for 119896 = 0 1 119899 + 1 where 0 lt 120572 le

1 then

119891 (119905) =

119899

sum119894=0

(119905 minus 119886)119894120572

Γ (119894120572 + 1)[

119888119863

119896120572

119886119891 (119905)]

119905=119886+ 119877

120572

119899(119905 119886) (21)

with

119877120572

119899(119905 119886) =

(119905 minus 119886)(119899+1)120572

Γ ((119899 + 1) 120572 + 1)[119888119863

(119899+1)120572

119886119891 (119905)]

119905=120585

119886 ⩽ 120585 ⩽ 119905 119905 isin [119886 119887]

(22)

where119888119863

119899120572

119886=

119888119863

120572119888

119886

119888119863

120572

119886

119888119863

120572

119886⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899 119905119894119898119890119904

(23)

Here 119888119863

120572 shows fractional derivative operator in theCaputo sense and 119891

119899119894

119879(119905) = sum

119899

119894=0((119905 minus 119886)

119894120572Γ(119894120572 +

1))[119888119863

119896120572

119886119891(119905)]

119905=119886is called fractional Taylor expansion with

degree 119899119894 of 119891 and 119877120572

119899(119905 119886) is reminder term in fractional

Taylor expansion of 119891Now we present some fuzzy fractional notations which

are used later in the paper


(i) 119871RF119901(119886 119887) 1 le 119901 le infin is the set of all fuzzy-valued

measurable functions 119891 on [119886 119887] where ||119891||119901

=

(int1

0(119889(119891(119905) 0))

119901119889119905)

1119901

(ii) 119862RF[119886 119887] is a space of fuzzy-valued functions whichare continuous on [119886 119887]

(iii) 119862RF119899[119886 119887] indicates the set of all fuzzy-valued func-

tions which are continuous up to order 119899(iv) 119860119862RF[119886 119887] denotes the set of all fuzzy-valued func-

tions which are absolutely continuousNote that one can easily find these notations in the crisp

context in [25 27] and references therein

Definition 11 (see [63]) Let 119891 isin 119862RF[119886 119887] cap 119871

RF[119886 119887] TheRiemann-Liouville integral of fuzzy-valued function 119891 isdefined as

(119877119871119868V119886+119891) (119909) =

1

Γ (V)int

119909

119886

119891 (119905) 119889119905

(119909 minus 119905)1minusV

119909 gt 119886 0 lt V le 1

(24)


RF[119886 119887] Then119891 is said to be Caputorsquos H-differentiable at 119909 when

(i) (119888119863

V119886+119891) (119909) = (

119877119871119863

120573

119886+[119891 (119905) ⊖ 119891 (119886)]) (119909)

(ii) (119888119863

V119886+119891) (119909) = (

119877119871119863

120573

119886+[minus119891 (119886) ⊖ (minus119891 (119905))]) (119909)

(25)

Definition 13 (see [63]) Let 119891 119871RF[119886119887]cap119862

RF[119886 119887] and 1199090isin

(119886 119887) and Φ(119909) = (1Γ(1 minus V)) int119909

119886(119891(119905)(119909 minus 119905)

V)119889119905 We say

that 119891(119909) is fuzzy Caputo fractional differentiable of order0 lt V le 1 at 119909

0 if there exists an element (119888119863V

119886+119891)(1199090

) isin

119862RF[119886119887]

[119886 119887] such that for all 0 le 119903 le 1 ℎ gt 0

(i) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090+ ℎ) ⊖ Φ (119909

0)

ℎ

= limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0minus ℎ)

ℎ

(26)

or

(ii) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0+ ℎ)

minusℎ

= limℎrarr0

+

Φ(1199090minus ℎ) ⊖ Φ (119909

0)

minusℎ

(27)

or

(iii) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090+ ℎ) ⊖ Φ (119909

0)

ℎ

= limℎrarr0

+

Φ(1199090minus ℎ) ⊖ Φ (119909

0)

minusℎ

(28)

or

(iv) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0+ ℎ)

minusℎ

= limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0minus ℎ)

ℎ

(29)

For the sake of simplicity we say that the fuzzy-valuedfunction 119891 is 119888

[(1)minusV]-differentiable if it is differentiable asin Definition 13 case (i) and 119891 is 119888

[(2) minus V]-differentiable ifit is differentiable as in Definition 13 case (ii) and so on forthe other cases

Theorem 14 (see [63]) Let 0 lt V le 1 and 119891 isin 119860119862RF[119886 119887]

then the fuzzy Caputo fractional derivative exists almosteverywhere on (119886 119887) and for all 0 le 119903 le 1 one has

(119888119863

V119886+119891) (119909 119903) = [

1

Γ (1 minus V)int

119909

119886

1198911015840119903

minus(119905) 119889119905

(119909 minus 119905)V

1

Γ (1 minus V)int

119909

119886

1198911015840119903

+(119905) 119889119905

(119909 minus 119905)V ]

= [(1198681minusV119886+

119863119891119903

minus) (119909) (119868

1minusV119886+

119863119891119903

+) (119909)]

(30)

when 119891 is (1)-differentiable and

(119888119863

V119886+119891) (119909 119903)

= [1

Γ (1 minus V)int

119909

119886

1198911015840119903

+(119905) 119889119905

(119909 minus 119905)V

1

Γ (1 minus V)int

119909

119886

1198911015840119903

minus(119905) 119889119905

(119909 minus 119905)V ]

= [(1198681minusV119886+

119863119891119903

+) (119909) (119868

1minusV119886+

119863119891119903

minus) (119909)]

(31)

when f is (2)-differentiable

Theorem 15 (fuzzy generalized Taylorrsquos formula [72])Let 119891(119909) isin 119860119862

RF[119886119887](0 119887] and suppose that 119888

119863119896120572119891(119909) isin

119862RF[119886119887]

(0 119887] for 119896 = 0 1 119899 + 1 where 0 lt 120572 lt 1 0 le

1199090le 119909 and 119909 isin (0 119887] Then one has

[119891 (119909)]119903= [119891

119903(119909) 119891

119903

(119909)]

119891119903(119909) =

119899

sum119894=0

119909119894120572

Γ (119894120572 + 1)

119888119863

119894120572119891

119903(0

+)

+

119888119863

(119899+1)120572119891

119903(119909

0)

Γ (119899120572 + 120572 + 1)119909(119899+1)120572

119891119903

(119909) =

119899

sum119894=0

119909119894120572

Γ (119894120572 + 1)

119888119863

119894120572119891

119903

(0+)

+

119888119863

(119899+1)120572119891

119903(119909

0)

Γ (119899120572 + 120572 + 1)119909(119899+1)120572

(32)

where 119888119863

120572119891

119903(0) =

119888119863

120572119891

119903(119909)|

119909=0119888119863

120572119891

119903

(0) =119888119863

120572119891

119903

(119909)|119909=0

22 Jacobi Polynomials The well-known Jacobi polynomialsassociated with the parameters (120572 gt minus1 120573 gt minus1)(see eg Luke [79] and Szego [80]) are a sequence ofpolynomials 119875(120572120573)

119894(119905) (119894 = 0 1 ) each respectively of

degree 119894 For using these polynomials on (0 119871) we presentthe shifted Jacobi polynomials by implementing the change


of variable 119905 = (2119909119871 minus 1) Let the shifted Jacobi polyno-mials 119875(120572120573)

119894(2119909119871 minus 1) be denoted by 119875(120572120573)

119871119894(119909) satisfying the

orthogonality relation

int119871

0

119875(120572120573)

119871119895(119909) 119875

(120572120573)

119871119896(119909) 119908

(120572120573)

119871(119909) 119889119909 = ℎ

119896 (33)

where 119908(120572120573)

119871(119909) = 119909

120573(119871 minus 119909)

120572 and

ℎ119896=

119871120572+120573+1

Γ (119896 + 120572 + 1) Γ (119896 + 120573 + 1)

(2119896 + 120572 + 120573 + 1) 119896Γ (119896 + 120572 + 120573 + 1)119894 = 119895

0 119894 = 119895

(34)

The shifted Jacobi polynomial 119875(120572120573)

119871119894(119909) of degree 119894 has the

form

119875(120572120573)

119871119894(119909)

=

119894

sum119896=0

(minus1)119894minus119896

Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

Γ (119896 + 120573 + 1) Γ (119894 + 120572 + 120573 + 1) (119894 minus 119896)119896119871119896119909119896

(35)

where

119875(120572120573)

119871119894(0) = (minus1)

119894Γ (119894 + 120573 + 1)

Γ (120573 + 1) 119894

119875(120572120573)

119871119894(119871) =

Γ (119894 + 120572 + 1)

Γ (120572 + 1) 119894

(36)

Also we can state the shifted Jacobi polynomial by thefollowing recurrence relation

119875(120572120573)

119871119894(119909)

= (120572 + 120573 + 2119894 minus 1) (1205722minus 120573

2+ (

2119909

119871minus 1))

times (120572 + 120573 + 2119894) (120572 + 120573 + 2119894 minus 2)

times (2119894 (120572 + 120573 + 119894) (120572 + 120573 + 2119894 minus 2))minus1

times 119875(120572120573)

119871119894minus1(119909)

minus(120572 + 119894 minus 1) (120573 + 119894 minus 1) (120572 + 120573 + 2119894)

119894 (120572 + 120573 + 119894) (120572 + 120573 + 2119894 minus 2)

times 119875(120572120573)

119871119894minus2(119909) 119894 = 2 3

(37)

where 119875(120572120573)

1198710(119909) = 1 and 119875

(120572120573)

1198711(119909) = ((120572 + 120573 + 2)2)(2119909119871 minus

1) + (120572 minus 120573)2We notice that a function 119906(119909) square integrable

in (0 119871) can be expanded in terms of shifted Jacobipolynomials as

119906 (119909) =

+infin

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (38)

where the coefficients 119886119895are

119886119895=

1

ℎ119895

int119871

0

119875(120572120573)

119871119895(119909) 119906 (119909)119908

(120572120573)

119871(119909) 119889119909 119895 = 0 1 (39)

For a given particular problem only the first (119873 + 1)-termsshifted Jacobi polynomials are used Therefore we have

119906119873(119909) ≃

119873

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (40)

Theorem 16 (see [59]) The Caputo fractional derivative oforder V of the shifted Jacobi polynomials of degree 119894 is obtainedfrom

119863V119875

(120572120573)

119871119894(119909) =

infin

sum119895=0

119878V (119894 119895 120572 120573) 119875(120572120573)

119871119894(119909)

119894 = lceilVrceil lceilVrceil + 1

(41)

where

119878V (119894 119895 120572 120573)

=

119894

sum119896=lceilVrceil

(minus1)119894minus119896119871120572+120573minusV+1

Γ (119895 + 120573 + 1)

times Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

times (ℎ119895Γ (119895 + 119896 + 120572 + 120573 + 1) Γ (119896 + 120573 + 1)

times Γ (119894 + 120572 + 120573 + 1) Γ (119896 minus V + 1) (119894 minus 119896))minus1

times

119895

sum119897=0

(minus1)119895minus119897Γ (119895 + 119897 + 120572 + 120573 + 1) Γ (120572 + 1)

times Γ (119897 + 119896 + 120573 minus V + 1)

times (Γ (119897 + 120573 + 1) Γ (119897 + 119896 + 120572 + 120573 minus V + 2) (119895 minus 119897)119897)minus1

(42)

3 Pharmacokinetics Model Equation

31 Drug Assimilation into the Blood The drug dissolves inthe gastrointestinal tract (GI) and each ingredient is diffusedinto the bloodstream They are carried to the locations inwhich they act and are removed from the blood by thekidneys and liver Generally the problem of drug assimilationinto the body can be considered as a two-compartmentmodel GI-tract and the bloodstream [16] Different com-partments and the inputoutput of the model are depicted inFigure 1


Drug intakeGI tract

DigestionBlood

Tissue

Figure 1 Schematic of inputoutput compartment for drug assimi-lation

For each compartment by applying the balance lawwe canobtain

rate of change ofdrug in GI tract

= rate of drug

intake minus rate drug leaves

GI-tract

rate of change ofdrug in blood

= rate drug

enters blood minus rate drug leaves

blood

(43)

In this study we consider a case of a single cold pill Also thereis no ingestion of the drug except that which occurs initially

32 Case of a Single Cold Pill Let us consider 119909(119905) to bethe amount of drug in the GI-tract at time 119905 and 119910(119905) theamount in the bloodstream at time 119905 In the GI-tract wesuppose that the pill is to be swallowed and so after this event(over subsequent time) we have nothing more entering theGI-tract The pill dissolves and diffuses into the bloodstreamfrom the GI-tract So the GI-tract is only an output termAssuming that the output rate is proportional to the amountof drug in the bloodstream which is in fact proportional tothe GI-tract drug concentration then [16]

119889119910

119889119905= minus119896

1119909 119909 (0) = 119909

0 (44)

where 1199090is the amount of a drug in the pill and 119896

1is a

positive coefficient We suppose that 119910(0) = 0 which meansthat the initial amount of the drug in the bloodstream is zeroAs the drug diffuses from theGI-tract the level increases andas the kidneys and liver remove it the level of drug graduallydecreases Thus

119889119910

119889119905= 119896

1119909 minus 119896

2119910 119910 (0) = 0 (45)

with 1198962another positive constant Decongestant and an

antihistamine are the component of the cold pill and thecoefficient of proportionality (119896

1and 119896

2) is different for the

different component drugs in the pillIn order to obtain the growth and decay of antihistamine

levels in the GI tract and bloodstream rearrange the first rateequation multiply by the integrating factor 1198901198961119905 integrateand then use the initial data to obtain

119883 = 119860119890(minus1198961119905) (46)

Insert this formula for 119909(119905) into the second rate equationwhich becomes

119889119910 (119905)

119889119905+ 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (47)

In this paper we are concerned with fractional time deriva-tives regarding (17) and (18) Having defined 119888

119863V we can

now turn to fractional differential equations and systemsof fractional differential equations which will be used tospecify PKPD models and will need to be solved over aninterval [0 119905] in accordance with appropriate initial condi-tions

A typical feature of differential equations (both classicaland fractional) is the need to specify additional conditionsin order to produce a unique solution For the case ofCaputo fractional differential equations these additionalconditions are just the initial conditions which are simi-larly required by classical ODEs In contrast for Riemann-Liouville fractional differential equations these additionalconditions constitute certain fractional derivatives (andorintegrals) of the unknown solution at the initial point 119905 =

0 [81] which are functions of 119905 These initial conditions arenot physical furthermore it is not clear how such quantitiesare to be measured from experiment say so that they canbe appropriately assigned in an analysis [82] If for no otherreason the need to solve fractional differential equationsis justification enough for choosing Caputorsquos definition forfractional differentiation over the more commonly used (atleast in mathematical analysis) definition of Liouville andRiemann and this is the operator that we choose to use inthe following

We start by representing drug concentration in theeffect compartment by the (Caputo) fractional differentialequation

119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (48)

In the standard direct action model the effect attime 119905 119884(119905) is expressed by an arbitrary (memory-less) func-tion of drug concentration in the effect site at time 119905 119866(119910(119905))however to generate a wider class of relationships we assumethat the effect at time 119905 isin [0 1] is related to the fuzzy Caputofractional derivative of 119910(119905) So we have

119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [119910119903

0 119910

119903

0] (49)

in which 119910(119909) 119871RF[0 1] cap 119862

RF[0 1] is a continuous fuzzy-valued function and 119888

119863V0+ denotes the fuzzy Caputo frac-

tional derivative of order V isin [0 1]

Remark 17 In this paper the drug concentration is modeledby an oscillation-relaxation fuzzy fractional differential equa-tion So the right hand side of (49) is determined based on themodel parameters It should mention here that the proposedmethod can be extended easily for solving other types ofFFDEs with more complicated right hand side functions Formore details one can refer to (33) in [65] and [57ndash59]

We have assumed that 1198961

= 1198962 an assumption that is

justified by the pharmaceutical data For the ldquoaveragerdquo person


a pharmaceutical company estimates that the values of therate constants for the antihistamine in the cold pills are 119896

1=

06931 (hour)minus1 and 1198962= 00231 (hour)minus1 It can be observed

from (46) that level of antihistamine in the bloodstreamincreases as the time increases and saturate in a maximumvalue of antihistamine however (47) can conclude that asthe time increases the amount of antihistamine in GI-tractdeclines and reaches a minimum value We now considertwo different cases of patients who are not average Weoften define 119896

2as clearance coefficient of medication from

the blood The value of 1198962often varies from old and sick

patients than young and healthy cases This means that thelevel of medication in the blood may become and thenremain excessively high with a standard dosage for the casesthat 119896

2is much lower which is normally observed in old and

sick casesWe investigate sensitivity of the medication over a 24-

hour period by keeping the value of 1198961fixed at some value

such as 1386 but setting 1198962

= 001386 006386 01386

06386 1386 (119860 = 1) Different values of 1198962correspond to

people of different ages and states of health For all casesas it is expected that for the first few hours the amountof antihistamine in the bloodstream increases and thendeclines gradually However for the cases with lower value ofclearance coefficient (old and sick) themaximumvalue of themedication in bloodstream was much higher than the caseswith high 119896

2 and it did not decline for remaining timesThis

means that the level of medication in the bloodstream stays athigh level for a long time and it could not be absorbed fromthe blood The same trend is observed for the case that 119896

2is

constant (1198962= 00231) and the value of 119896

1varies (119896

1=

006931 011 03 06931 10 and 15) In this case the 1198961is

a constant for the GI tract which is analogous to the 1198962for

the bloodstream

4 Description of the Method

In this study by developing the Jacobi polynomial approxi-mation [57 58 65] with the help of the matrix operationsthe tau method and the fuzzy Caputo fractional derivativewe obtain an approximate solution of the problem (49) fordifferent values of 119896

1and 119896

2 As it is known the existence

and uniqueness of the FFDEs have been introduced in [6061 63] Here we assume that the shifted Jacobi polynomialsare defined on (0 1) so we have [65]

119906 (119909) =

+infin

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

1119894(119909) (50)

where the fuzzy coefficients 119886119894are gained by

119886119894=1

ℎ119894

int1

0

119875(120572120573)

1119894(119909) ⊙ 119906 (119909) ⊙ 119908

(120572120573)

1(119909) 119889119909 119894 = 0 1

(51)

and 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573 119906 isin 119871RF119901[0 1] cap 119862

RF[0 1]and 119875

(120572120573)

1119894(119909) is as the same as the shifted Jacobi polyno-

mials presented in Section 22 and sumlowast means addition with

respect to oplus in RF

Remark 18 In the remainder of paper formore simplicity weconsider 119875(120572120573)

119894(119909) instead of 119875(120572120573)

1119894(119909) and 119908

(120572120573)(119909) = (1 minus

119909)120572⊙ 119909

120573 instead of 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573

Remark 19 Practically only the first (119873 + 1)-terms shiftedJacobi polynomials are taken into consideration So we have

