research article a fractional anomalous diffusion...

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 479634 8 pageshttpdxdoiorg1011552013479634

Research ArticleA Fractional Anomalous Diffusion Model and NumericalSimulation for Sodium Ion Transport in the Intestinal Wall

Bo Yu and Xiaoyun Jiang

School of Mathematics Shandong University Jinan 250100 China

Correspondence should be addressed to Xiaoyun Jiang wqjxyfsdueducn

Received 17 May 2013 Accepted 1 July 2013

Academic Editor Changpin Li

Copyright copy 2013 B Yu and X Jiang 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

The authors present a fractional anomalous diffusion model to describe the uptake of sodium ions across the epitheliumof gastrointestinal mucosa and their subsequent diffusion in the underlying blood capillaries using fractional Fickrsquos law Aheterogeneous two-phase model of the gastrointestinal mucosa is considered consisting of a continuous extracellular phase anda dispersed cellular phase The main mode of uptake is considered to be a fractional anomalous diffusion under concentrationgradient and potential gradient Appropriate partial differential equations describing the variation with time of concentrations ofsodium ions in both the two phases across the intestinal wall are obtained using Riemann-Liouville space-fractional derivative andare solved by finite differencemethodsThe concentrations of sodium ions in the interstitial space and in the cells have been studiedas a function of time and the mean concentration of sodium ions available for absorption by the blood capillaries has also beenstudied Finally numerical results are presented graphically for various values of different parameters This study demonstratesthat fractional anomalous diffusion model is appropriate for describing the uptake of sodium ions across the epithelium ofgastrointestinal mucosa

1 Introduction

The intestinal wall represents a complex system which allowsthe passage of substances either through the cells or inbetween the cells The luminal surface of the intestine iscovered with a typically leaky epithelium which enables thepassage of ions via the intercellular routeThe substance to beabsorbed either penetrates into the intercellular space directlythrough the tight junction or enters the cell cytoplasmthrough the apical plasma membrane from the lumen ofthe intestine and then penetrates through the lateral plasmamembrane to enter the intercellular space The latter routeleads to the underlying lamina propria which consists ofconnective tissue blood vessels and lymph capillaries andthus the substance enters the circulation (Figure 1) [1] Theprocess in which the ions enter the cell is passive diffusionunder concentration gradient and potential gradient This ismainly because transmural electrical potential differences of5ndash12mV have been reported from a variety of species during

recent years [2 3] Although the potential differences acrossthe intestinal wall are relatively small they cannot be ignoredin the studies of the intestinal transport of charged species [4]

Numerous techniques involving both in vivo and in vitropreparations have been employed in the study of intestinaltransport But because the cells are too small to providecontinuous sections large enough for steady-state determi-nations of their transmission properties in actual physicalsituations the distribution of the ions in the cellular andextracellular phases cannot be determined experimentallyTherefore the idea of analysing such physiological problemsusing a theoretical approach has arisen Fadali et al [5]proposed an analytical model for water absorption in theintestine based on an integration of mass balance equationfor active contact area for absorption The model gave asolution for the amount of water absorbed in the intestine asa function of time following water ingestion using data fromthe physiological literature Hills [6] proposed a two-phasemodel to study linear bulk diffusion into a continuous fluid

2 Advances in Mathematical Physics

Lumen

MucosaEpithelium

Blood capillary

Cellular phase

Intercellular phase

Serosa

x = 0

x = L1

x = L

Figure 1 Schematic diagram of the intestinal wall

in which a less permeable phase was distributed as particlesof irregular profile the overall uptake of solute by a parallel-faced section of tissue could be expressed as the sum of aninfinite number of exponential terms The advantage of thismodel was that it represented the histology of cellular tissuein the most realistic manner Karmakar and Jayaraman [1]presented a linear diffusion model of the intestinal wall todescribe the uptake of lead ions across the epithelium ofgastrointestinal mucosa and their subsequent diffusion in theunderlying blood capillariesThemodel studied the variationof concentration with time in the extracellular phase andthe cellular phase and the mean concentration available forabsorption by the blood capillaries as a function of time andit also reported the determination of membrane permeabil-ity for lead through theoretical analysis Varadharajan andJayaraman [7] presented a theoretical approach to study theuptake of sodium ions across the gastrointestinal mucosaand the concentrations at which they were taken up into theunderlying blood capillaries The model took into accountboth the diffusion under concentration gradient and poten-tial gradient and active transport which was ATPase enzymemediated and appropriate partial differential equations forthe two mechanisms of transport had been derived and weresolved by iterative methods