119906 (119909) ≃ 119906119873(119909) =

119873

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

119894(119909) = a119879 ⊙ Φ

119873(119909) (52)

where the fuzzy shifted Jacobi coefficient vector 119860119879 andshifted Jacobi polynomials vector Φ

119873+1(119909) are presented by

a119879 = [1198860 119886

1 119886

119873]

Φ119873(119909) = [119875

(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)]

119879

(53)

We can state the fuzzy approximate function (52) in theparametric form as follows

Definition 20 (see [65]) Let 119906(119909) isin 119871RF119901[0 1] cap 119862

RF[0 1]the approximation of fuzzy-valued function 119906(119909) in the para-metric form is

119906119903(119909) ≃ 119906

119903

119873(119909) = [

119873

sum119894=0

119886119903

119894minus119875

(120572120573)

119894(119909)

119873

sum119894=0

119886119903

119894+119875

(120572120573)

119894(119909)]

0 le 119903 le 1

(54)

We call that Span119875(120572120573)

119899(119909) 119899 ge 0 forms a complete

orthogonal system in 119871RF

2119908(120572120573)

(0 1) Hence we set

S119873(0 1) = Span 119875(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)

(55)

The shifted Jacobi tau method to (49) is to obtain 119906119873

isin

119878119873(0 1) such that

(119863V119906119873 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119906

119873 119875

(120572120573)

119896(119909))

119908(120572120573)

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1 119906119873 (0) = 119889

0

(56)

where 119908(120572120573)(119909) = (1 minus 119909)

120572⊙ 119909

120573 and (119906 V)119908(120572120573) = int

1

0119908

(120572120573)

(119909) ⊙ 119906(119909) ⊙ V(119909)119889119909 denotes the fuzzy inner productin 119871

RF

2119908(120572120573)

(0 1) and 1198890is the fuzzy initial value of the prob-

lemLet us define the following notations

119891119896= (119896

1119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

f = (1198910 119891

1 119891

119873minus1 119889

0)119879

(57)


Hence the variational formulation of (56) according toRelation (14) in [65] by means of a typical tau method likein the crisp context [29] and (52) is equivalent to

119873

sum119895=0

119886119895⊙ [(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895⊙ 119875

(120572120573)

119895(0) = 119889

0

(58)

Denoting

119860 = (119886119896119895)0lt119896119895lt119873

119862 = (119888119896119895)0lt119896119895lt119873

(59)

we investigate that (58) is equivalent to the matrix system

(119860 + 1198962119862) ⊙ a = f (60)

The elements of the matrices mentioned above are deter-mined in the next theorem

Theorem 21 (see [59]) Let us denote 119886119896119895

= (119863(V)119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))

119908(120572120573) (0 ⩽ 119896 ⩽ 119873 minus 1 0 ⩽ 119895 ⩽ 119873) 119886

119896119895=

119863119896minus119873

119875(120572120573)

119895(0) (119896 = 119873 0 ⩽ 119895 ⩽ 119873) and 119888

119896119895= (119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))

119908(120572120573) (0 ⩽ 119896 ⩽ 119873 minus 1 0 ⩽ 119895 ⩽ 119873) then the nonzero

elements of 119886119896119895and 119888

119896119895are given as

119886119896119895= ℎ

119896119878V (119895 119896 120572 120573) 0 ⩽ 119896 ⩽ 119873 minus 1 1 ⩽ 119895 ⩽ 119873 119886

119896119895

=(minus1)

119895minus119896+119873Γ (119895 + 120573 + 1) (119895 + 120572 + 120573 + 1)

119896minus119873

Γ (119895 minus 119896 + 119873 + 1) Γ (119896 minus 119873 + 1 + 120573)

119896 = 119873 0 ⩽ 119895 ⩽ 119873 119888119896119895

= ℎ119896 0 ⩽ 119896 = 119895 ⩽ 119873 minus 1

(61)

41 Error Analysis In this section error analysis of themethod will be presented for the FFDEs Firstly an upperbound of the absolute errors will be given for the techniqueby using generalized Taylor formula Secondly an errorbound will be introduced for the approximation of the fuzzyfractional Caputorsquos derivative using Jacobi polynomials

Lemma 22 Let 119910119873V(119905) and 119891(119905) be the Jacobi approximate

solution (40) and the exact solution of (48) respectivelyIf 119863119896V

0+119891(119905) isin 119862(0 119887] 119896 = 0 1 119873 + 1 then

1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816 (62)

where 119891119873119894

119879and 119877

V119873are defined according to Theorem 10

Proof Since 119863119896V0+119891(119905) isin 119862(0 119887] 119896 = 0 1 119873 + 1 119891 can be

expanded to the fractional Taylor series

119891119873119894

119879(119905) =

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+ (63)

and its reminder term is

119877V119873(119905 0) =

119905(119873+1)V

Γ ((119873 + 1) V + 1)[119863

(119873+1)V0+

119891 (119905)]119905=120585

0 ⩽ 120585 ⩽ 119905 forall119905 isin (0 119887]

(64)

Therefore

119891 (119905) minus 119891119873119894

119879(119905) = 119877

V119873(119905 0) (65)

From (65) and triangle inequality we obtain

1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 =

10038161003816100381610038161003816119891 (119905) minus 119910

119873V (119905) + 119891119873119894

119879(119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816

⩽10038161003816100381610038161003816119891 (119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816+10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

=1003816100381610038161003816119877

V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

(66)

Therefore an upper bound of the absolute errors is obtainedfor the method in the crisp cases

Now we provide an upper bound for the absolute errorsof the fuzzy approximate solution by using the proposedmethod

Theorem 23 Let 119906(119905) isin 119871RF119901[0 1] cap 119862

RF[0 1] be the fuzzyexact solution of (49) and 119906

119873(119905) is the best fuzzy Jacobi

approximate function (52) and suppose that 119888119863

119896V119906(119905) isin

119862RF[0 119887] for 119896 = 0 1 119873 + 1 where 0 lt V lt 1 and 119905 isin

[0 119887] Then one has

[119906 (119905)]119903= [119906

119903(119905) 119906

119903(119905)] [119906

119873(119905)]

119903= [119906

119903

119873(119905) 119906

119903

119873(119905)]

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

10038161003816100381610038161003816119877V

119873(119905 0)

10038161003816100381610038161003816+10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

(67)

where regarding Theorem 15 one has

[119891119873119894

119879(119905)]

119903

= [119891119873119894

119879(119905 119903) 119891

119873119894

119879(119905 119903)]

= [

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+]


[119877V119873(119905 0)]

119903= [119877

V119903119873(119905 0) 119877

V119903

119873(119905 0)]

= [

[

119888119863

(119873+1)V119891

119903(0+)

Γ (119873V + V + 1)119905(119873+1)V

119888119863

(119873+1)V119891

119903

(0+)

Γ (119873V + V + 1)119905(119873+1)V]

]

(68)

Proof It is straightforward from Lemma 22 and the fuzzyfunction definition

Also in the following theorem according to the Relation(14) in [65] an upper bound for the absolute errors of theapproximate function of fuzzy fractional Caputorsquos derivativeis provided

Theorem 24 (see [65]) Assume that the error function offuzzy Caputo fractional derivative operator by using the shiftedJacobi polynomials 119864

119896V is continuously fuzzy differentiablefor 0 lt 119909

0le 119909 119909 isin (0 1] Additionally 119864

119896V isin

119862RF[119909

0 1] and 0 lt V lt 1 then the error bound is given by

119863lowast(RF119863

VΦ (119909)

RF119863(V)Φ (119909))

le119909minusV0

(2) |Γ (1 minus V)|(119878)

2(120573 + 2

2)radic119861 (120572 + 1 120573 + 1)

(69)

5 Numerical Results

In this section the fuzzy fractional PKPD model (49) issolved for different values of 119896

1and 119896

2by using the JT

method presented in Section 4 We have performed allnumerical computations with a computer programwritten inMATLAB Also absolute errors between fuzzy approximatesolution [119910

119873]119903= [119910

119903

119873 119910

119903

119873] and the corresponding exact solu-

tions 119910(119905 119903) = [119910(119905 119903) 119910(119905 119903)] that is [119873119890]119903= [|119910

119903

119873minus 119910

119903|

|119910119903

119873minus 119910

119903|] are considered

Now we recall the FFDE (49) as follows119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (70)

in which 119910(119909) 119871RF[0 1] cap 119862



tional derivative of order V isin [0 1]Two cases are considered for solving by the proposed

technique In the first case we assume that 1198962is unchange-

able and different values of 1198961are substituted in (49) to get

the fuzzy approximate solution Conversely in the secondcase we try to solve the problem such that 119896

1has an invari-

able value and 1198962is varied The problem is analyzed for both

cases in details

Case I Let us consider from Section 3 that 1198962= 00231

but let 1198961vary (eg 06931 011 and 03) So (49) with the

assumption namely 1198962= 00231 and Dose 119860 = 1 is as

follows119888119863

V119910 (119905) + 00231119910 (119905) = 06931119890

minus06931119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (71)

By usingTheorem 7 for the above equation under 119888[(1)minusV]

differentiability we have the following systems119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = minus1 + 119903 0 lt 119903 le 1

119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = 1 minus 119903 0 lt 119903 le 1

(72)

Solving (72) leads to determining the exact solution of(71) as follows

119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus00231119905V]

+ int119905

0

(119905 minus 119909)Vminus1119864VV [minus00231(119905 minus 119909)

V]

times (06931119890minus06931119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt 119903 le 1

(73)

We seek the fuzzy approximate solutions by applying thetechnique described in Section 4 with 119873 = 2 and as

1199102(119905) = 119886

0⊙ 119875

(120572120573)

0(119905) + 119886

1⊙ 119875

(120572120573)

1(119905) + 119886

2⊙ 119875

(120572120573)

0(119905)

(74)

Here we have

119863085

= (

0 0 0

18639 03901 minus01755

minus03901 45267 08696

) (75)

where 120572 = 120573 = 0 Regarding (54) we have

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

(76)

By solving the fuzzy linear system (60) in the parametricform one can get fuzzy unknown coefficients in (76) as

1198860= minus05757 119886

1= 02624 119886

2= minus00619

1198860= 12008 119886

1= 02408 119886

2= minus00600

(77)

where we assume that 119903-cut = 01 in (77)


Table 1 The results of the proposed method for Case I with V = 085 120572 = 120573 = 0 and119873 = 8

1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890

0 90140119890 minus 5 16623119890 minus 5 40452119890 minus 5 84377119890 minus 5 10860119890 minus 5 34689119890 minus 5











10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0

Figure 2 The absolute errors for different 120572 and 120573 with 119873 = 9 1198961= 03 and V = 075 Case I

10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11

Figure 3 The absolute errors for different values 119873 with V = 095 1198961= 011 and 120572 = 120573 = 0 Case I


Table 2 The fuzzy coefficients for Case II with V = 075 120572 = 120573 = 05119873 = 2 and 1198962= 001386

119903 1198860

1198861

1198862

1198860

1198861

1198862

0 minus042793 024218 minus011135 155482 023234 minus01100301 minus032879 024169 minus011128 145568 023283 minus01100902 minus022965 024120 minus011121 135654 023332 minus01101603 minus013052 024071 minus011115 125741 023382 minus01102204 minus003138 024021 minus011108 115827 023431 minus01102905 006775 023972 minus011102 105913 023480 minus01103606 016689 023923 minus011095 095999 023529 minus01104207 026603 023874 minus011088 086085 023578 minus01104908 036516 023824 minus011082 076172 023628 minus01105509 046430 023775 minus011075 066258 023677 minus0110621 056344 023726 minus011069 056344 023726 minus011069

Table 3 The results of the proposed method for Case II with V = 095 120572 = 120573 = 05 and119873 = 7

1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03

Figure 4 The fuzzy approximate solution of Case I for different value of 1198961 120572 = 120573 = 05119873 = 8


10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0

Figure 5 The absolute errors for different 120572 and 120573 with 119873 =

8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11

Figure 6 The absolute errors for different values 119873 with V =

085 1198962= 01386 and 120572 = 120573 = 0 Case II

The comparison between absolute errors of different 1198961

obtained by our method is shown in Table 1 In Figure 2logarithmic plot of absolute error with different valuesof 120572 and 120573 was obtained and different number of Jacobifunctions was experienced for solving Case I by usingthe proposed method in Figure 3 Finally the approximatefuzzy solutions are shown for different values of 119896

1with the

fractional order V = 085 in Figure 4The absolute error value for 119903-cut varied from 0 to 1

for different value of 1198961which is calculated in Table 1 As

it can be observed at a constant 119903-cut by increasing thevalue of 119896

1which is the ratio of variation of the drug in the

bloodstream to the amount of drug in the GI-tract the valueof absolute error increases This is analogous to the previous

reports for various values of the 1198961[12] It should be noted

here that the value of 119873119903

119890is directly proportional to the vari-

ation of exact solutions and corresponding fuzzy approximatesolution As a result increase of 119873119903

119890with increasing 119896

1is also

expected

Case II In this case we assume that 1198961

= 1386 andis not variable but 119896

2takes different values (ie 119896

2=

001386 006386 and 01386) and Dose 119860 = 1 So (70) alterswith for example 119896

2= 001386 as follows

119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)

with the exact solution as119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus001386119905

V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1

(79)By applying the JT method in Section 4 we can get the

fuzzy unknown coefficients 119886119895119873

119895=0by using (58) which is in

the parametric form as follows119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386

Figure 7 The fuzzy approximate solution of Case II for different value of 1198962 120572 = 0 120573 = 05 and 119873 = 9

where 119863(V) is given as

119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)

with 120572 = 120573 = 05 and 119873 = 2 By substituting the abovematric in (80) we can reach the fuzzy linear algebraic system(60) that can solve easily to determine the fuzzy coefficientsas shown in Table 2

We compared JT method results for different valuesof 119896

2and the outcomes are tabulated in Table 3The absolute

errors of the proposed method for this case are exhibitedin Figure 5 with four choices of 120572 and 120573 Clearly the bestapproximation is achieved when the values of 120572 = 0 and 120573 =

0 Additionally we compared the approximate solutionsobtained by the present method at 119873 = 4 7 9 and 11 whichis shown in Figure 6 From Figure 6 one can concludethat with increasing the number of Jacobi polynomials theabsolute errors are decreasing dramatically Finally Figure 7shows the numerical results for different values of 119896

2at V =

075 It can be seen that for all the values of 1198962the approxi-

mate solution is the fuzzy numberIn Table 3 the value of absolute error for 119903-cut varied

from 0 to 1 for different value of 1198962is calculated At a

constant 119903-cut value a descending trend is observed withincreasing the value of 119896

2 This trend is opposed to the trend

observed in Table 1 for different value of 1198961 119896

2is clearance

constant and it is a coefficient for 119910(119905) which is the amountof drug in the bloodstream in (44) We already explainedthat by decreasing the value of 119896

2which happens in the

case of old and sick patient the drug absorption from thebloodstream by kidney and liver also decreases and the

drug stays in the bloodstream for longer time Since theconstant value of 119896

2appears with negative sign in the main

differential equation as we expected the value of 119873119903

119890which is

directly proportional to the exact solutions is decreased withincreasing the value of 119896

2

6 Conclusion

Fuzzy theory provides a suitable way to objectively accountfor parameter uncertainty in models Fuzzy logic approachesappear promising in preclinical applications and might beuseful in drug discovery and design Considerable progresshas been made in the last few years in the developmentof computational approaches for prediction of drug absorp-tion distribution metabolism and excretion Whilst severalapproaches have been developed in pharmacokinetics mostof these approaches have not yet been adequately used in thecomplex process such as prediction of metabolism and theyrequire further improvement

In summary in this research a tau method based onthe Jacobi operational matrix was utilized to numericallysolve the PKPD equation arising from drug assimilationinto the bloodstream The comparison of the results showsthat the present method is a powerful mathematical tool forfinding the numerical solutions of a generalized linear fuzzyfractional PKPD equation

Although we concentrated on applying our algorithm tosolve fuzzy fractional PKPD equation we show that suchalgorithm can be applied to solve other types of fractionalequations models in science and engineering fields Ouralgorithm for the fuzzy fractional PKPD equations is efficient


and numerically stable Numerical results are presentedwhich exhibit the high accuracy of the proposed algorithm

By using the obtained operational matrix the followingobjectives were achieved

(i) The given fuzzy fractional differential equation wasconverted into a fuzzy algebraic system of equationssimplifying the solution procedure

(ii) The method is computer oriented thus solving fuzzyfractional differential equations of different fractionalorders V becomes a matter of changing V only

References

[1] B G Katzung S B Masters and A J Trevor Basic amp ClinicalPharmacology McGraw-Hill 2004

[2] H Peng and B Cheung ldquoA review on pharmacokinetic model-ing and the effects of environmental stressors on pharmacoki-netics for operationalmedicine operational pharmacokineticsrdquoTech Rep ADA509469 DTIC 2009

[3] A Suzuki W I Higuchi and N F Ho ldquoTheoretical modelstudies of drug absorption and transport in the gastrointestinaltract Irdquo Journal of Pharmaceutical Sciences vol 59 no 5 pp644ndash651 1970

[4] H Boxenbaum ldquoAbsorption potential and its variantsrdquo Phar-maceutical Research vol 16 no 12 p 1893 1999

[5] A Boobis U Gundert-Remy P Kremers P Macheras andO Pelkonen ldquoIn silico prediction of ADME and pharmacoki-netics report of an expert meeting organised by COST B15rdquoEuropean Journal of Pharmaceutical Sciences vol 17 no 4-5 pp183ndash193 2002

[6] P Poulin K Schoenlein and F P Theil ldquoPrediction of adiposetissue plasma partition coefcients for structurally unrelateddrugsrdquo Journal of Pharmaceutical Sciences vol 90 pp 436ndash4472001

[7] F Lombardo R S Obach M Y Shalaeva and F Gao ldquoPredic-tion of volume of distribution values in humans for neutral andbasic drugs using physicochemical measurements and plasmaprotein binding datardquo Journal of Medicinal Chemistry vol 45no 13 pp 2867ndash2876 2002

[8] P Poulin and F P Theil ldquoA priori prediction of tissueplasma partition coefcients of drugs to facilitate the use ofphysiologically-based pharmacokinetic models in drug discov-eryrdquo Journal of Pharmaceutical Sciences vol 89 pp 16ndash35 1999

[9] P Poulin and F-P Theil ldquoPrediction of pharmacokinetics priorto in vivo studies 1 Mechanism-based prediction of volume ofdistributionrdquo Journal of Pharmaceutical Sciences vol 91 no 1pp 129ndash156 2002

[10] P Poulin and F-P Theil ldquoPrediction of pharmacokinetics priorto in vivo studies II Generic physiologically based pharma-cokinetic models of drug dispositionrdquo Journal of Pharmaceu-tical Sciences vol 91 no 5 pp 1358ndash1370 2002

[11] L Mohler D Flockerzi H Sann and U Reichl ldquoMathematicalmodel of influenza a virus production in large-scale microcar-rier culturerdquo Biotechnology and Bioengineering vol 90 no 1 pp46ndash58 2005

[12] P L Bonate Pharmacokinetic-Pharmacodynamic Modeling andSimulation Springer 2011

[13] MHolz andA Fahr ldquoCompartmentmodelingrdquoAdvancedDrugDelivery Reviews vol 48 no 2-3 pp 249ndash264 2001

[14] P Veng-Pedersen ldquoNoncompartmentally-based pharmacoki-netic modelingrdquo Advanced Drug Delivery Reviews vol 48 no2-3 pp 265ndash300 2001

[15] D EMager ldquoQuantitative structure-pharmacokineticpharma-codynamic relationshipsrdquoAdvancedDrugDelivery Reviews vol58 no 12-13 pp 1326ndash1356 2006

[16] B Barnes and G R FulfordMathematical Modelling with CaseStudies A Differential Equation Approach Using Maple andMATLAB CRC Press 2011

[17] S Dhillon and K Gill ldquoBasic pharmacokineticsrdquo in ClinicalPharmacokinetics pp 1ndash44 Pharmaceutical Press 2006

[18] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991

[19] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[20] R J Marks II andMWHall ldquoDifferintegral interpolation froma bandlimited signalrsquos samplesrdquo IEEE Transactions on AcousticsSpeech and Signal Processing vol 29 no 4 pp 872ndash877 1981

[21] A Atangana and D Baleanu ldquoNonlinear fractional Jaulent-Miodek and Whitham-Broer-Kaup equations within Sumudutransformrdquo Abstract and Applied Analysis vol 2013 Article ID160681 8 pages 2013

[22] B Mehdinejadiani A A Naseri H Jafari A Ghanbarzadehand D Baleanu ldquoA mathematical model for simulation ofa water table profile between two parallel subsurface drainsusing fractional derivativesrdquo Computers amp Mathematics withApplications vol 66 no 5 pp 785ndash794 2013

[23] C Csajka andDVerotta ldquoPharmacokinetic-pharmacodynamicmodelling history and perspectivesrdquo Journal of Pharmacokinet-ics and Pharmacodynamics vol 33 no 3 pp 227ndash279 2006

[24] D Verotta ldquoFractional dynamics pharmacokinetics-pharmaco-dynamic modelsrdquo Journal of Pharmacokinetics and Pharmaco-dynamics vol 37 no 3 pp 257ndash276 2010

[25] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Publish-ing Hackensack NJ USA 2012

[26] D Baleanu Z B G Guvenc and J A Tenreiro Machado NewTrends in Nanotechnology and Fractional Calculus ApplicationsSpringer New York NY USA 2010

[27] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[28] M Gulsu Y Ozturk and A Anapalı ldquoNumerical approachfor solving fractional relaxation-oscillation equationrdquo AppliedMathematical Modelling vol 37 no 8 pp 5927ndash5937 2013

[29] M Dehghan and A Saadatmandi ldquoA tau method for the one-dimensional parabolic inverse problem subject to temperatureoverspecificationrdquo Computers and Mathematics with Applica-tions vol 52 no 6-7 pp 933ndash940 2006

[30] H Jafari H Tajadodi and D Baleanu ldquoA modified variationaliterationmethod for solving fractional Riccati differential equa-tion by Adomian polynomialsrdquo Fractional Calculus and AppliedAnalysis vol 16 no 1 pp 109ndash122 2013

[31] G-C Wu and D Baleanu ldquoVariational iteration method forfractional calculusmdasha universal approach by Laplace trans-formrdquo Advances in Difference Equations vol 2013 article 182013


[32] Y Hu Y Luo and Z Lu ldquoAnalytical solution of the linearfractional differential equation by Adomian decompositionmethodrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 220ndash229 2008

[33] S Saha Ray and R K Bera ldquoAnalytical solution of the BagleyTorvik equation by Adomian decomposition methodrdquo AppliedMathematics andComputation vol 168 no 1 pp 398ndash410 2005

[34] A Arikoglu and I Ozkol ldquoSolution of fractional differentialequations by using differential transform methodrdquo ChaosSolitons amp Fractals vol 34 no 5 pp 1473ndash1481 2007

[35] P Darania and A Ebadian ldquoA method for the numerical solu-tion of the integro-differential equationsrdquo Applied Mathematicsand Computation vol 188 no 1 pp 657ndash668 2007

[36] I Hashim O Abdulaziz and S Momani ldquoHomotopy analysismethod for fractional IVPsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 3 pp 674ndash6842009

[37] HM Jaradat F Awawdeh and E A Rawashdeh ldquoAn analyticalscheme for multi-order fractional differential equationsrdquo Tam-sui Oxford Journal of Mathematical Sciences vol 26 no 3 pp305ndash320 2010

[38] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988

[39] B Fornberg A Practical Guide to Pseudospectral MethodsCambridge University Press Cambridge Mass USA 1998

[40] E L Ortiz and H Samara ldquoNumerical solution of differentialeigenvalue problems with an operational approach to the Taumethodrdquo Computing vol 31 no 2 pp 95ndash103 1983

[41] Y Li and N Sun ldquoNumerical solution of fractional differentialequations using the generalized block pulse operationalmatrixrdquoComputers andMathematics with Applications vol 62 no 3 pp1046ndash1054 2011

[42] M Yi J Huang and J Wei ldquoBlock pulse operational matrixmethod for solving fractional partial differential equationrdquoApplied Mathematics and Computation vol 221 pp 121ndash1312013

[43] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo ComputersandMathematics with Applications vol 59 no 3 pp 1326ndash13362010

[44] S Kazem S Abbasbandy and S Kumar ldquoFractional-orderLegendre functions for solving fractional-order differentialequationsrdquo Applied Mathematical Modelling vol 37 no 7 pp5498ndash5510 2013

[45] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legen-dre spectral method for fractional-order multipoint boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 2012

[46] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011

[47] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011

[48] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers andMathematics with Applications vol 62 no 5 pp 2364ndash23732011

[49] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013

[50] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

[51] D Baleanu A H Bhrawy and T M Taha ldquoA modifiedgeneralized Laguerre spectral method for fractional differentialequations on the half linerdquo Abstract and Applied Analysis vol2013 Article ID 413529 12 pages 2013

[52] A H Bhrawy and T M Taha ldquoAn operational matrix offractional integration of the Laguerre polynomials and itsapplication on a semi-infinite intervalrdquo Mathematical Sciencesvol 6 article 41 2012

[53] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquoAdvances in Difference Equations vol 2012 article 1792012

[54] D Rostamy M Alipour H Jafari and D Baleanu ldquoSolvingmulti-term orders fractional differential equations by opera-tional matrices of BPs with convergence analysisrdquo RomanianReports in Physics vol 65 pp 334ndash349 2013

[55] S Yuzbası ldquoNumerical solutions of fractional Riccati typedifferential equations by means of the Bernstein polynomialsrdquoApplied Mathematics and Computation vol 219 no 11 pp6328ndash6343 2013

[56] D Baleanu M Alipour and H Jafari ldquoThe Bernstein oper-ational matrices for solving the fractional quadratic Riccatidifferential equations with the Riemann-Liouville derivativerdquoAbstract and Applied Analysis vol 2013 Article ID 461970 7pages 2013

[57] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[58] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[59] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[60] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 6 pp 2859ndash2862 2010

[61] T Allahviranloo S Salahshour and S Abbasbandy ldquoExplicitsolutions of fractional differential equations with uncertaintyrdquoSoft Computing vol 16 no 2 pp 297ndash302 2012

[62] T Allahviranloo Z Gouyandeh and A Armand ldquoFuzzyfractionaldifferential equations under generalized fuzzy Caputoderivativerdquo Journal of Intelligent and Fuzzy Systems In press

[63] S Salahshour T Allahviranloo S Abbasbandy and D BaleanuldquoExistence and uniqueness results for fractional differentialequations with uncertaintyrdquo Advances in Difference Equationsvol 2012 article 112 2012


[64] S Salahshour T Allahviranloo and S Abbasbandy ldquoSolvingfuzzy fractional differential equations by fuzzy Laplace trans-formsrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 3 pp 1372ndash1381 2012

[65] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 2013

[66] A Ahmadian M Suleiman and S Salahshour ldquoAn opera-tional matrix based on Legendre polynomials for solving fuzzyfractional-order differential equationsrdquo Abstract and AppliedAnalysis vol 2013 Article ID 505903 29 pages 2013

[67] M R Balooch Shahriyar F Ismail S Aghabeigi A Ahma-dian and S Salahshour ldquoAn eigenvalue-eigenvector methodfor solving a system of fractional differential equations withuncertaintyrdquo Mathematical Problems in Engineering vol 2013Article ID 579761 11 pages 2013

[68] F Ghaemi R Yunus A Ahmadian S SalahshourM Suleimanand Sh Faridah Saleh ldquoApplication of fuzzy fractional kineticequations tomodelling of the acid hydrolysis reactionrdquoAbstractand Applied Analysis vol 2013 Article ID 610314 19 pages 2013

[69] MMazandarani andAVahidianKamyad ldquoModified fractionalEulermethod for solving fuzzy fractional initial value problemrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 1 pp 12ndash21 2013

[70] D Dubois and H Prade ldquoTowards fuzzy differential calculusmdashpart 3 differentiationrdquo Fuzzy Sets and Systems vol 8 no 3 pp225ndash233 1982

[71] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986

[72] G A Anastassiou Fuzzy Mathematics Approximation Theoryvol 251 of Studies in Fuzziness and Soft Computing SpringerBerlin Germany 2010

[73] G A Anastassiou and S G Gal ldquoOn a fuzzy trigonometricapproximation theorem of Weierstrass-typerdquo Journal of FuzzyMathematics vol 9 no 3 pp 701ndash708 2001

[74] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[75] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons amp Fractals vol 38no 1 pp 112ndash119 2008

[76] T Allahviranloo and M Afshar Kermani ldquoSolution of a fuzzysystem of linear equationrdquo Applied Mathematics and Computa-tion vol 175 no 1 pp 519ndash531 2006

[77] S Abbasbandy and R Ezzati ldquoNewtonrsquos method for solving asystem of fuzzy nonlinear equationsrdquo Applied Mathematics andComputation vol 175 no 2 pp 1189ndash1199 2006

[78] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[79] Y LukeThe Special Functions andTheir Approximations vol 2Academic Press New York NY USA 1969

[80] G Szego Orthogonal Polynomials vol 23 of Colloquium Publi-cations American Mathematical Society Providence RI USA1985

[81] A A Kilbas and J J Trujillo ldquoDifferential equations offractional order methods results and problems Irdquo ApplicableAnalysis vol 78 no 1-2 pp 153ndash192 2001

[82] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


composition the blood composition and blood flow tothe tissue is utilized to develop a partitioning model [8ndash10] Mohler et al [11] proposed a model that describes cellgrowth and detailed investigations on the metabolism andthe kinetics of the influenza infection cycle which allowfor the optimization of influenza virus vaccine productionIn addition to these simple models more complex math-ematical manipulations called mathematical models havealso been used to describe pharmacokinetics [12] Thereare many classifications of pharmacokinetics models In aspecific classification for pharmacokinetics modeling twogeneral approaches can be considered compartment-basedmodeling [13] and noncompartment-based modeling [14]Both types of modeling take advantage of the quantitativestructure-pharmacokinetics relationships that are describedby empirical mathematical algorithms They can be used toestimate the activity of a compound based on its chemicalstructure in a numeric format [15]

In this study our focus is to find the approximate solutionof the fuzzy fractional model of the compartment model forthe flow of antihistamine in the blood To this end we usethe operational matrix of the Caputo fractional derivative ofthe Jacobi tau approximation based on the Jacobi polynomialsto derive the fuzzy fractional approximate solution underHukuhara differentiability (H-differentiability) To the bestof our knowledge there are no results on the JT method forsolving pharmacokinetics equations arising in mathematicalphysics This partially motivated our interest in such modeland solution method

11 Compartment Model The compartment model frame-work is an extremely natural and valuable means with whichto formulate models for processes which have inputs andoutputs over time [16] Compartment model itself can beclassified in one-compartment model and two-compartmentmodel In the former one the body is depicted as a kineticallyhomogeneous unit while in the latter one the body resolvesinto a central compartment and a peripheral compartment[17] According to the one-compartment model the druginstantaneously distributes throughout the body and at thesame time it equilibrates between tissues Thus the drugconcentration time profile shows a monophasic responseOn the other hand in two-compartment model the drugdoes not achieve instantaneous equilibration between the twocompartments Although in this model two compartmentshave no anatomical or physiological meaning it is usuallysurmised that the central compartment is composed of tissuesthat are highly perfused such as heart lungs kidneys liverand brain while the peripheral compartment comprises lesswell-perfused tissues such as muscle fat and skin [17]

12 Methods Have Been Proposed to Solve Fractional Pharma-cokinetics Models While an increasing number of fractionalorder integrals and differential equations applications havebeen reported in the physics [18 19] signal processing[20] engineering and bioengineering literatures [21 22]little attention has been paid to this class of models in thepharmacokinetics-pharmacodynamic (PKPD) literatureThereasons for the lack of application to pharmacodynamics are

mainly two first and foremost PKPD models incorporatingfractional calculus have not been proposed second is compu-tational while the analytical solution of fractional differentialequations is available in special cases it turns out that eventhe simplest PKPD models that can be constructed usingfractional calculus do not allow analytical solutions [23 24]

In this paper we propose new category of PKPD mod-els incorporating fuzzy fractional calculus and investigatetheir behavior using purposely written algorithm The mainpurpose of this paper is to attract the attention of thefield into these possibly interesting families of PKPD mod-els In particular we will focus on compartment modelsdirect concentration-response relationships describing mod-els relating drug concentration (in plasma or biophase) to apharmacodynamic effect

The concept of fractional or noninteger order derivationand integration can be traced back to the genesis of integerorder calculus itself [25ndash27] Due to its tremendous scope andapplications in several disciplines a considerable attentionhas been given to exact and numerical solutions of fractionaldifferential equations which is an extremely difficult taskMoreover the solution techniques and their reliability arerather more important aspects Several methods have beenproposed by many researchers to solve the fractional orderdifferential equations such as Taylor series [28] variationaliteration method [29ndash31] adomian decomposition method[32 33] fractional differential transform method (FDTM)[34 35] and homotopy analysis method [36 37]

As it is known the spectral method is one of the flexiblemethods of discretization for most types of differentialequations [38ndash40] Historically spectral method has beenrelegated to fractional calculus but in few years it has beensuccessfully applied for the fractional equation models basedon the different types of orthogonal polynomials such asBlock pulse functions [41 42] Legendre polynomials [43ndash46] Chebyshev polynomials [47ndash49] Laguerre polynomials[50ndash53] and Bernstein polynomials [54ndash56] Doha et al[57] introduced the shifted Jacobi operational matrix offractional derivative which is based on Jacobi tau method forsolving numerically linear multiterm fractional differentialequations with initial or boundary conditions SubsequentlyKazem [58] generalized Jacobi integral operational matrix tofractional calculus Afterwards Doha et al [59] proposed adirect solution techniques for solving the linear multiorderfractional initial value problems with constant and variablecoefficients using shifted Jacobi tau method and quadratureshifted Jacobi tau method respectively In this paper weintend to extend the application of the Jacobi polynomi-als to solve fuzzy fractional pharmacokinetics model oforder [0 1]

On the other hand a considerable attention has beenmade to the fractional differential equations in the senseof fuzzy setting theory Agarwal et al [60] is among thepioneerswhopresented the concept of solutions for fractionaldifferential equations with uncertainty Thereafter the exis-tence and uniqueness of the solution of the fuzzy fractionaldifferential equations (FFDEs) under the Riemann-Liouvilleand Caputorsquos fuzzy fractional differentiability were investi-gated in the literature [61ndash64] However the application of










0) = 1


119906 (120582119909 + (1 minus 120582) 119910 ge min 119906 (119909) 119906 (119910)) (1)


that

119906 (1199090) ge lim


0

119906 (119909) (2)




[119906]119903= 119909 isin R | 119906 (119909) ge 119903 (3)


[119906]119903= [119906

119903

1 119906

119903

2] (4)


(119906 + V) (119909) = sup119910isinR

min 119906 (119910) V (119909 minus 119910) 119909 isin R (5)


(119896 ⊙ 119906) (119909) =

119906(119909

119896) 119896 gt 0

0 119896 = 0

(6)

where 0 isin RF


119863 (119906 V) = sup119903isin[01]

119889119867([119906]

119903 [V]

119903) (7)

where

119889119867([119906]

119903 [V]

119903) = max 1003816100381610038161003816119906

119903

1minus V

119903

1

1003816100381610038161003816 1003816100381610038161003816119906

119903

2minus V

119903

2

1003816100381610038161003816 (8)





119863lowast(119891 119892) = sup

119909isin[119886119887]

119863(119891 (119909) 119892 (119909)) (9)


119863(119891 (119909) 0) = sup119903isin[01]

max 1003816100381610038161003816119891119903

1(119909)

1003816100381610038161003816 1003816100381610038161003816119891

119903

2(119909)

1003816100381610038161003816 forall119909 isin [119886 119887]

(10)






119903(119905) 119892



119903(119905) are dif-


[1198651015840(119905)]

119903

= [1198911015840

119903(119905) 119892

1015840

119903(119905)] (11)


119903(119905) are dif-


[1198651015840(119905)]

119903

= [1198921015840

119903(119905) 119891

1015840

119903(119905)] (12)


119886111199091+ 119886

121199092+ sdot sdot sdot + 119886

1119899119909119899= 119910

1

119886211199091+ 119886

221199092+ sdot sdot sdot + 119886

2119899119909119899= 119910

2

11988611989911199091+ 119886

11989921199092+ sdot sdot sdot + 119886

119899119899119909119899= 119910

119899

(13)


119860119883 = 119884 (14)





2

119909119899)119905 given by 119909

119894= (119909

119894

119903

minus 119909

119894

119903



(

119899

sum119895=1

119886119894119895119909119895)

119903

minus

=

119899

sum119895=1

(119886119894119895119909119895)119903

minus= 119910

119903

119894minus

(

119899

sum119895=1

119886119894119895119909119895)

119903

+

=

119899

sum119895=1

(119886119894119895119909119895)119903

+= 119910

119903

119894+

(15)


119899

sum119895=1

119886119896119895119909119903

119895minus

= 119910119903

119896minus

119899

sum119895=1

119886119896119895119909119903

119895+

= 119910119903

119896+

(16)




119888119863

V119891 (119909) =

119869119898minusV

119863119898119891 (119909) if 119898 minus 1 lt V lt 119898

119863119898119891 (119909) if V = 119898 119898 isin N

(17)

where

119869V119891 (119909) =

1

Γ (V)int

119909

0



V119862 = 0 (119862 is a constant)