Recently fractional calculus has been a subject of world-wide attention due to its surprisingly broad range of applica-tions in physics chemistry engineering economics biologyand so forth [8ndash11] In particular fractional calculus is akey tool to study anomalous diffusion in transport processeswhich implies a fractional Fickrsquos law for the flux that accountsfor spatial and temporal nonlocality Many literatures haveshown that the power-law behavior is a hallmark of manybiological phenomena observed at different scales and atvarious levels of organization and that a rheological behaviorthat conforms to the power law can be described by usingmethods of fractional calculus [12ndash16] For example Maginet al [12] describe the formulation of the bioheat transfer inone dimension in terms of the fractional order differentiationwith respect to time His study demonstrates that fractionalcalculus can provide a unified approach to examine theperiodic heat transfer in peripheral tissue regions In thispaper according to Maginrsquos idea based on the previousanalysis and fractional calculus theory we consider a frac-tional anomalous diffusion model to describe the uptakeof sodium ions across the epithelium of gastrointestinalmucosa using fractional Fickrsquos law In Section 2 we presentthe fractional anomalous diffusion model and appropriate

partial differential equations describing the variation withtime of concentrations of sodium ions in both the interstitialphase and the intracellular phase across the intestinal wall areobtained using Riemann-Liouville space-fractional deriva-tive and are solved by numerical computation In Section 3numerical results are presented graphically for various valuesof different parameters In Section 4 we have presented ourconclusions

2 Materials and Methods

The fractional anomalous diffusion model considers a two-phase structure of the intestinal wall in which the epitheliumis treated as a thin layer The apical plasma membrane isadjacent to the lumen of the intestine and at the origin of aone-dimensional coordinate system The rest of the cellularelements form a uniformly distributed array of identical cellsIn between are the intercellular spaces which correspond tointerstitial phase (Figure 1)

21 Fractional Fickrsquos Law Fickrsquos law is extensively adopted asa model for standard diffusion processes For example thesimplest reaction diffusion model in spherical coordinatescan be expressed as

120597119862 (119903 119905)

120597119905= minus

1

1199032

120597 (1199032119869 (119903 119905))

120597119903+ 119891 (119903 119905) (1)

where 119862(119903 119905) is the concentration of solute (with radialsymmetry) 119891(119903 119905) represents reaction kinetics and 119869(119903 119905)

is dispersive flux Generally Fickrsquos law is used in normaldiffusion for dispersive flux based on empirical observations

119869 (119903 119905) = minus119863120597119862 (119903 119905)

120597119903 (2)

where119863 is the diffusion coefficientHowever requiring separation of scales it is not suitable

for describing nonlocal transport process In order to studythe anomalous diffusion the fractional Fickrsquos law has beenproposed [17] where the gradient of the solute concentrationin the empirical flux equation is replaced by a fractional-orderderivative

119869 (119903 119905) = minus1198631205971minus120582

1205971199051minus120582(120597120572minus1

119862 (119903 119905)

120597119903120572minus1) (3)

where 0 lt 120582 le 1 1 lt 120572 le 2 and 119863 is the anomalousdiffusion coefficient 1205971minus1205821205971199051minus120582 and 120597

120572120597119903120572 are Riemann-

Liouville operators which are defined as follows

1205971minus120582

119862 (119903 119905)

1205971199051minus120582=

1

Γ (120582)

120597

120597119905int

119905

0

119862 (119903 120591)

(119905 minus 120591)1minus120582

119889120591

120597120572119862 (119903 119905)

120597119903120572=

1

Γ (2 minus 120572)

1205972

1205971199032int

119903

0

119862 (120591 119905)

(119903 minus 120591)120572minus1

119889120591

(4)

where 0 lt 120582 le 1 1 lt 120572 le 2 We name it as the time-spacefractional Fickrsquos law [17] Some special cases of this equation

Advances in Mathematical Physics 3

are as follows when 120582 = 1 120572 = 2 it gives the classical Fickrsquoslaw when 120572 = 2 it gives time fractional Fickrsquos law when 120582 =1 it gives space fractional Fickrsquos law Here we only considerthe case of 120582 = 1 that is the space fractional Fickrsquos law

22 Fractional Anomalous Diffusion Model The diffusion ofsodium ions is complicated because its flux is determined byboth the concentration gradient and the electrical gradientConsidering the motion of sodium ions under all forcesMacey [18] proposes that the flux equation is written asfollows