119888119863

V119909120573

=


Γ (120573 + 1)

Γ (120573 + 1 minus V)119909120573minusV

for 120573 isin N



(19)


119888119863

V(120582119891 (119909) + 120583119892 (119909)) = 120582

119888119863

V119891 (119909) + 120583

119888119863

V119892 (119909) (20)




1 then

119891 (119905) =

119899

sum119894=0

(119905 minus 119886)119894120572

Γ (119894120572 + 1)[

119888119863

119896120572

119886119891 (119905)]

119905=119886+ 119877

120572

119899(119905 119886) (21)

with

119877120572

119899(119905 119886) =

(119905 minus 119886)(119899+1)120572

Γ ((119899 + 1) 120572 + 1)[119888119863

(119899+1)120572

119886119891 (119905)]

119905=120585

119886 ⩽ 120585 ⩽ 119905 119905 isin [119886 119887]

(22)

where119888119863

119899120572

119886=

119888119863

120572119888

119886

119888119863

120572

119886

119888119863

120572

119886⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899 119905119894119898119890119904

(23)

Here 119888119863


119899119894

119879(119905) = sum

119899

119894=0((119905 minus 119886)

119894120572Γ(119894120572 +

1))[119888119863

119896120572

119886119891(119905)]


degree 119899119894 of 119891 and 119877120572







=

(int1

0(119889(119891(119905) 0))

119901119889119905)

1119901








(119877119871119868V119886+119891) (119909) =

1

Γ (V)int

119909

119886

119891 (119905) 119889119905

(119909 minus 119905)1minusV

119909 gt 119886 0 lt V le 1

(24)



(i) (119888119863

V119886+119891) (119909) = (

119877119871119863

120573

119886+[119891 (119905) ⊖ 119891 (119886)]) (119909)

(ii) (119888119863

V119886+119891) (119909) = (

119877119871119863

120573

119886+[minus119891 (119886) ⊖ (minus119891 (119905))]) (119909)

(25)


RF[119886 119887] and 1199090isin

(119886 119887) and Φ(119909) = (1Γ(1 minus V)) int119909

119886(119891(119905)(119909 minus 119905)

V)119889119905 We say



119886+119891)(1199090

) isin

119862RF[119886119887]


(i) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090+ ℎ) ⊖ Φ (119909

0)

ℎ

= limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0minus ℎ)

ℎ

(26)

or

(ii) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0+ ℎ)

minusℎ

= limℎrarr0

+

Φ(1199090minus ℎ) ⊖ Φ (119909

0)

minusℎ

(27)

or

(iii) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090+ ℎ) ⊖ Φ (119909

0)

ℎ

= limℎrarr0

+

Φ(1199090minus ℎ) ⊖ Φ (119909

0)

minusℎ

(28)

or

(iv) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0+ ℎ)

minusℎ

= limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0minus ℎ)

ℎ

(29)






(119888119863

V119886+119891) (119909 119903) = [

1

Γ (1 minus V)int

119909

119886

1198911015840119903

minus(119905) 119889119905

(119909 minus 119905)V

1

Γ (1 minus V)int

119909

119886

1198911015840119903

+(119905) 119889119905

(119909 minus 119905)V ]

= [(1198681minusV119886+

119863119891119903

minus) (119909) (119868

1minusV119886+

119863119891119903

+) (119909)]

(30)


(119888119863

V119886+119891) (119909 119903)

= [1

Γ (1 minus V)int

119909

119886

1198911015840119903

+(119905) 119889119905

(119909 minus 119905)V

1

Γ (1 minus V)int

119909

119886

1198911015840119903

minus(119905) 119889119905

(119909 minus 119905)V ]

= [(1198681minusV119886+

119863119891119903

+) (119909) (119868

1minusV119886+

119863119891119903

minus) (119909)]

(31)




119863119896120572119891(119909) isin

119862RF[119886119887]

(0 119887] for 119896 = 0 1 119899 + 1 where 0 lt 120572 lt 1 0 le


[119891 (119909)]119903= [119891

119903(119909) 119891

119903

(119909)]

119891119903(119909) =

119899

sum119894=0

119909119894120572

Γ (119894120572 + 1)

119888119863

119894120572119891

119903(0

+)

+

119888119863

(119899+1)120572119891

119903(119909

0)

Γ (119899120572 + 120572 + 1)119909(119899+1)120572

119891119903

(119909) =

119899

sum119894=0

119909119894120572

Γ (119894120572 + 1)

119888119863

119894120572119891

119903

(0+)

+

119888119863

(119899+1)120572119891

119903(119909

0)

Γ (119899120572 + 120572 + 1)119909(119899+1)120572

(32)

where 119888119863

120572119891

119903(0) =

119888119863

120572119891

119903(119909)|

119909=0119888119863

120572119891

119903

(0) =119888119863

120572119891

119903

(119909)|119909=0







119871119894(119909) satisfying the


int119871

0

119875(120572120573)

119871119895(119909) 119875

(120572120573)

119871119896(119909) 119908

(120572120573)

119871(119909) 119889119909 = ℎ

119896 (33)

where 119908(120572120573)

119871(119909) = 119909

120573(119871 minus 119909)

120572 and

ℎ119896=

119871120572+120573+1

Γ (119896 + 120572 + 1) Γ (119896 + 120573 + 1)

(2119896 + 120572 + 120573 + 1) 119896Γ (119896 + 120572 + 120573 + 1)119894 = 119895

0 119894 = 119895

(34)


119871119894(119909) of degree 119894 has the

form

119875(120572120573)

119871119894(119909)

=

119894

sum119896=0


Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

Γ (119896 + 120573 + 1) Γ (119894 + 120572 + 120573 + 1) (119894 minus 119896)119896119871119896119909119896

(35)

where

119875(120572120573)

119871119894(0) = (minus1)

119894Γ (119894 + 120573 + 1)

Γ (120573 + 1) 119894

119875(120572120573)

119871119894(119871) =

Γ (119894 + 120572 + 1)

Γ (120572 + 1) 119894

(36)


119875(120572120573)

119871119894(119909)

= (120572 + 120573 + 2119894 minus 1) (1205722minus 120573

2+ (

2119909

119871minus 1))

times (120572 + 120573 + 2119894) (120572 + 120573 + 2119894 minus 2)


times 119875(120572120573)

119871119894minus1(119909)


119894 (120572 + 120573 + 119894) (120572 + 120573 + 2119894 minus 2)

times 119875(120572120573)

119871119894minus2(119909) 119894 = 2 3

(37)

where 119875(120572120573)

1198710(119909) = 1 and 119875

(120572120573)

1198711(119909) = ((120572 + 120573 + 2)2)(2119909119871 minus



119906 (119909) =

+infin

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (38)


119886119895=

1

ℎ119895

int119871

0

119875(120572120573)

119871119895(119909) 119906 (119909)119908

(120572120573)

119871(119909) 119889119909 119895 = 0 1 (39)


119906119873(119909) ≃

119873

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (40)


119863V119875

(120572120573)

119871119894(119909) =

infin

sum119895=0

119878V (119894 119895 120572 120573) 119875(120572120573)

119871119894(119909)


(41)

where

119878V (119894 119895 120572 120573)

=

119894



Γ (119895 + 120573 + 1)

times Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

times (ℎ119895Γ (119895 + 119896 + 120572 + 120573 + 1) Γ (119896 + 120573 + 1)


times

119895

sum119897=0

(minus1)119895minus119897Γ (119895 + 119897 + 120572 + 120573 + 1) Γ (120572 + 1)

times Γ (119897 + 119896 + 120573 minus V + 1)


(42)




Drug intakeGI tract

DigestionBlood

Tissue




= rate of drug


GI-tract


= rate drug


blood

(43)



119889119910

119889119905= minus119896

1119909 119909 (0) = 119909

0 (44)


1is a


119889119910

119889119905= 119896

1119909 minus 119896

2119910 119910 (0) = 0 (45)



1and 119896




119883 = 119860119890(minus1198961119905) (46)


119889119910 (119905)

119889119905+ 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (47)


119863V we can





119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (48)


119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [119910119903

0 119910

119903

0] (49)

in which 119910(119909) 119871RF[0 1] cap 119862










1=










= 001386 006386 01386





2is


1varies (119896

1=



the bloodstream



1and 119896



119906 (119909) =

+infin

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

1119894(119909) (50)


119886119894=1

ℎ119894

int1

0

119875(120572120573)

1119894(119909) ⊙ 119906 (119909) ⊙ 119908

(120572120573)

1(119909) 119889119909 119894 = 0 1

(51)

and 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573 119906 isin 119871RF119901[0 1] cap 119862

RF[0 1]and 119875

(120572120573)





119894(119909) instead of 119875(120572120573)

1119894(119909) and 119908

(120572120573)(119909) = (1 minus

119909)120572⊙ 119909

120573 instead of 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573


119906 (119909) ≃ 119906119873(119909) =

119873

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

119894(119909) = a119879 ⊙ Φ

119873(119909) (52)



a119879 = [1198860 119886

1 119886

119873]

Φ119873(119909) = [119875

(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)]

119879

(53)




119906119903(119909) ≃ 119906

119903

119873(119909) = [

119873

sum119894=0

119886119903

119894minus119875

(120572120573)

119894(119909)

119873

sum119894=0

119886119903

119894+119875

(120572120573)

119894(119909)]

0 le 119903 le 1

(54)

We call that Span119875(120572120573)



2119908(120572120573)

(0 1) Hence we set

S119873(0 1) = Span 119875(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)

(55)


isin

119878119873(0 1) such that

(119863V119906119873 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119906

119873 119875

(120572120573)

119896(119909))

119908(120572120573)

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1 119906119873 (0) = 119889

0

(56)

where 119908(120572120573)(119909) = (1 minus 119909)

120572⊙ 119909

120573 and (119906 V)119908(120572120573) = int

1

0119908

(120572120573)


RF

2119908(120572120573)



119891119896= (119896

1119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

f = (1198910 119891

1 119891

119873minus1 119889

0)119879

(57)



119873

sum119895=0

119886119895⊙ [(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895⊙ 119875

(120572120573)

119895(0) = 119889

0

(58)

Denoting

119860 = (119886119896119895)0lt119896119895lt119873

119862 = (119888119896119895)0lt119896119895lt119873

(59)


(119860 + 1198962119862) ⊙ a = f (60)



= (119863(V)119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))

119908(120572120573) (0 ⩽ 119896 ⩽ 119873 minus 1 0 ⩽ 119895 ⩽ 119873) 119886

119896119895=

119863119896minus119873

119875(120572120573)

119895(0) (119896 = 119873 0 ⩽ 119895 ⩽ 119873) and 119888

119896119895= (119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))


elements of 119886119896119895and 119888


119886119896119895= ℎ

119896119878V (119895 119896 120572 120573) 0 ⩽ 119896 ⩽ 119873 minus 1 1 ⩽ 119895 ⩽ 119873 119886

119896119895

=(minus1)

119895minus119896+119873Γ (119895 + 120573 + 1) (119895 + 120572 + 120573 + 1)

119896minus119873

Γ (119895 minus 119896 + 119873 + 1) Γ (119896 minus 119873 + 1 + 120573)

119896 = 119873 0 ⩽ 119895 ⩽ 119873 119888119896119895

= ℎ119896 0 ⩽ 119896 = 119895 ⩽ 119873 minus 1

(61)




0+119891(119905) isin 119862(0 119887] 119896 = 0 1 119873 + 1 then

1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816 (62)

where 119891119873119894

119879and 119877




119891119873119894

119879(119905) =

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+ (63)


119877V119873(119905 0) =

119905(119873+1)V

Γ ((119873 + 1) V + 1)[119863

(119873+1)V0+

119891 (119905)]119905=120585

0 ⩽ 120585 ⩽ 119905 forall119905 isin (0 119887]

(64)

Therefore

119891 (119905) minus 119891119873119894

119879(119905) = 119877

V119873(119905 0) (65)


1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 =

10038161003816100381610038161003816119891 (119905) minus 119910

119873V (119905) + 119891119873119894

119879(119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816

⩽10038161003816100381610038161003816119891 (119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816+10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

=1003816100381610038161003816119877

V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

(66)







119896V119906(119905) isin



[119906 (119905)]119903= [119906

119903(119905) 119906

119903(119905)] [119906

119873(119905)]

119903= [119906

119903

119873(119905) 119906

119903

119873(119905)]

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

10038161003816100381610038161003816119877V

119873(119905 0)

10038161003816100381610038161003816+10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

(67)


[119891119873119894

119879(119905)]

119903

= [119891119873119894

119879(119905 119903) 119891

119873119894

119879(119905 119903)]

= [

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+]


[119877V119873(119905 0)]

119903= [119877

V119903119873(119905 0) 119877

V119903

119873(119905 0)]

= [

[

119888119863

(119873+1)V119891

119903(0+)

Γ (119873V + V + 1)119905(119873+1)V

119888119863

(119873+1)V119891

119903

(0+)

Γ (119873V + V + 1)119905(119873+1)V]

]

(68)






119896V isin

119862RF[119909


119863lowast(RF119863

VΦ (119909)

RF119863(V)Φ (119909))

le119909minusV0

(2) |Γ (1 minus V)|(119878)

2(120573 + 2

2)radic119861 (120572 + 1 120573 + 1)

(69)

5 Numerical Results


1and 119896

2by using the JT


119873]119903= [119910

119903

119873 119910

119903


tions 119910(119905 119903) = [119910(119905 119903) 119910(119905 119903)] that is [119873119890]119903= [|119910

119903

119873minus 119910

119903|

|119910119903

119873minus 119910



V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (70)

in which 119910(119909) 119871RF[0 1] cap 119862







1has an invari-


cases in details




follows119888119863

V119910 (119905) + 00231119910 (119905) = 06931119890

minus06931119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (71)



V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = minus1 + 119903 0 lt 119903 le 1

119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = 1 minus 119903 0 lt 119903 le 1

(72)


119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt 119903 le 1

(73)


1199102(119905) = 119886

0⊙ 119875

(120572120573)

0(119905) + 119886

1⊙ 119875

(120572120573)

1(119905) + 119886

2⊙ 119875

(120572120573)

0(119905)

(74)

Here we have

119863085

= (

0 0 0

18639 03901 minus01755

minus03901 45267 08696

) (75)


119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

(76)


1198860= minus05757 119886

1= 02624 119886

2= minus00619

1198860= 12008 119886

1= 02408 119886

2= minus00600

(77)




1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11




119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















0) = 1


119906 (120582119909 + (1 minus 120582) 119910 ge min 119906 (119909) 119906 (119910)) (1)


that

119906 (1199090) ge lim


0

119906 (119909) (2)




[119906]119903= 119909 isin R | 119906 (119909) ge 119903 (3)


[119906]119903= [119906

119903

1 119906

119903

2] (4)


(119906 + V) (119909) = sup119910isinR

min 119906 (119910) V (119909 minus 119910) 119909 isin R (5)


(119896 ⊙ 119906) (119909) =

119906(119909

119896) 119896 gt 0

0 119896 = 0

(6)

where 0 isin RF


119863 (119906 V) = sup119903isin[01]

119889119867([119906]

119903 [V]

119903) (7)

where

119889119867([119906]

119903 [V]

119903) = max 1003816100381610038161003816119906

119903

1minus V

119903

1

1003816100381610038161003816 1003816100381610038161003816119906

119903

2minus V

119903

2

1003816100381610038161003816 (8)





119863lowast(119891 119892) = sup

119909isin[119886119887]

119863(119891 (119909) 119892 (119909)) (9)


119863(119891 (119909) 0) = sup119903isin[01]

max 1003816100381610038161003816119891119903

1(119909)

1003816100381610038161003816 1003816100381610038161003816119891

119903

2(119909)

1003816100381610038161003816 forall119909 isin [119886 119887]

(10)






119903(119905) 119892



119903(119905) are dif-


[1198651015840(119905)]

119903

= [1198911015840

119903(119905) 119892

1015840

119903(119905)] (11)


119903(119905) are dif-


[1198651015840(119905)]

119903

= [1198921015840

119903(119905) 119891

1015840

119903(119905)] (12)


119886111199091+ 119886

121199092+ sdot sdot sdot + 119886

1119899119909119899= 119910

1

119886211199091+ 119886

221199092+ sdot sdot sdot + 119886

2119899119909119899= 119910

2

11988611989911199091+ 119886

11989921199092+ sdot sdot sdot + 119886

119899119899119909119899= 119910

119899

(13)


119860119883 = 119884 (14)





2

119909119899)119905 given by 119909

119894= (119909

119894

119903

minus 119909

119894

119903



(

119899

sum119895=1

119886119894119895119909119895)

119903

minus

=

119899

sum119895=1

(119886119894119895119909119895)119903

minus= 119910

119903

119894minus

(

119899

sum119895=1

119886119894119895119909119895)

119903

+

=

119899

sum119895=1

(119886119894119895119909119895)119903

+= 119910

119903

119894+

(15)


119899

sum119895=1

119886119896119895119909119903

119895minus

= 119910119903

119896minus

119899

sum119895=1

119886119896119895119909119903

119895+

= 119910119903

119896+

(16)




119888119863

V119891 (119909) =

119869119898minusV

119863119898119891 (119909) if 119898 minus 1 lt V lt 119898

119863119898119891 (119909) if V = 119898 119898 isin N

(17)

where

119869V119891 (119909) =

1

Γ (V)int

119909

0



V119862 = 0 (119862 is a constant)

119888119863

V119909120573

=


Γ (120573 + 1)

Γ (120573 + 1 minus V)119909120573minusV

for 120573 isin N



(19)


119888119863

V(120582119891 (119909) + 120583119892 (119909)) = 120582

119888119863

V119891 (119909) + 120583

119888119863

V119892 (119909) (20)




1 then

119891 (119905) =

119899

sum119894=0

(119905 minus 119886)119894120572

Γ (119894120572 + 1)[

119888119863

119896120572

119886119891 (119905)]

119905=119886+ 119877

120572

119899(119905 119886) (21)

with

119877120572

119899(119905 119886) =

(119905 minus 119886)(119899+1)120572

Γ ((119899 + 1) 120572 + 1)[119888119863

(119899+1)120572

119886119891 (119905)]

119905=120585

119886 ⩽ 120585 ⩽ 119905 119905 isin [119886 119887]

(22)

where119888119863

119899120572

119886=

119888119863

120572119888

119886

119888119863

120572

119886

119888119863

120572

119886⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899 119905119894119898119890119904

(23)

Here 119888119863


119899119894

119879(119905) = sum

119899

119894=0((119905 minus 119886)

119894120572Γ(119894120572 +

1))[119888119863

119896120572

119886119891(119905)]


degree 119899119894 of 119891 and 119877120572







=

(int1

0(119889(119891(119905) 0))

119901119889119905)

1119901








(119877119871119868V119886+119891) (119909) =

1

Γ (V)int

119909

119886

119891 (119905) 119889119905

(119909 minus 119905)1minusV

119909 gt 119886 0 lt V le 1

(24)



(i) (119888119863

V119886+119891) (119909) = (

119877119871119863

120573

119886+[119891 (119905) ⊖ 119891 (119886)]) (119909)

(ii) (119888119863

V119886+119891) (119909) = (

119877119871119863

120573

119886+[minus119891 (119886) ⊖ (minus119891 (119905))]) (119909)

(25)


RF[119886 119887] and 1199090isin

(119886 119887) and Φ(119909) = (1Γ(1 minus V)) int119909

119886(119891(119905)(119909 minus 119905)

V)119889119905 We say



119886+119891)(1199090

) isin

119862RF[119886119887]


(i) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090+ ℎ) ⊖ Φ (119909

0)

ℎ

= limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0minus ℎ)

ℎ

(26)

or

(ii) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0+ ℎ)

minusℎ

= limℎrarr0

+

Φ(1199090minus ℎ) ⊖ Φ (119909

0)

minusℎ

(27)

or

(iii) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090+ ℎ) ⊖ Φ (119909

0)

ℎ

= limℎrarr0

+

Φ(1199090minus ℎ) ⊖ Φ (119909

0)

minusℎ

(28)

or

(iv) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0+ ℎ)

minusℎ

= limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0minus ℎ)

ℎ

(29)






(119888119863

V119886+119891) (119909 119903) = [

1

Γ (1 minus V)int

119909

119886

1198911015840119903

minus(119905) 119889119905

(119909 minus 119905)V

1

Γ (1 minus V)int

119909

119886

1198911015840119903

+(119905) 119889119905

(119909 minus 119905)V ]

= [(1198681minusV119886+

119863119891119903

minus) (119909) (119868

1minusV119886+

119863119891119903

+) (119909)]

(30)


(119888119863

V119886+119891) (119909 119903)

= [1

Γ (1 minus V)int

119909

119886

1198911015840119903

+(119905) 119889119905

(119909 minus 119905)V

1

Γ (1 minus V)int

119909

119886

1198911015840119903

minus(119905) 119889119905

(119909 minus 119905)V ]

= [(1198681minusV119886+

119863119891119903

+) (119909) (119868

1minusV119886+

119863119891119903

minus) (119909)]

(31)




119863119896120572119891(119909) isin

119862RF[119886119887]

(0 119887] for 119896 = 0 1 119899 + 1 where 0 lt 120572 lt 1 0 le


[119891 (119909)]119903= [119891

119903(119909) 119891

119903

(119909)]

119891119903(119909) =

119899

sum119894=0

119909119894120572

Γ (119894120572 + 1)

119888119863

119894120572119891

119903(0

+)

+

119888119863

(119899+1)120572119891

119903(119909

0)

Γ (119899120572 + 120572 + 1)119909(119899+1)120572

119891119903

(119909) =

119899

sum119894=0

119909119894120572

Γ (119894120572 + 1)

119888119863

119894120572119891

119903

(0+)

+

119888119863

(119899+1)120572119891

119903(119909

0)

Γ (119899120572 + 120572 + 1)119909(119899+1)120572

(32)

where 119888119863

120572119891

119903(0) =

119888119863

120572119891

119903(119909)|

119909=0119888119863

120572119891

119903

(0) =119888119863

120572119891

119903

(119909)|119909=0







119871119894(119909) satisfying the


int119871

0

119875(120572120573)

119871119895(119909) 119875

(120572120573)

119871119896(119909) 119908

(120572120573)

119871(119909) 119889119909 = ℎ

119896 (33)

where 119908(120572120573)

119871(119909) = 119909

120573(119871 minus 119909)

120572 and

ℎ119896=

119871120572+120573+1

Γ (119896 + 120572 + 1) Γ (119896 + 120573 + 1)

(2119896 + 120572 + 120573 + 1) 119896Γ (119896 + 120572 + 120573 + 1)119894 = 119895

0 119894 = 119895

(34)


119871119894(119909) of degree 119894 has the

form

119875(120572120573)

119871119894(119909)

=

119894

sum119896=0


Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

Γ (119896 + 120573 + 1) Γ (119894 + 120572 + 120573 + 1) (119894 minus 119896)119896119871119896119909119896

(35)

where

119875(120572120573)

119871119894(0) = (minus1)

119894Γ (119894 + 120573 + 1)

Γ (120573 + 1) 119894

119875(120572120573)

119871119894(119871) =

Γ (119894 + 120572 + 1)

Γ (120572 + 1) 119894

(36)


119875(120572120573)

119871119894(119909)

= (120572 + 120573 + 2119894 minus 1) (1205722minus 120573

2+ (

2119909

119871minus 1))

times (120572 + 120573 + 2119894) (120572 + 120573 + 2119894 minus 2)


times 119875(120572120573)

119871119894minus1(119909)


119894 (120572 + 120573 + 119894) (120572 + 120573 + 2119894 minus 2)

times 119875(120572120573)

119871119894minus2(119909) 119894 = 2 3

(37)

where 119875(120572120573)

1198710(119909) = 1 and 119875

(120572120573)

1198711(119909) = ((120572 + 120573 + 2)2)(2119909119871 minus



119906 (119909) =

+infin

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (38)


119886119895=

1

ℎ119895

int119871

0

119875(120572120573)

119871119895(119909) 119906 (119909)119908

(120572120573)

119871(119909) 119889119909 119895 = 0 1 (39)


119906119873(119909) ≃

119873

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (40)


119863V119875

(120572120573)

119871119894(119909) =

infin

sum119895=0

119878V (119894 119895 120572 120573) 119875(120572120573)

119871119894(119909)


(41)

where

119878V (119894 119895 120572 120573)

=

119894



Γ (119895 + 120573 + 1)

times Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

times (ℎ119895Γ (119895 + 119896 + 120572 + 120573 + 1) Γ (119896 + 120573 + 1)


times

119895

sum119897=0

(minus1)119895minus119897Γ (119895 + 119897 + 120572 + 120573 + 1) Γ (120572 + 1)

times Γ (119897 + 119896 + 120573 minus V + 1)


(42)




Drug intakeGI tract

DigestionBlood

Tissue




= rate of drug


GI-tract


= rate drug


blood

(43)



119889119910

119889119905= minus119896

1119909 119909 (0) = 119909

0 (44)


1is a


119889119910

119889119905= 119896

1119909 minus 119896

2119910 119910 (0) = 0 (45)



1and 119896




119883 = 119860119890(minus1198961119905) (46)


119889119910 (119905)

119889119905+ 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (47)


119863V we can





119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (48)


119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [119910119903

0 119910

119903

0] (49)

in which 119910(119909) 119871RF[0 1] cap 119862










1=










= 001386 006386 01386





2is


1varies (119896

1=



the bloodstream



1and 119896



119906 (119909) =

+infin

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

1119894(119909) (50)


119886119894=1

ℎ119894

int1

0

119875(120572120573)

1119894(119909) ⊙ 119906 (119909) ⊙ 119908

(120572120573)

1(119909) 119889119909 119894 = 0 1

(51)

and 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573 119906 isin 119871RF119901[0 1] cap 119862

RF[0 1]and 119875

(120572120573)





119894(119909) instead of 119875(120572120573)

1119894(119909) and 119908

(120572120573)(119909) = (1 minus

119909)120572⊙ 119909

120573 instead of 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573


119906 (119909) ≃ 119906119873(119909) =

119873

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

119894(119909) = a119879 ⊙ Φ

119873(119909) (52)



a119879 = [1198860 119886

1 119886

119873]

Φ119873(119909) = [119875

(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)]

119879

(53)




119906119903(119909) ≃ 119906

119903

119873(119909) = [

119873

sum119894=0

119886119903

119894minus119875

(120572120573)

119894(119909)

119873

sum119894=0

119886119903

119894+119875

(120572120573)

119894(119909)]

0 le 119903 le 1

(54)

We call that Span119875(120572120573)



2119908(120572120573)

(0 1) Hence we set

S119873(0 1) = Span 119875(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)

(55)


isin

119878119873(0 1) such that

(119863V119906119873 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119906

119873 119875

(120572120573)

119896(119909))

119908(120572120573)

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1 119906119873 (0) = 119889

0

(56)

where 119908(120572120573)(119909) = (1 minus 119909)

120572⊙ 119909

120573 and (119906 V)119908(120572120573) = int

1

0119908

(120572120573)


RF

2119908(120572120573)



119891119896= (119896

1119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

f = (1198910 119891

1 119891

119873minus1 119889

0)119879

(57)



119873

sum119895=0

119886119895⊙ [(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895⊙ 119875

(120572120573)

119895(0) = 119889

0

(58)

Denoting

119860 = (119886119896119895)0lt119896119895lt119873

119862 = (119888119896119895)0lt119896119895lt119873

(59)


(119860 + 1198962119862) ⊙ a = f (60)



= (119863(V)119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))

119908(120572120573) (0 ⩽ 119896 ⩽ 119873 minus 1 0 ⩽ 119895 ⩽ 119873) 119886

119896119895=

119863119896minus119873

119875(120572120573)

119895(0) (119896 = 119873 0 ⩽ 119895 ⩽ 119873) and 119888

119896119895= (119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))


elements of 119886119896119895and 119888


119886119896119895= ℎ

119896119878V (119895 119896 120572 120573) 0 ⩽ 119896 ⩽ 119873 minus 1 1 ⩽ 119895 ⩽ 119873 119886

119896119895

=(minus1)

119895minus119896+119873Γ (119895 + 120573 + 1) (119895 + 120572 + 120573 + 1)

119896minus119873

Γ (119895 minus 119896 + 119873 + 1) Γ (119896 minus 119873 + 1 + 120573)

119896 = 119873 0 ⩽ 119895 ⩽ 119873 119888119896119895

= ℎ119896 0 ⩽ 119896 = 119895 ⩽ 119873 minus 1

(61)




0+119891(119905) isin 119862(0 119887] 119896 = 0 1 119873 + 1 then

1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816 (62)

where 119891119873119894

119879and 119877




119891119873119894

119879(119905) =

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+ (63)


119877V119873(119905 0) =

119905(119873+1)V

Γ ((119873 + 1) V + 1)[119863

(119873+1)V0+

119891 (119905)]119905=120585

0 ⩽ 120585 ⩽ 119905 forall119905 isin (0 119887]

(64)

Therefore

119891 (119905) minus 119891119873119894

119879(119905) = 119877

V119873(119905 0) (65)


1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 =

10038161003816100381610038161003816119891 (119905) minus 119910

119873V (119905) + 119891119873119894

119879(119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816

⩽10038161003816100381610038161003816119891 (119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816+10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

=1003816100381610038161003816119877

V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

(66)







119896V119906(119905) isin



[119906 (119905)]119903= [119906

119903(119905) 119906

119903(119905)] [119906

119873(119905)]

119903= [119906

119903

119873(119905) 119906

119903

119873(119905)]

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

10038161003816100381610038161003816119877V

119873(119905 0)

10038161003816100381610038161003816+10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

(67)


[119891119873119894

119879(119905)]

119903

= [119891119873119894

119879(119905 119903) 119891

119873119894

119879(119905 119903)]

= [

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+]


[119877V119873(119905 0)]

119903= [119877

V119903119873(119905 0) 119877

V119903

119873(119905 0)]

= [

[

119888119863

(119873+1)V119891

119903(0+)

Γ (119873V + V + 1)119905(119873+1)V

119888119863

(119873+1)V119891

119903

(0+)

Γ (119873V + V + 1)119905(119873+1)V]

]

(68)






119896V isin

119862RF[119909


119863lowast(RF119863

VΦ (119909)

RF119863(V)Φ (119909))

le119909minusV0

(2) |Γ (1 minus V)|(119878)

2(120573 + 2

2)radic119861 (120572 + 1 120573 + 1)

(69)

5 Numerical Results


1and 119896

2by using the JT


119873]119903= [119910

119903

119873 119910

119903


tions 119910(119905 119903) = [119910(119905 119903) 119910(119905 119903)] that is [119873119890]119903= [|119910

119903

119873minus 119910

119903|

|119910119903

119873minus 119910



V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (70)

in which 119910(119909) 119871RF[0 1] cap 119862







1has an invari-


cases in details




follows119888119863

V119910 (119905) + 00231119910 (119905) = 06931119890

minus06931119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (71)



V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = minus1 + 119903 0 lt 119903 le 1

119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = 1 minus 119903 0 lt 119903 le 1

(72)


119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt 119903 le 1

(73)


1199102(119905) = 119886

0⊙ 119875

(120572120573)

0(119905) + 119886

1⊙ 119875

(120572120573)

1(119905) + 119886

2⊙ 119875

(120572120573)

0(119905)

(74)

Here we have

119863085

= (

0 0 0

18639 03901 minus01755

minus03901 45267 08696

) (75)


119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

(76)


1198860= minus05757 119886

1= 02624 119886

2= minus00619

1198860= 12008 119886

1= 02408 119886

2= minus00600

(77)




1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11




119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119903(119905) 119892



119903(119905) are dif-


[1198651015840(119905)]

119903

= [1198911015840

119903(119905) 119892

1015840

119903(119905)] (11)


119903(119905) are dif-


[1198651015840(119905)]

119903

= [1198921015840

119903(119905) 119891

1015840

119903(119905)] (12)


119886111199091+ 119886

121199092+ sdot sdot sdot + 119886

1119899119909119899= 119910

1

119886211199091+ 119886

221199092+ sdot sdot sdot + 119886

2119899119909119899= 119910

2

11988611989911199091+ 119886

11989921199092+ sdot sdot sdot + 119886

119899119899119909119899= 119910

119899

(13)


119860119883 = 119884 (14)





2

119909119899)119905 given by 119909

119894= (119909

119894

119903

minus 119909

119894

119903



(

119899

sum119895=1

119886119894119895119909119895)

119903

minus

=

119899

sum119895=1

(119886119894119895119909119895)119903

minus= 119910

119903

119894minus

(

119899

sum119895=1

119886119894119895119909119895)

119903

+

=

119899

sum119895=1

(119886119894119895119909119895)119903

+= 119910

119903

119894+

(15)


119899

sum119895=1

119886119896119895119909119903

119895minus

= 119910119903

119896minus

119899

sum119895=1

119886119896119895119909119903

119895+

= 119910119903

119896+

(16)




119888119863

V119891 (119909) =

119869119898minusV

119863119898119891 (119909) if 119898 minus 1 lt V lt 119898

119863119898119891 (119909) if V = 119898 119898 isin N

(17)

where

119869V119891 (119909) =

1

Γ (V)int

119909

0



V119862 = 0 (119862 is a constant)

119888119863

V119909120573

=


Γ (120573 + 1)

Γ (120573 + 1 minus V)119909120573minusV

for 120573 isin N



(19)


119888119863

V(120582119891 (119909) + 120583119892 (119909)) = 120582

119888119863

V119891 (119909) + 120583

119888119863

V119892 (119909) (20)




1 then

119891 (119905) =

119899

sum119894=0

(119905 minus 119886)119894120572

Γ (119894120572 + 1)[

119888119863

119896120572

119886119891 (119905)]

119905=119886+ 119877

120572

119899(119905 119886) (21)

with

119877120572

119899(119905 119886) =

(119905 minus 119886)(119899+1)120572

Γ ((119899 + 1) 120572 + 1)[119888119863

(119899+1)120572

119886119891 (119905)]

119905=120585

119886 ⩽ 120585 ⩽ 119905 119905 isin [119886 119887]

(22)

where119888119863

119899120572

119886=

119888119863

120572119888

119886

119888119863

120572

119886

119888119863

120572

119886⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899 119905119894119898119890119904

(23)

Here 119888119863


119899119894

119879(119905) = sum

119899

119894=0((119905 minus 119886)

119894120572Γ(119894120572 +

1))[119888119863

119896120572

119886119891(119905)]


degree 119899119894 of 119891 and 119877120572







=

(int1

0(119889(119891(119905) 0))

119901119889119905)

1119901








(119877119871119868V119886+119891) (119909) =

1

Γ (V)int

119909

119886

119891 (119905) 119889119905

(119909 minus 119905)1minusV

119909 gt 119886 0 lt V le 1

(24)



(i) (119888119863

V119886+119891) (119909) = (

119877119871119863

120573

119886+[119891 (119905) ⊖ 119891 (119886)]) (119909)

(ii) (119888119863

V119886+119891) (119909) = (

119877119871119863

120573

119886+[minus119891 (119886) ⊖ (minus119891 (119905))]) (119909)

(25)


RF[119886 119887] and 1199090isin

(119886 119887) and Φ(119909) = (1Γ(1 minus V)) int119909

119886(119891(119905)(119909 minus 119905)

V)119889119905 We say



119886+119891)(1199090

) isin

119862RF[119886119887]


(i) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090+ ℎ) ⊖ Φ (119909

0)

ℎ

= limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0minus ℎ)

ℎ

(26)

or

(ii) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0+ ℎ)

minusℎ

= limℎrarr0

+

Φ(1199090minus ℎ) ⊖ Φ (119909

0)

minusℎ

(27)

or

(iii) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090+ ℎ) ⊖ Φ (119909

0)

ℎ

= limℎrarr0

+

Φ(1199090minus ℎ) ⊖ Φ (119909

0)

minusℎ

(28)

or

(iv) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0+ ℎ)

minusℎ

= limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0minus ℎ)

ℎ

(29)






(119888119863

V119886+119891) (119909 119903) = [

1

Γ (1 minus V)int

119909

119886

1198911015840119903

minus(119905) 119889119905

(119909 minus 119905)V

1

Γ (1 minus V)int

119909

119886

1198911015840119903

+(119905) 119889119905

(119909 minus 119905)V ]

= [(1198681minusV119886+

119863119891119903

minus) (119909) (119868

1minusV119886+

119863119891119903

+) (119909)]

(30)


(119888119863

V119886+119891) (119909 119903)

= [1

Γ (1 minus V)int

119909

119886

1198911015840119903

+(119905) 119889119905

(119909 minus 119905)V

1

Γ (1 minus V)int

119909

119886

1198911015840119903

minus(119905) 119889119905

(119909 minus 119905)V ]

= [(1198681minusV119886+

119863119891119903

+) (119909) (119868

1minusV119886+

119863119891119903

minus) (119909)]

(31)




119863119896120572119891(119909) isin

119862RF[119886119887]

(0 119887] for 119896 = 0 1 119899 + 1 where 0 lt 120572 lt 1 0 le


[119891 (119909)]119903= [119891

119903(119909) 119891

119903

(119909)]

119891119903(119909) =

119899

sum119894=0

119909119894120572

Γ (119894120572 + 1)

119888119863

119894120572119891

119903(0

+)

+

119888119863

(119899+1)120572119891

119903(119909

0)

Γ (119899120572 + 120572 + 1)119909(119899+1)120572

119891119903

(119909) =

119899

sum119894=0

119909119894120572

Γ (119894120572 + 1)

119888119863

119894120572119891

119903

(0+)