119869 = minus119863(120597119862

120597119909+119885119865

119877119879

119862120597120595

120597119909) (5)

where 119863 is the diffusion coefficient 120595 is the electricalpotential 119862 is the concentration of the sodium ions 119909 is thedistance across the wall measured from the lumen 119885 is thecharge on the ion (+1 for the sodium ion) 119865 is Faradayrsquosconstant (96500Cmolminus1)119877 is the universal gas constant and119879 is the absolute temperature

Here according to Maginrsquos idea [12] based on the spacefractional Fickrsquos law the flux equation is expressed in thefollowing form

119869 = minus119863120572(120597120572minus1

119862

120597119909120572minus1+119885119865

119877119879

119862120597120595

120597119909) (6)

where 1 lt 120572 le 2 119863120572is the anomalous diffusion coefficient

The first term on the right stands for the concentrationgradient and the second term on the right stands for theelectrical gradient

We consider a two-phase model consisting of the inter-stitial phase and the intracellular phase The mass balanceequation in the interstitial phase which accounts for themolecular diffusion flux and a uniformly distributed contin-uumof point sinks whose strength is proportional to the localconcentration differences between the two phases [1] is

1205971198621015840

1

120597119905= minusnabla sdot 119869 + 119875 (119862

1015840

2minus 1198621015840

1) (7)

where11986210158401and1198621015840

2are the concentrations of sodium ions in the

interstitial phase and in the intracellular phase respectivelyand 119875 is the membrane permeability coefficient for themolecular diffusion of sodium ions into the cellular phaseSubstituting (6) into (7) we can get the following equation

1205971198621015840

1

120597119905= 119863120572[1205971205721198621015840

1

120597119909120572+119885119865

119877119879(1205971198621015840

1

120597119909

120597120595

120597119909+1205972120595

12059711990921198621015840

1)]

+ 119875 (1198621015840

2minus 1198621015840

1)

(8)

And based on the assumption that diffusion does not con-tribute significantly to the totalmolecular transport inside thecell [1] the mass balance equation in the cellular phase is

1205971198621015840

2

120597119905= 119875 (119862

1015840

1minus 1198621015840

2) (9)

which is justified by the fact that the dimensions of the cellsare small compared to the thickness of the intestinal walltherefore the flux through them is independent of distance

Meanwhile we assume that 120595 = 1198601015840119909 where 119860

1015840 is aconstant to be determined A justification for the constantfield assumption can be found in the observation that if amembrane contains a large number of dipolar ions close totheir isoelectric point these dipoles will tend to alter theirorientation in such a way that they tend to smooth out anyirregularities and maintain a constant field [7] The validityof this assumption has also been discussed by Goldman [19]and Cole [20] Hence (8) can be reduced to

1205971198621015840

1

120597119905= 119863120572(1205971205721198621015840

1

120597119909120572+119885119865

119877119879

1205971198621015840

1

1205971199091198601015840) + 119875 (119862

1015840

2minus 1198621015840

1) (10)

that is

1205971198621015840

1

120597119905= 119863120572(1205971205721198621015840

1

120597119909120572+ 119860

1205971198621015840

1

120597119909) + 119875 (119862

1015840

2minus 1198621015840

1) (11)

where 119860 = 1198601015840(119885119865119877119879)

The value 119909 = 0 corresponds to the lumen of the intestineand 119909 = 119871 corresponds to serosa We are interested in findingthe ion concentration at 119909 = 119871

1 which corresponds to the

blood capillary at which it is absorbed (Figure 1) In ratsthe mucosal epithelium is approximately 014 of the totalintestinal wall thickness Equations (9) and (11) are solvedto obtain 119862

1015840

1and 119862

1015840

2as functions of 119909 and 119905 and the mean

concentration of sodium ions at 119909 = 1198711is calculated from

119872119862 =12057411198621(1198711) + 12057421198622(1198711)

1205741+ 1205742

(12)

where 1205741and 1205742are the interstitial and intracellular volume

fractions respectivelyThen we introduce dimensionless parameters

119905lowast=119905119863120572

1198712 119909

lowast=119909

119871 120573 =

1198751198712

119863120572

1198621=1198621015840

1

1198621015840119871

1198622=1198621015840

2

1198621015840119871

119871lowast

1=1198711

119871

(13)

to reduce (11) and (9) to the nondimensional form (with thelowast notation dropped for convenience)