+

119888119863

(119899+1)120572119891

119903(119909

0)

Γ (119899120572 + 120572 + 1)119909(119899+1)120572

(32)

where 119888119863

120572119891

119903(0) =

119888119863

120572119891

119903(119909)|

119909=0119888119863

120572119891

119903

(0) =119888119863

120572119891

119903

(119909)|119909=0







119871119894(119909) satisfying the


int119871

0

119875(120572120573)

119871119895(119909) 119875

(120572120573)

119871119896(119909) 119908

(120572120573)

119871(119909) 119889119909 = ℎ

119896 (33)

where 119908(120572120573)

119871(119909) = 119909

120573(119871 minus 119909)

120572 and

ℎ119896=

119871120572+120573+1

Γ (119896 + 120572 + 1) Γ (119896 + 120573 + 1)

(2119896 + 120572 + 120573 + 1) 119896Γ (119896 + 120572 + 120573 + 1)119894 = 119895

0 119894 = 119895

(34)


119871119894(119909) of degree 119894 has the

form

119875(120572120573)

119871119894(119909)

=

119894

sum119896=0


Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

Γ (119896 + 120573 + 1) Γ (119894 + 120572 + 120573 + 1) (119894 minus 119896)119896119871119896119909119896

(35)

where

119875(120572120573)

119871119894(0) = (minus1)

119894Γ (119894 + 120573 + 1)

Γ (120573 + 1) 119894

119875(120572120573)

119871119894(119871) =

Γ (119894 + 120572 + 1)

Γ (120572 + 1) 119894

(36)


119875(120572120573)

119871119894(119909)

= (120572 + 120573 + 2119894 minus 1) (1205722minus 120573

2+ (

2119909

119871minus 1))

times (120572 + 120573 + 2119894) (120572 + 120573 + 2119894 minus 2)


times 119875(120572120573)

119871119894minus1(119909)


119894 (120572 + 120573 + 119894) (120572 + 120573 + 2119894 minus 2)

times 119875(120572120573)

119871119894minus2(119909) 119894 = 2 3

(37)

where 119875(120572120573)

1198710(119909) = 1 and 119875

(120572120573)

1198711(119909) = ((120572 + 120573 + 2)2)(2119909119871 minus



119906 (119909) =

+infin

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (38)


119886119895=

1

ℎ119895

int119871

0

119875(120572120573)

119871119895(119909) 119906 (119909)119908

(120572120573)

119871(119909) 119889119909 119895 = 0 1 (39)


119906119873(119909) ≃

119873

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (40)


119863V119875

(120572120573)

119871119894(119909) =

infin

sum119895=0

119878V (119894 119895 120572 120573) 119875(120572120573)

119871119894(119909)


(41)

where

119878V (119894 119895 120572 120573)

=

119894



Γ (119895 + 120573 + 1)

times Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

times (ℎ119895Γ (119895 + 119896 + 120572 + 120573 + 1) Γ (119896 + 120573 + 1)


times

119895

sum119897=0

(minus1)119895minus119897Γ (119895 + 119897 + 120572 + 120573 + 1) Γ (120572 + 1)

times Γ (119897 + 119896 + 120573 minus V + 1)


(42)




Drug intakeGI tract

DigestionBlood

Tissue




= rate of drug


GI-tract


= rate drug


blood

(43)



119889119910

119889119905= minus119896

1119909 119909 (0) = 119909

0 (44)


1is a


119889119910

119889119905= 119896

1119909 minus 119896

2119910 119910 (0) = 0 (45)



1and 119896




119883 = 119860119890(minus1198961119905) (46)


119889119910 (119905)

119889119905+ 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (47)


119863V we can





119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (48)


119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [119910119903

0 119910

119903

0] (49)

in which 119910(119909) 119871RF[0 1] cap 119862










1=










= 001386 006386 01386





2is


1varies (119896

1=



the bloodstream



1and 119896



119906 (119909) =

+infin

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

1119894(119909) (50)


119886119894=1

ℎ119894

int1

0

119875(120572120573)

1119894(119909) ⊙ 119906 (119909) ⊙ 119908

(120572120573)

1(119909) 119889119909 119894 = 0 1

(51)

and 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573 119906 isin 119871RF119901[0 1] cap 119862

RF[0 1]and 119875

(120572120573)





119894(119909) instead of 119875(120572120573)

1119894(119909) and 119908

(120572120573)(119909) = (1 minus

119909)120572⊙ 119909

120573 instead of 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573


119906 (119909) ≃ 119906119873(119909) =

119873

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

119894(119909) = a119879 ⊙ Φ

119873(119909) (52)



a119879 = [1198860 119886

1 119886

119873]

Φ119873(119909) = [119875

(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)]

119879

(53)




119906119903(119909) ≃ 119906

119903

119873(119909) = [

119873

sum119894=0

119886119903

119894minus119875

(120572120573)

119894(119909)

119873

sum119894=0

119886119903

119894+119875

(120572120573)

119894(119909)]

0 le 119903 le 1

(54)

We call that Span119875(120572120573)



2119908(120572120573)

(0 1) Hence we set

S119873(0 1) = Span 119875(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)

(55)


isin

119878119873(0 1) such that

(119863V119906119873 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119906

119873 119875

(120572120573)

119896(119909))

119908(120572120573)

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1 119906119873 (0) = 119889

0

(56)

where 119908(120572120573)(119909) = (1 minus 119909)

120572⊙ 119909

120573 and (119906 V)119908(120572120573) = int

1

0119908

(120572120573)


RF

2119908(120572120573)



119891119896= (119896

1119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

f = (1198910 119891

1 119891

119873minus1 119889

0)119879

(57)



119873

sum119895=0

119886119895⊙ [(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895⊙ 119875

(120572120573)

119895(0) = 119889

0

(58)

Denoting

119860 = (119886119896119895)0lt119896119895lt119873

119862 = (119888119896119895)0lt119896119895lt119873

(59)


(119860 + 1198962119862) ⊙ a = f (60)



= (119863(V)119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))

119908(120572120573) (0 ⩽ 119896 ⩽ 119873 minus 1 0 ⩽ 119895 ⩽ 119873) 119886

119896119895=

119863119896minus119873

119875(120572120573)

119895(0) (119896 = 119873 0 ⩽ 119895 ⩽ 119873) and 119888

119896119895= (119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))


elements of 119886119896119895and 119888


119886119896119895= ℎ

119896119878V (119895 119896 120572 120573) 0 ⩽ 119896 ⩽ 119873 minus 1 1 ⩽ 119895 ⩽ 119873 119886

119896119895

=(minus1)

119895minus119896+119873Γ (119895 + 120573 + 1) (119895 + 120572 + 120573 + 1)

119896minus119873

Γ (119895 minus 119896 + 119873 + 1) Γ (119896 minus 119873 + 1 + 120573)

119896 = 119873 0 ⩽ 119895 ⩽ 119873 119888119896119895

= ℎ119896 0 ⩽ 119896 = 119895 ⩽ 119873 minus 1

(61)




0+119891(119905) isin 119862(0 119887] 119896 = 0 1 119873 + 1 then

1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816 (62)

where 119891119873119894

119879and 119877




119891119873119894

119879(119905) =

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+ (63)


119877V119873(119905 0) =

119905(119873+1)V

Γ ((119873 + 1) V + 1)[119863

(119873+1)V0+

119891 (119905)]119905=120585

0 ⩽ 120585 ⩽ 119905 forall119905 isin (0 119887]

(64)

Therefore

119891 (119905) minus 119891119873119894

119879(119905) = 119877

V119873(119905 0) (65)


1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 =

10038161003816100381610038161003816119891 (119905) minus 119910

119873V (119905) + 119891119873119894

119879(119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816

⩽10038161003816100381610038161003816119891 (119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816+10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

=1003816100381610038161003816119877

V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

(66)







119896V119906(119905) isin



[119906 (119905)]119903= [119906

119903(119905) 119906

119903(119905)] [119906

119873(119905)]

119903= [119906

119903

119873(119905) 119906

119903

119873(119905)]

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

10038161003816100381610038161003816119877V

119873(119905 0)

10038161003816100381610038161003816+10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

(67)


[119891119873119894

119879(119905)]

119903

= [119891119873119894

119879(119905 119903) 119891

119873119894

119879(119905 119903)]

= [

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+]


[119877V119873(119905 0)]

119903= [119877

V119903119873(119905 0) 119877

V119903

119873(119905 0)]

= [

[

119888119863

(119873+1)V119891

119903(0+)

Γ (119873V + V + 1)119905(119873+1)V

119888119863

(119873+1)V119891

119903

(0+)

Γ (119873V + V + 1)119905(119873+1)V]

]

(68)






119896V isin

119862RF[119909


119863lowast(RF119863

VΦ (119909)

RF119863(V)Φ (119909))

le119909minusV0

(2) |Γ (1 minus V)|(119878)

2(120573 + 2

2)radic119861 (120572 + 1 120573 + 1)

(69)

5 Numerical Results


1and 119896

2by using the JT


119873]119903= [119910

119903

119873 119910

119903


tions 119910(119905 119903) = [119910(119905 119903) 119910(119905 119903)] that is [119873119890]119903= [|119910

119903

119873minus 119910

119903|

|119910119903

119873minus 119910



V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (70)

in which 119910(119909) 119871RF[0 1] cap 119862







1has an invari-


cases in details




follows119888119863

V119910 (119905) + 00231119910 (119905) = 06931119890

minus06931119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (71)



V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = minus1 + 119903 0 lt 119903 le 1

119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = 1 minus 119903 0 lt 119903 le 1

(72)


119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt 119903 le 1

(73)


1199102(119905) = 119886

0⊙ 119875

(120572120573)

0(119905) + 119886

1⊙ 119875

(120572120573)

1(119905) + 119886

2⊙ 119875

(120572120573)

0(119905)

(74)

Here we have

119863085

= (

0 0 0

18639 03901 minus01755

minus03901 45267 08696

) (75)


119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

(76)


1198860= minus05757 119886

1= 02624 119886

2= minus00619

1198860= 12008 119886

1= 02408 119886

2= minus00600

(77)




1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11




119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












=

(int1

0(119889(119891(119905) 0))

119901119889119905)

1119901








(119877119871119868V119886+119891) (119909) =

1

Γ (V)int

119909

119886

119891 (119905) 119889119905

(119909 minus 119905)1minusV

119909 gt 119886 0 lt V le 1

(24)



(i) (119888119863

V119886+119891) (119909) = (

119877119871119863

120573

119886+[119891 (119905) ⊖ 119891 (119886)]) (119909)

(ii) (119888119863

V119886+119891) (119909) = (

119877119871119863

120573

119886+[minus119891 (119886) ⊖ (minus119891 (119905))]) (119909)

(25)


RF[119886 119887] and 1199090isin

(119886 119887) and Φ(119909) = (1Γ(1 minus V)) int119909

119886(119891(119905)(119909 minus 119905)

V)119889119905 We say



119886+119891)(1199090

) isin

119862RF[119886119887]


(i) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090+ ℎ) ⊖ Φ (119909

0)

ℎ

= limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0minus ℎ)

ℎ

(26)

or

(ii) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0+ ℎ)

minusℎ

= limℎrarr0

+

Φ(1199090minus ℎ) ⊖ Φ (119909

0)

minusℎ

(27)

or

(iii) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090+ ℎ) ⊖ Φ (119909

0)

ℎ

= limℎrarr0

+

Φ(1199090minus ℎ) ⊖ Φ (119909

0)

minusℎ

(28)

or

(iv) (119888119863

V119886+119891) (1199090

) = limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0+ ℎ)

minusℎ

= limℎrarr0

+

Φ(1199090) ⊖ Φ (119909

0minus ℎ)

ℎ

(29)






(119888119863

V119886+119891) (119909 119903) = [

1

Γ (1 minus V)int

119909

119886

1198911015840119903

minus(119905) 119889119905

(119909 minus 119905)V

1

Γ (1 minus V)int

119909

119886

1198911015840119903

+(119905) 119889119905

(119909 minus 119905)V ]

= [(1198681minusV119886+

119863119891119903

minus) (119909) (119868

1minusV119886+

119863119891119903

+) (119909)]

(30)


(119888119863

V119886+119891) (119909 119903)

= [1

Γ (1 minus V)int

119909

119886

1198911015840119903

+(119905) 119889119905

(119909 minus 119905)V

1

Γ (1 minus V)int

119909

119886

1198911015840119903

minus(119905) 119889119905

(119909 minus 119905)V ]

= [(1198681minusV119886+

119863119891119903

+) (119909) (119868

1minusV119886+

119863119891119903

minus) (119909)]

(31)




119863119896120572119891(119909) isin

119862RF[119886119887]

(0 119887] for 119896 = 0 1 119899 + 1 where 0 lt 120572 lt 1 0 le


[119891 (119909)]119903= [119891

119903(119909) 119891

119903

(119909)]

119891119903(119909) =

119899

sum119894=0

119909119894120572

Γ (119894120572 + 1)

119888119863

119894120572119891

119903(0

+)

+

119888119863

(119899+1)120572119891

119903(119909

0)

Γ (119899120572 + 120572 + 1)119909(119899+1)120572

119891119903

(119909) =

119899

sum119894=0

119909119894120572

Γ (119894120572 + 1)

119888119863

119894120572119891

119903

(0+)

+

119888119863

(119899+1)120572119891

119903(119909

0)

Γ (119899120572 + 120572 + 1)119909(119899+1)120572

(32)

where 119888119863

120572119891

119903(0) =

119888119863

120572119891

119903(119909)|

119909=0119888119863

120572119891

119903

(0) =119888119863

120572119891

119903

(119909)|119909=0







119871119894(119909) satisfying the


int119871

0

119875(120572120573)

119871119895(119909) 119875

(120572120573)

119871119896(119909) 119908

(120572120573)

119871(119909) 119889119909 = ℎ

119896 (33)

where 119908(120572120573)

119871(119909) = 119909

120573(119871 minus 119909)

120572 and

ℎ119896=

119871120572+120573+1

Γ (119896 + 120572 + 1) Γ (119896 + 120573 + 1)

(2119896 + 120572 + 120573 + 1) 119896Γ (119896 + 120572 + 120573 + 1)119894 = 119895

0 119894 = 119895

(34)


119871119894(119909) of degree 119894 has the

form

119875(120572120573)

119871119894(119909)

=

119894

sum119896=0


Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

Γ (119896 + 120573 + 1) Γ (119894 + 120572 + 120573 + 1) (119894 minus 119896)119896119871119896119909119896

(35)

where

119875(120572120573)

119871119894(0) = (minus1)

119894Γ (119894 + 120573 + 1)

Γ (120573 + 1) 119894

119875(120572120573)

119871119894(119871) =

Γ (119894 + 120572 + 1)

Γ (120572 + 1) 119894

(36)


119875(120572120573)

119871119894(119909)

= (120572 + 120573 + 2119894 minus 1) (1205722minus 120573

2+ (

2119909

119871minus 1))

times (120572 + 120573 + 2119894) (120572 + 120573 + 2119894 minus 2)


times 119875(120572120573)

119871119894minus1(119909)


119894 (120572 + 120573 + 119894) (120572 + 120573 + 2119894 minus 2)

times 119875(120572120573)

119871119894minus2(119909) 119894 = 2 3

(37)

where 119875(120572120573)

1198710(119909) = 1 and 119875

(120572120573)

1198711(119909) = ((120572 + 120573 + 2)2)(2119909119871 minus



119906 (119909) =

+infin

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (38)


119886119895=

1

ℎ119895

int119871

0

119875(120572120573)

119871119895(119909) 119906 (119909)119908

(120572120573)

119871(119909) 119889119909 119895 = 0 1 (39)


119906119873(119909) ≃

119873

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (40)


119863V119875

(120572120573)

119871119894(119909) =

infin

sum119895=0

119878V (119894 119895 120572 120573) 119875(120572120573)

119871119894(119909)


(41)

where

119878V (119894 119895 120572 120573)

=

119894



Γ (119895 + 120573 + 1)

times Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

times (ℎ119895Γ (119895 + 119896 + 120572 + 120573 + 1) Γ (119896 + 120573 + 1)


times

119895

sum119897=0

(minus1)119895minus119897Γ (119895 + 119897 + 120572 + 120573 + 1) Γ (120572 + 1)

times Γ (119897 + 119896 + 120573 minus V + 1)


(42)




Drug intakeGI tract

DigestionBlood

Tissue




= rate of drug


GI-tract


= rate drug


blood

(43)



119889119910

119889119905= minus119896

1119909 119909 (0) = 119909

0 (44)


1is a


119889119910

119889119905= 119896

1119909 minus 119896

2119910 119910 (0) = 0 (45)



1and 119896




119883 = 119860119890(minus1198961119905) (46)


119889119910 (119905)

119889119905+ 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (47)


119863V we can





119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (48)


119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [119910119903

0 119910

119903

0] (49)

in which 119910(119909) 119871RF[0 1] cap 119862










1=










= 001386 006386 01386





2is


1varies (119896

1=



the bloodstream



1and 119896



119906 (119909) =

+infin

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

1119894(119909) (50)


119886119894=1

ℎ119894

int1

0

119875(120572120573)

1119894(119909) ⊙ 119906 (119909) ⊙ 119908

(120572120573)

1(119909) 119889119909 119894 = 0 1

(51)

and 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573 119906 isin 119871RF119901[0 1] cap 119862

RF[0 1]and 119875

(120572120573)





119894(119909) instead of 119875(120572120573)

1119894(119909) and 119908

(120572120573)(119909) = (1 minus

119909)120572⊙ 119909

120573 instead of 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573


119906 (119909) ≃ 119906119873(119909) =

119873

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

119894(119909) = a119879 ⊙ Φ

119873(119909) (52)



a119879 = [1198860 119886

1 119886

119873]

Φ119873(119909) = [119875

(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)]

119879

(53)




119906119903(119909) ≃ 119906

119903

119873(119909) = [

119873

sum119894=0

119886119903

119894minus119875

(120572120573)

119894(119909)

119873

sum119894=0

119886119903

119894+119875

(120572120573)

119894(119909)]

0 le 119903 le 1

(54)

We call that Span119875(120572120573)



2119908(120572120573)

(0 1) Hence we set

S119873(0 1) = Span 119875(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)

(55)


isin

119878119873(0 1) such that

(119863V119906119873 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119906

119873 119875

(120572120573)

119896(119909))

119908(120572120573)

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1 119906119873 (0) = 119889

0

(56)

where 119908(120572120573)(119909) = (1 minus 119909)

120572⊙ 119909

120573 and (119906 V)119908(120572120573) = int

1

0119908

(120572120573)


RF

2119908(120572120573)



119891119896= (119896

1119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

f = (1198910 119891

1 119891

119873minus1 119889

0)119879

(57)



119873

sum119895=0

119886119895⊙ [(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895⊙ 119875

(120572120573)

119895(0) = 119889

0

(58)

Denoting

119860 = (119886119896119895)0lt119896119895lt119873

119862 = (119888119896119895)0lt119896119895lt119873

(59)


(119860 + 1198962119862) ⊙ a = f (60)



= (119863(V)119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))

119908(120572120573) (0 ⩽ 119896 ⩽ 119873 minus 1 0 ⩽ 119895 ⩽ 119873) 119886

119896119895=

119863119896minus119873

119875(120572120573)

119895(0) (119896 = 119873 0 ⩽ 119895 ⩽ 119873) and 119888

119896119895= (119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))


elements of 119886119896119895and 119888


119886119896119895= ℎ

119896119878V (119895 119896 120572 120573) 0 ⩽ 119896 ⩽ 119873 minus 1 1 ⩽ 119895 ⩽ 119873 119886

119896119895

=(minus1)

119895minus119896+119873Γ (119895 + 120573 + 1) (119895 + 120572 + 120573 + 1)

119896minus119873

Γ (119895 minus 119896 + 119873 + 1) Γ (119896 minus 119873 + 1 + 120573)

119896 = 119873 0 ⩽ 119895 ⩽ 119873 119888119896119895

= ℎ119896 0 ⩽ 119896 = 119895 ⩽ 119873 minus 1

(61)




0+119891(119905) isin 119862(0 119887] 119896 = 0 1 119873 + 1 then

1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816 (62)

where 119891119873119894

119879and 119877




119891119873119894

119879(119905) =

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+ (63)


119877V119873(119905 0) =

119905(119873+1)V

Γ ((119873 + 1) V + 1)[119863

(119873+1)V0+

119891 (119905)]119905=120585

0 ⩽ 120585 ⩽ 119905 forall119905 isin (0 119887]

(64)

Therefore

119891 (119905) minus 119891119873119894

119879(119905) = 119877

V119873(119905 0) (65)


1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 =

10038161003816100381610038161003816119891 (119905) minus 119910

119873V (119905) + 119891119873119894

119879(119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816

⩽10038161003816100381610038161003816119891 (119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816+10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

=1003816100381610038161003816119877

V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

(66)







119896V119906(119905) isin



[119906 (119905)]119903= [119906

119903(119905) 119906

119903(119905)] [119906

119873(119905)]

119903= [119906

119903

119873(119905) 119906

119903

119873(119905)]

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

10038161003816100381610038161003816119877V

119873(119905 0)

10038161003816100381610038161003816+10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

(67)


[119891119873119894

119879(119905)]

119903

= [119891119873119894

119879(119905 119903) 119891

119873119894

119879(119905 119903)]

= [

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+]


[119877V119873(119905 0)]

119903= [119877

V119903119873(119905 0) 119877

V119903

119873(119905 0)]

= [

[

119888119863

(119873+1)V119891

119903(0+)

Γ (119873V + V + 1)119905(119873+1)V

119888119863

(119873+1)V119891

119903

(0+)

Γ (119873V + V + 1)119905(119873+1)V]

]

(68)






119896V isin

119862RF[119909


119863lowast(RF119863

VΦ (119909)

RF119863(V)Φ (119909))

le119909minusV0

(2) |Γ (1 minus V)|(119878)

2(120573 + 2

2)radic119861 (120572 + 1 120573 + 1)

(69)

5 Numerical Results


1and 119896

2by using the JT


119873]119903= [119910

119903

119873 119910

119903


tions 119910(119905 119903) = [119910(119905 119903) 119910(119905 119903)] that is [119873119890]119903= [|119910

119903

119873minus 119910

119903|

|119910119903

119873minus 119910



V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (70)

in which 119910(119909) 119871RF[0 1] cap 119862







1has an invari-


cases in details




follows119888119863

V119910 (119905) + 00231119910 (119905) = 06931119890

minus06931119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (71)



V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = minus1 + 119903 0 lt 119903 le 1

119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = 1 minus 119903 0 lt 119903 le 1

(72)


119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt 119903 le 1

(73)


1199102(119905) = 119886

0⊙ 119875

(120572120573)

0(119905) + 119886

1⊙ 119875

(120572120573)

1(119905) + 119886

2⊙ 119875

(120572120573)

0(119905)

(74)

Here we have

119863085

= (

0 0 0

18639 03901 minus01755

minus03901 45267 08696

) (75)


119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

(76)


1198860= minus05757 119886

1= 02624 119886

2= minus00619

1198860= 12008 119886

1= 02408 119886

2= minus00600

(77)




1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11




119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119871119894(119909) satisfying the


int119871

0

119875(120572120573)

119871119895(119909) 119875

(120572120573)

119871119896(119909) 119908

(120572120573)

119871(119909) 119889119909 = ℎ

119896 (33)

where 119908(120572120573)

119871(119909) = 119909

120573(119871 minus 119909)

120572 and

ℎ119896=

119871120572+120573+1

Γ (119896 + 120572 + 1) Γ (119896 + 120573 + 1)

(2119896 + 120572 + 120573 + 1) 119896Γ (119896 + 120572 + 120573 + 1)119894 = 119895

0 119894 = 119895

(34)


119871119894(119909) of degree 119894 has the

form

119875(120572120573)

119871119894(119909)

=

119894

sum119896=0


Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

Γ (119896 + 120573 + 1) Γ (119894 + 120572 + 120573 + 1) (119894 minus 119896)119896119871119896119909119896

(35)

where

119875(120572120573)

119871119894(0) = (minus1)

119894Γ (119894 + 120573 + 1)

Γ (120573 + 1) 119894

119875(120572120573)

119871119894(119871) =

Γ (119894 + 120572 + 1)

Γ (120572 + 1) 119894

(36)


119875(120572120573)

119871119894(119909)

= (120572 + 120573 + 2119894 minus 1) (1205722minus 120573

2+ (

2119909

119871minus 1))

times (120572 + 120573 + 2119894) (120572 + 120573 + 2119894 minus 2)


times 119875(120572120573)

119871119894minus1(119909)


119894 (120572 + 120573 + 119894) (120572 + 120573 + 2119894 minus 2)

times 119875(120572120573)

119871119894minus2(119909) 119894 = 2 3

(37)

where 119875(120572120573)

1198710(119909) = 1 and 119875

(120572120573)

1198711(119909) = ((120572 + 120573 + 2)2)(2119909119871 minus



119906 (119909) =

+infin

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (38)


119886119895=

1

ℎ119895

int119871

0

119875(120572120573)

119871119895(119909) 119906 (119909)119908

(120572120573)

119871(119909) 119889119909 119895 = 0 1 (39)


119906119873(119909) ≃

119873

sum119895=0

119886119895119875

(120572120573)

119871119895(119909) (40)


119863V119875

(120572120573)

119871119894(119909) =

infin

sum119895=0

119878V (119894 119895 120572 120573) 119875(120572120573)

119871119894(119909)


(41)

where

119878V (119894 119895 120572 120573)

=

119894



Γ (119895 + 120573 + 1)

times Γ (119894 + 120573 + 1) Γ (119894 + 119896 + 120572 + 120573 + 1)

times (ℎ119895Γ (119895 + 119896 + 120572 + 120573 + 1) Γ (119896 + 120573 + 1)


times

119895

sum119897=0

(minus1)119895minus119897Γ (119895 + 119897 + 120572 + 120573 + 1) Γ (120572 + 1)

times Γ (119897 + 119896 + 120573 minus V + 1)


(42)




Drug intakeGI tract

DigestionBlood

Tissue




= rate of drug


GI-tract


= rate drug


blood

(43)



119889119910

119889119905= minus119896

1119909 119909 (0) = 119909

0 (44)


1is a


119889119910

119889119905= 119896

1119909 minus 119896

2119910 119910 (0) = 0 (45)



1and 119896




119883 = 119860119890(minus1198961119905) (46)


119889119910 (119905)

119889119905+ 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (47)


119863V we can





119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (48)


119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [119910119903

0 119910

119903

0] (49)

in which 119910(119909) 119871RF[0 1] cap 119862










1=










= 001386 006386 01386





2is


1varies (119896

1=



the bloodstream



1and 119896



119906 (119909) =

+infin

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

1119894(119909) (50)


119886119894=1

ℎ119894

int1

0

119875(120572120573)

1119894(119909) ⊙ 119906 (119909) ⊙ 119908

(120572120573)

1(119909) 119889119909 119894 = 0 1

(51)

and 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573 119906 isin 119871RF119901[0 1] cap 119862

RF[0 1]and 119875

(120572120573)





119894(119909) instead of 119875(120572120573)

1119894(119909) and 119908

(120572120573)(119909) = (1 minus

119909)120572⊙ 119909

120573 instead of 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573


119906 (119909) ≃ 119906119873(119909) =

119873

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

119894(119909) = a119879 ⊙ Φ

119873(119909) (52)



a119879 = [1198860 119886

1 119886

119873]

Φ119873(119909) = [119875

(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)]

119879

(53)




119906119903(119909) ≃ 119906

119903

119873(119909) = [

119873

sum119894=0

119886119903

119894minus119875

(120572120573)

119894(119909)

119873

sum119894=0

119886119903

119894+119875

(120572120573)

119894(119909)]

0 le 119903 le 1

(54)

We call that Span119875(120572120573)



2119908(120572120573)

(0 1) Hence we set

S119873(0 1) = Span 119875(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)

(55)


isin

119878119873(0 1) such that

(119863V119906119873 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119906

119873 119875

(120572120573)

119896(119909))

119908(120572120573)

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1 119906119873 (0) = 119889

0

(56)

where 119908(120572120573)(119909) = (1 minus 119909)

120572⊙ 119909

120573 and (119906 V)119908(120572120573) = int

1

0119908

(120572120573)


RF

2119908(120572120573)



119891119896= (119896

1119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

f = (1198910 119891

1 119891

119873minus1 119889

0)119879

(57)



119873

sum119895=0

119886119895⊙ [(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895⊙ 119875

(120572120573)

119895(0) = 119889

0

(58)

Denoting

119860 = (119886119896119895)0lt119896119895lt119873

119862 = (119888119896119895)0lt119896119895lt119873

(59)


(119860 + 1198962119862) ⊙ a = f (60)



= (119863(V)119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))

119908(120572120573) (0 ⩽ 119896 ⩽ 119873 minus 1 0 ⩽ 119895 ⩽ 119873) 119886

119896119895=

119863119896minus119873

119875(120572120573)

119895(0) (119896 = 119873 0 ⩽ 119895 ⩽ 119873) and 119888

119896119895= (119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))


elements of 119886119896119895and 119888


119886119896119895= ℎ

119896119878V (119895 119896 120572 120573) 0 ⩽ 119896 ⩽ 119873 minus 1 1 ⩽ 119895 ⩽ 119873 119886

119896119895

=(minus1)

119895minus119896+119873Γ (119895 + 120573 + 1) (119895 + 120572 + 120573 + 1)

119896minus119873

Γ (119895 minus 119896 + 119873 + 1) Γ (119896 minus 119873 + 1 + 120573)

119896 = 119873 0 ⩽ 119895 ⩽ 119873 119888119896119895

= ℎ119896 0 ⩽ 119896 = 119895 ⩽ 119873 minus 1

(61)




0+119891(119905) isin 119862(0 119887] 119896 = 0 1 119873 + 1 then

1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816 (62)

where 119891119873119894

119879and 119877




119891119873119894

119879(119905) =

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+ (63)


119877V119873(119905 0) =

119905(119873+1)V

Γ ((119873 + 1) V + 1)[119863

(119873+1)V0+

119891 (119905)]119905=120585

0 ⩽ 120585 ⩽ 119905 forall119905 isin (0 119887]

(64)

Therefore

119891 (119905) minus 119891119873119894

119879(119905) = 119877

V119873(119905 0) (65)


1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 =

10038161003816100381610038161003816119891 (119905) minus 119910

119873V (119905) + 119891119873119894

119879(119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816

⩽10038161003816100381610038161003816119891 (119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816+10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

=1003816100381610038161003816119877

V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

(66)







119896V119906(119905) isin



[119906 (119905)]119903= [119906

119903(119905) 119906

119903(119905)] [119906

119873(119905)]

119903= [119906

119903

119873(119905) 119906

119903

119873(119905)]

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

10038161003816100381610038161003816119877V

119873(119905 0)

10038161003816100381610038161003816+10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

(67)


[119891119873119894

119879(119905)]

119903

= [119891119873119894

119879(119905 119903) 119891

119873119894

119879(119905 119903)]

= [

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+]


[119877V119873(119905 0)]

119903= [119877

V119903119873(119905 0) 119877

V119903

119873(119905 0)]

= [

[

119888119863

(119873+1)V119891

119903(0+)

Γ (119873V + V + 1)119905(119873+1)V

119888119863

(119873+1)V119891

119903

(0+)

Γ (119873V + V + 1)119905(119873+1)V]

]

(68)






119896V isin

119862RF[119909


119863lowast(RF119863

VΦ (119909)

RF119863(V)Φ (119909))

le119909minusV0

(2) |Γ (1 minus V)|(119878)

2(120573 + 2

2)radic119861 (120572 + 1 120573 + 1)

(69)

5 Numerical Results


1and 119896

2by using the JT


119873]119903= [119910

119903

119873 119910

119903


tions 119910(119905 119903) = [119910(119905 119903) 119910(119905 119903)] that is [119873119890]119903= [|119910

119903

119873minus 119910

119903|

|119910119903

119873minus 119910



V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (70)

in which 119910(119909) 119871RF[0 1] cap 119862







1has an invari-


cases in details




follows119888119863

V119910 (119905) + 00231119910 (119905) = 06931119890

minus06931119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (71)



V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = minus1 + 119903 0 lt 119903 le 1

119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = 1 minus 119903 0 lt 119903 le 1

(72)


119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt 119903 le 1

(73)


1199102(119905) = 119886

0⊙ 119875

(120572120573)

0(119905) + 119886

1⊙ 119875

(120572120573)

1(119905) + 119886

2⊙ 119875

(120572120573)

0(119905)

(74)

Here we have

119863085

= (

0 0 0

18639 03901 minus01755

minus03901 45267 08696

) (75)


119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

(76)


1198860= minus05757 119886

1= 02624 119886

2= minus00619

1198860= 12008 119886

1= 02408 119886

2= minus00600

(77)




1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11




119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Drug intakeGI tract

DigestionBlood

Tissue




= rate of drug


GI-tract


= rate drug


blood

(43)



119889119910

119889119905= minus119896

1119909 119909 (0) = 119909

0 (44)


1is a


119889119910

119889119905= 119896

1119909 minus 119896

2119910 119910 (0) = 0 (45)



1and 119896




119883 = 119860119890(minus1198961119905) (46)


119889119910 (119905)

119889119905+ 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (47)


119863V we can





119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0) = 0 (48)


119888119863

V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [119910119903

0 119910

119903

0] (49)

in which 119910(119909) 119871RF[0 1] cap 119862










1=










= 001386 006386 01386





2is


1varies (119896

1=



the bloodstream



1and 119896



119906 (119909) =

+infin

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

1119894(119909) (50)


119886119894=1

ℎ119894

int1

0

119875(120572120573)

1119894(119909) ⊙ 119906 (119909) ⊙ 119908

(120572120573)

1(119909) 119889119909 119894 = 0 1

(51)

and 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573 119906 isin 119871RF119901[0 1] cap 119862

RF[0 1]and 119875

(120572120573)





119894(119909) instead of 119875(120572120573)

1119894(119909) and 119908

(120572120573)(119909) = (1 minus

119909)120572⊙ 119909

120573 instead of 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573


119906 (119909) ≃ 119906119873(119909) =

119873

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

119894(119909) = a119879 ⊙ Φ

119873(119909) (52)



a119879 = [1198860 119886

1 119886

119873]

Φ119873(119909) = [119875

(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)]

119879

(53)




119906119903(119909) ≃ 119906

119903

119873(119909) = [

119873

sum119894=0

119886119903

119894minus119875

(120572120573)

119894(119909)

119873

sum119894=0

119886119903

119894+119875

(120572120573)

119894(119909)]

0 le 119903 le 1

(54)

We call that Span119875(120572120573)



2119908(120572120573)

(0 1) Hence we set

S119873(0 1) = Span 119875(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)

(55)


isin

119878119873(0 1) such that

(119863V119906119873 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119906

119873 119875

(120572120573)

119896(119909))

119908(120572120573)

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1 119906119873 (0) = 119889

0

(56)

where 119908(120572120573)(119909) = (1 minus 119909)

120572⊙ 119909

120573 and (119906 V)119908(120572120573) = int

1

0119908

(120572120573)


RF

2119908(120572120573)



119891119896= (119896

1119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

f = (1198910 119891

1 119891

119873minus1 119889

0)119879

(57)



119873

sum119895=0

119886119895⊙ [(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895⊙ 119875

(120572120573)

119895(0) = 119889

0

(58)

Denoting

119860 = (119886119896119895)0lt119896119895lt119873

119862 = (119888119896119895)0lt119896119895lt119873

(59)


(119860 + 1198962119862) ⊙ a = f (60)



= (119863(V)119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))

119908(120572120573) (0 ⩽ 119896 ⩽ 119873 minus 1 0 ⩽ 119895 ⩽ 119873) 119886

119896119895=

119863119896minus119873

119875(120572120573)

119895(0) (119896 = 119873 0 ⩽ 119895 ⩽ 119873) and 119888

119896119895= (119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))


elements of 119886119896119895and 119888


119886119896119895= ℎ

119896119878V (119895 119896 120572 120573) 0 ⩽ 119896 ⩽ 119873 minus 1 1 ⩽ 119895 ⩽ 119873 119886

119896119895

=(minus1)

119895minus119896+119873Γ (119895 + 120573 + 1) (119895 + 120572 + 120573 + 1)

119896minus119873

Γ (119895 minus 119896 + 119873 + 1) Γ (119896 minus 119873 + 1 + 120573)

119896 = 119873 0 ⩽ 119895 ⩽ 119873 119888119896119895

= ℎ119896 0 ⩽ 119896 = 119895 ⩽ 119873 minus 1

(61)




0+119891(119905) isin 119862(0 119887] 119896 = 0 1 119873 + 1 then

1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816 (62)

where 119891119873119894

119879and 119877




119891119873119894

119879(119905) =

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+ (63)


119877V119873(119905 0) =

119905(119873+1)V

Γ ((119873 + 1) V + 1)[119863

(119873+1)V0+

119891 (119905)]119905=120585

0 ⩽ 120585 ⩽ 119905 forall119905 isin (0 119887]

(64)

Therefore

119891 (119905) minus 119891119873119894

119879(119905) = 119877

V119873(119905 0) (65)


1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 =

10038161003816100381610038161003816119891 (119905) minus 119910

119873V (119905) + 119891119873119894

119879(119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816

⩽10038161003816100381610038161003816119891 (119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816+10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

=1003816100381610038161003816119877

V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

(66)







119896V119906(119905) isin



[119906 (119905)]119903= [119906

119903(119905) 119906

119903(119905)] [119906

119873(119905)]

119903= [119906

119903

119873(119905) 119906

119903

119873(119905)]

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

10038161003816100381610038161003816119877V

119873(119905 0)

10038161003816100381610038161003816+10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

(67)


[119891119873119894

119879(119905)]

119903

= [119891119873119894

119879(119905 119903) 119891

119873119894

119879(119905 119903)]

= [

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+]


[119877V119873(119905 0)]

119903= [119877

V119903119873(119905 0) 119877

V119903

119873(119905 0)]

= [

[

119888119863

(119873+1)V119891

119903(0+)

Γ (119873V + V + 1)119905(119873+1)V

119888119863

(119873+1)V119891

119903

(0+)

Γ (119873V + V + 1)119905(119873+1)V]

]

(68)






119896V isin

119862RF[119909


119863lowast(RF119863

VΦ (119909)

RF119863(V)Φ (119909))

le119909minusV0

(2) |Γ (1 minus V)|(119878)

2(120573 + 2

2)radic119861 (120572 + 1 120573 + 1)

(69)

5 Numerical Results


1and 119896

2by using the JT


119873]119903= [119910

119903

119873 119910

119903


tions 119910(119905 119903) = [119910(119905 119903) 119910(119905 119903)] that is [119873119890]119903= [|119910

119903

119873minus 119910

119903|

|119910119903

119873minus 119910



V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (70)

in which 119910(119909) 119871RF[0 1] cap 119862







1has an invari-


cases in details




follows119888119863

V119910 (119905) + 00231119910 (119905) = 06931119890

minus06931119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (71)



V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = minus1 + 119903 0 lt 119903 le 1

119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = 1 minus 119903 0 lt 119903 le 1

(72)


119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt 119903 le 1

(73)


1199102(119905) = 119886

0⊙ 119875

(120572120573)

0(119905) + 119886

1⊙ 119875

(120572120573)

1(119905) + 119886

2⊙ 119875

(120572120573)

0(119905)

(74)

Here we have

119863085

= (

0 0 0

18639 03901 minus01755

minus03901 45267 08696

) (75)


119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

(76)


1198860= minus05757 119886

1= 02624 119886

2= minus00619

1198860= 12008 119886

1= 02408 119886

2= minus00600

(77)




1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11




119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1=










= 001386 006386 01386





2is


1varies (119896

1=



the bloodstream



1and 119896



119906 (119909) =

+infin

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

1119894(119909) (50)


119886119894=1

ℎ119894

int1

0

119875(120572120573)

1119894(119909) ⊙ 119906 (119909) ⊙ 119908

(120572120573)

1(119909) 119889119909 119894 = 0 1

(51)

and 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573 119906 isin 119871RF119901[0 1] cap 119862

RF[0 1]and 119875

(120572120573)





119894(119909) instead of 119875(120572120573)

1119894(119909) and 119908

(120572120573)(119909) = (1 minus

119909)120572⊙ 119909

120573 instead of 119908(120572120573)

1(119909) = (1 minus 119909)

120572⊙ 119909

120573


119906 (119909) ≃ 119906119873(119909) =

119873

sum119894=0

lowast119886119894⊙ 119875

(120572120573)

119894(119909) = a119879 ⊙ Φ

119873(119909) (52)



a119879 = [1198860 119886

1 119886

119873]

Φ119873(119909) = [119875

(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)]

119879

(53)




119906119903(119909) ≃ 119906

119903

119873(119909) = [

119873

sum119894=0

119886119903

119894minus119875

(120572120573)

119894(119909)

119873

sum119894=0

119886119903

119894+119875

(120572120573)

119894(119909)]

0 le 119903 le 1

(54)

We call that Span119875(120572120573)



2119908(120572120573)

(0 1) Hence we set

S119873(0 1) = Span 119875(120572120573)

0(119909) 119875

(120572120573)

1(119909) 119875

(120572120573)

119873(119909)

(55)


isin

119878119873(0 1) such that

(119863V119906119873 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119906

119873 119875

(120572120573)

119896(119909))

119908(120572120573)

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1 119906119873 (0) = 119889

0

(56)

where 119908(120572120573)(119909) = (1 minus 119909)

120572⊙ 119909

120573 and (119906 V)119908(120572120573) = int

1

0119908

(120572120573)


RF

2119908(120572120573)



119891119896= (119896

1119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

f = (1198910 119891

1 119891

119873minus1 119889

0)119879

(57)



119873

sum119895=0

119886119895⊙ [(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895⊙ 119875

(120572120573)

119895(0) = 119889

0

(58)

Denoting

119860 = (119886119896119895)0lt119896119895lt119873

119862 = (119888119896119895)0lt119896119895lt119873

(59)


(119860 + 1198962119862) ⊙ a = f (60)



= (119863(V)119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))

119908(120572120573) (0 ⩽ 119896 ⩽ 119873 minus 1 0 ⩽ 119895 ⩽ 119873) 119886

119896119895=

119863119896minus119873

119875(120572120573)

119895(0) (119896 = 119873 0 ⩽ 119895 ⩽ 119873) and 119888

119896119895= (119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))


elements of 119886119896119895and 119888


119886119896119895= ℎ

119896119878V (119895 119896 120572 120573) 0 ⩽ 119896 ⩽ 119873 minus 1 1 ⩽ 119895 ⩽ 119873 119886

119896119895

=(minus1)

119895minus119896+119873Γ (119895 + 120573 + 1) (119895 + 120572 + 120573 + 1)

119896minus119873

Γ (119895 minus 119896 + 119873 + 1) Γ (119896 minus 119873 + 1 + 120573)

119896 = 119873 0 ⩽ 119895 ⩽ 119873 119888119896119895

= ℎ119896 0 ⩽ 119896 = 119895 ⩽ 119873 minus 1

(61)




0+119891(119905) isin 119862(0 119887] 119896 = 0 1 119873 + 1 then

1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816 (62)

where 119891119873119894

119879and 119877




119891119873119894

119879(119905) =

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+ (63)


119877V119873(119905 0) =

119905(119873+1)V

Γ ((119873 + 1) V + 1)[119863

(119873+1)V0+

119891 (119905)]119905=120585

0 ⩽ 120585 ⩽ 119905 forall119905 isin (0 119887]

(64)

Therefore

119891 (119905) minus 119891119873119894

119879(119905) = 119877

V119873(119905 0) (65)


1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 =

10038161003816100381610038161003816119891 (119905) minus 119910

119873V (119905) + 119891119873119894

119879(119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816

⩽10038161003816100381610038161003816119891 (119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816+10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

=1003816100381610038161003816119877

V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

(66)







119896V119906(119905) isin



[119906 (119905)]119903= [119906

119903(119905) 119906

119903(119905)] [119906

119873(119905)]

119903= [119906

119903

119873(119905) 119906

119903

119873(119905)]

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

10038161003816100381610038161003816119877V

119873(119905 0)

10038161003816100381610038161003816+10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

(67)


[119891119873119894

119879(119905)]

119903

= [119891119873119894

119879(119905 119903) 119891

119873119894

119879(119905 119903)]

= [

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+]


[119877V119873(119905 0)]

119903= [119877

V119903119873(119905 0) 119877

V119903

119873(119905 0)]

= [

[

119888119863

(119873+1)V119891

119903(0+)

Γ (119873V + V + 1)119905(119873+1)V

119888119863

(119873+1)V119891

119903

(0+)

Γ (119873V + V + 1)119905(119873+1)V]

]

(68)






119896V isin

119862RF[119909


119863lowast(RF119863

VΦ (119909)

RF119863(V)Φ (119909))

le119909minusV0

(2) |Γ (1 minus V)|(119878)

2(120573 + 2

2)radic119861 (120572 + 1 120573 + 1)

(69)

5 Numerical Results


1and 119896

2by using the JT


119873]119903= [119910

119903

119873 119910

119903


tions 119910(119905 119903) = [119910(119905 119903) 119910(119905 119903)] that is [119873119890]119903= [|119910

119903

119873minus 119910

119903|

|119910119903

119873minus 119910



V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (70)

in which 119910(119909) 119871RF[0 1] cap 119862







1has an invari-


cases in details




follows119888119863

V119910 (119905) + 00231119910 (119905) = 06931119890

minus06931119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (71)



V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = minus1 + 119903 0 lt 119903 le 1

119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = 1 minus 119903 0 lt 119903 le 1

(72)


119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt 119903 le 1

(73)


1199102(119905) = 119886

0⊙ 119875

(120572120573)

0(119905) + 119886

1⊙ 119875

(120572120573)

1(119905) + 119886

2⊙ 119875

(120572120573)

0(119905)

(74)

Here we have

119863085

= (

0 0 0

18639 03901 minus01755

minus03901 45267 08696

) (75)


119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

(76)


1198860= minus05757 119886

1= 02624 119886

2= minus00619

1198860= 12008 119886

1= 02408 119886

2= minus00600

(77)




1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11




119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873

sum119895=0

119886119895⊙ [(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 1198962(119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1198961119860119890

minus1198961119905 119875

(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895⊙ 119875

(120572120573)

119895(0) = 119889

0

(58)

Denoting

119860 = (119886119896119895)0lt119896119895lt119873

119862 = (119888119896119895)0lt119896119895lt119873

(59)


(119860 + 1198962119862) ⊙ a = f (60)



= (119863(V)119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))

119908(120572120573) (0 ⩽ 119896 ⩽ 119873 minus 1 0 ⩽ 119895 ⩽ 119873) 119886

119896119895=

119863119896minus119873

119875(120572120573)

119895(0) (119896 = 119873 0 ⩽ 119895 ⩽ 119873) and 119888

119896119895= (119875

(120572120573)

119895(119909)

119875(120572120573)

119896(119909))


elements of 119886119896119895and 119888


119886119896119895= ℎ

119896119878V (119895 119896 120572 120573) 0 ⩽ 119896 ⩽ 119873 minus 1 1 ⩽ 119895 ⩽ 119873 119886

119896119895

=(minus1)

119895minus119896+119873Γ (119895 + 120573 + 1) (119895 + 120572 + 120573 + 1)

119896minus119873

Γ (119895 minus 119896 + 119873 + 1) Γ (119896 minus 119873 + 1 + 120573)

119896 = 119873 0 ⩽ 119895 ⩽ 119873 119888119896119895

= ℎ119896 0 ⩽ 119896 = 119895 ⩽ 119873 minus 1

(61)




0+119891(119905) isin 119862(0 119887] 119896 = 0 1 119873 + 1 then

1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816 (62)

where 119891119873119894

119879and 119877




119891119873119894

119879(119905) =

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+ (63)


119877V119873(119905 0) =

119905(119873+1)V

Γ ((119873 + 1) V + 1)[119863

(119873+1)V0+

119891 (119905)]119905=120585

0 ⩽ 120585 ⩽ 119905 forall119905 isin (0 119887]

(64)

Therefore

119891 (119905) minus 119891119873119894

119879(119905) = 119877

V119873(119905 0) (65)


1003816100381610038161003816119891 (119905) minus 119910119873V (119905)1003816100381610038161003816 =

10038161003816100381610038161003816119891 (119905) minus 119910

119873V (119905) + 119891119873119894

119879(119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816

⩽10038161003816100381610038161003816119891 (119905) minus 119891

119873119894

119879(119905)10038161003816100381610038161003816+10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

=1003816100381610038161003816119877

V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119891

119873119894

119879(119905) minus 119910

119873V (119905)10038161003816100381610038161003816

(66)







119896V119906(119905) isin



[119906 (119905)]119903= [119906

119903(119905) 119906

119903(119905)] [119906

119873(119905)]

119903= [119906

119903

119873(119905) 119906

119903

119873(119905)]

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

1003816100381610038161003816119877V119873(119905 0)

1003816100381610038161003816 +10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

1003816100381610038161003816119906 (119905) minus 119906119873(119905)1003816100381610038161003816 ⩽

10038161003816100381610038161003816119877V

119873(119905 0)

10038161003816100381610038161003816+10038161003816100381610038161003816119906119873119894

119879(119905) minus 119906

119873(119905)10038161003816100381610038161003816

(67)


[119891119873119894

119879(119905)]

119903

= [119891119873119894

119879(119905 119903) 119891

119873119894

119879(119905 119903)]

= [

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+

119873

sum119894=0

119905119894V

Γ (119894V + 1)[119863

119894V0+119891 (119905)]

119905=0+]


[119877V119873(119905 0)]

119903= [119877

V119903119873(119905 0) 119877

V119903

119873(119905 0)]

= [

[

119888119863

(119873+1)V119891

119903(0+)

Γ (119873V + V + 1)119905(119873+1)V

119888119863

(119873+1)V119891

119903

(0+)

Γ (119873V + V + 1)119905(119873+1)V]

]

(68)






119896V isin

119862RF[119909


119863lowast(RF119863

VΦ (119909)

RF119863(V)Φ (119909))

le119909minusV0

(2) |Γ (1 minus V)|(119878)

2(120573 + 2

2)radic119861 (120572 + 1 120573 + 1)

(69)

5 Numerical Results


1and 119896

2by using the JT


119873]119903= [119910

119903

119873 119910

119903


tions 119910(119905 119903) = [119910(119905 119903) 119910(119905 119903)] that is [119873119890]119903= [|119910

119903

119873minus 119910

119903|

|119910119903

119873minus 119910



V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (70)

in which 119910(119909) 119871RF[0 1] cap 119862







1has an invari-


cases in details




follows119888119863

V119910 (119905) + 00231119910 (119905) = 06931119890

minus06931119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (71)



V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = minus1 + 119903 0 lt 119903 le 1

119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = 1 minus 119903 0 lt 119903 le 1

(72)


119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt 119903 le 1

(73)


1199102(119905) = 119886

0⊙ 119875

(120572120573)

0(119905) + 119886

1⊙ 119875

(120572120573)

1(119905) + 119886

2⊙ 119875

(120572120573)

0(119905)

(74)

Here we have

119863085

= (

0 0 0

18639 03901 minus01755

minus03901 45267 08696

) (75)


119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

(76)


1198860= minus05757 119886

1= 02624 119886

2= minus00619

1198860= 12008 119886

1= 02408 119886

2= minus00600

(77)




1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11




119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[119877V119873(119905 0)]

119903= [119877

V119903119873(119905 0) 119877

V119903

119873(119905 0)]

= [

[

119888119863

(119873+1)V119891

119903(0+)

Γ (119873V + V + 1)119905(119873+1)V

119888119863

(119873+1)V119891

119903

(0+)

Γ (119873V + V + 1)119905(119873+1)V]

]

(68)






119896V isin

119862RF[119909


119863lowast(RF119863

VΦ (119909)

RF119863(V)Φ (119909))

le119909minusV0

(2) |Γ (1 minus V)|(119878)

2(120573 + 2

2)radic119861 (120572 + 1 120573 + 1)

(69)

5 Numerical Results


1and 119896

2by using the JT


119873]119903= [119910

119903

119873 119910

119903


tions 119910(119905 119903) = [119910(119905 119903) 119910(119905 119903)] that is [119873119890]119903= [|119910

119903

119873minus 119910

119903|

|119910119903

119873minus 119910



V119910 (119905) + 119896

2119910 (119905) = 119896

1119860119890

minus1198961119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (70)

in which 119910(119909) 119871RF[0 1] cap 119862







1has an invari-


cases in details




follows119888119863

V119910 (119905) + 00231119910 (119905) = 06931119890

minus06931119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (71)



V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = minus1 + 119903 0 lt 119903 le 1

119888119863

V119910 (119905 119903) + 00231119910 (119905 119903)

= 06931119890minus06931119905

0 lt V le 1

119910 (0 119903) = 1 minus 119903 0 lt 119903 le 1

(72)


119884 (119905 119903) = (minus1 + 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus00231119905V]

+ int119905

0


V]

times (06931119890minus06931119909

) 119889119909 0 lt 119903 le 1

(73)


1199102(119905) = 119886

0⊙ 119875

(120572120573)

0(119905) + 119886

1⊙ 119875

(120572120573)

1(119905) + 119886

2⊙ 119875

(120572120573)

0(119905)

(74)

Here we have

119863085

= (

0 0 0

18639 03901 minus01755

minus03901 45267 08696

) (75)


119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

119910119903

2(119905) = 119886

119903

0+ 119886

119903

1(2119909 minus 1) + 119886

119903

2(6119909

2minus 6119909 + 1)

(76)


1198860= minus05757 119886

1= 02624 119886

2= minus00619

1198860= 12008 119886

1= 02408 119886

2= minus00600

(77)




1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11




119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199031198961= 06931 119896

1= 011 119896

1= 03 119896

1= 06931 119896

1= 011 119896

1= 03

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












10minus34

10minus35

10minus36

10minus37

10minus38

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


10minus5

10minus6

10minus7

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

N = 4

N = 7

N = 9

N = 11




119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903 1198860

1198861

1198862

1198860

1198861

1198862



1199031198962= 001386 119896

2= 006386 119896

2= 01386 119896

2= 001386 119896

2= 06386 119896

2= 01386

119873119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890119873

119903

119890












1

2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

k1 = 011

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

k1 = 06931

k = 03



10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10minus3

10minus4

10minus5

10minus6

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

120572 = 120573 = 0

120572 = 120573 = 05

120572 = 0 120573 = 05

120572 = 05 120573 = 0


8 1198962= 01386 and V = 095 Case II

10minus3

10minus2

10minus4

10minus5

Abso

lute

erro

r (Ne)

0 01 02 03 04 05 06 07 08 09 1r-cuts

m = 4

m = 7

m = 9

m = 11


085 1198962= 01386 and 120572 = 120573 = 0 Case II



1with the











1is also

expected




2=



119888119863

V119910 (119905) + 001386119910 (119905) = 1386119890

minus1386119905

119910 (0 119903) = [minus1 + 119903 1 minus 119903] (78)


V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt V le 1

119884 (119905 119903) = (1 minus 119903) 119864V1 [minus001386119905V]

+ int119905

0


V]

times (1386119890minus1386119909

) 119889119909 0 lt 119903 le 1





sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = minus1 + 119903

119873

sum119895=0

119886119895[(119863

(V)119875

(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

+ 001386(119875(120572120573)

119895(119909) 119875

(120572120573)

119896(119909))

119908(120572120573)

]

= (1386119860119890minus1386119905

119875(120572120573)

119896(119909))

119908(120572120573)

119896 = 0 1 119873 minus 1

119873

sum119895=0

119886119895119875

(120572120573)

119895(0) = 1 minus 119903

(80)


2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

0

minus2

Appr

oxim

ate s

olut

ion

1

050x 0

051

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 005

1

r-cuts

r-cuts

2

0

minus2

Appr

oxim

ate s

olut

ion

105

0x 0 02 04 06 08 1

2k2 = 006386

k2 = 001386

k = 01386



119863075

= (

0 0 0

26929 05524 minus01755

minus12429 42241 11048

) (81)






2at V =







2is clearance







119890which is


2

6 Conclusion









References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














References




























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article numerical solution of fuzzy fractional...

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