1205971198621

120597119905=

1

119871120572minus2

1205971205721198621

120597119909120572+ 119860

1205971198621

120597119909+ 120573 (119862

2minus 1198621)

1205971198622

120597119905= 120573 (119862

1minus 1198622)

(14)

where 1198621015840119871is the concentration of sodium ions in the lumen

The parameter 120573 = 1198751198712119863 could be considered as the ratio

of the membrane diffusion flux into the cellular phase to themolecular diffusion flux in the interstitial phase

Based on the assumption that the concentration ofsodium ions in the intestinal lumina surface is equal to the


concentration at its abluminal surface for the epithelial isspecially thin the boundary conditions are given by

1198621(0 119905) = 1 119862

1(1 119905) = 0 (15)

which mean that the concentration of sodium ions in thelumen is set to 1 whereas at the serosa it is set to 0 at all timesFurther the initial concentration is taken to be 0 a conditionjustified in the case of in vitro experiments Both 119862

1and 119862

2

are set to 0 at any point within the tissue when time equals 0Therefore the initial conditions are given by

1198621(119909 0) = 0 119862

2(119909 0) = 0 (16)

23 Numerical Computation For the numerical solution ofthe problem above we introduce a uniform grid of meshpoints (119909

119895 119905119896) with 119909

119895= 119895ℎ 119895 = 0 1 119873 and 119905

119896=

119896120591 119896 = 0 1 119872 where119872 and119873 are two positive integersℎ = 1119873 and 120591 = 119879119872 are the uniform spatial andtemporal mesh size respectively The theoretical solution 119862

1

at the point (119909119895 119905119896) is denoted by 119862

1(119909119895 119905119896) the solution of

an approximating difference scheme at the point (119909119895 119905119896) will

be denoted by 1198621198961119895 Similarly the theoretical solution 119862

2at

the point (119909119895 119905119896) is denoted by 119862

2(119909119895 119905119896) the solution of an

approximating difference scheme at the point (119909119895 119905119896) will be

denoted by 1198621198962119895

Then we start to introduce the discretization of thedifferential operatorsThe first-order derivatives with respectto the temporal variable 120597119862

1120597119905 and 120597119862

2120597119905 are approximated

by the following Euler backward difference respectively

1205971198621(119909119895 119905119896)

120597119905asymp1198621(119909119895 119905119896) minus 1198621(119909119895 119905119896minus1

)

120591

1205971198622(119909119895 119905119896)


)

120591

(17)

and the first-order derivative with respect to the spatialvariable 120597119862

1120597119909 is approximated by Euler forward difference

1205971198621(119909119895 119905119896minus1

)

120597119909asymp1198621(119909119895+1 119905119896minus1

) minus 1198621(119909119895 119905119896minus1

)

ℎ (18)

As for the Riemann-Liouville fractional derivative usingthe relationship between the Grunwald-Letnikov formulaand Riemann-Liouville fractional derivative we can approx-imate the fractional derivative by [21 22]

1205971205721198621(119909119895 119905119896minus1

)

120597119909120572asymp ℎminus120572

119895+1

sum

119897=0

120596(120572)

1198971198621(119909119895minus (119897 minus 1) ℎ 119905

119896minus1) (19)

where 120596(120572)0

= 1 120596(120572)119896

= (minus1)119896(120572(120572 minus 1) sdot sdot sdot (120572 minus 119896 + 1)119896) for

119896 ge 1There we have adopted the shiftedGrunwald-Letnikovformula for 1 lt 120572 le 2

Finally the finite difference method for the above prob-lem is given as follows

119862119896

1119895minus 119862119896minus1

1119895

120591=

ℎminus120572

119871120572minus2

119895+1

sum

119897=0

120596(120572)

119897119862119896minus1

1119895minus119897+1

+ 119860119862119896minus1

1119895+1minus 119862119896minus1

1119895

ℎ+ 120573 (119862

119896minus1

2119895minus 119862119896minus1

1119895)

119896 = 1 2 119872 119895 = 1 2 119873 minus 1

119862119896

2119895minus 119862119896minus1

2119895

120591= 120573 (119862

119896minus1

1119895minus 119862119896minus1

2119895)

119896 = 1 2 119872 119895 = 0 1 2 119873 minus 1119873

(20)

The boundary and initial conditions can be discretized by

119862119896

10= 1 119862

119896

1119873= 0 119896 = 0 1 119872

1198620

1119895= 0 119895 = 1 2 119873

1198620

2119895= 0 119895 = 0 1 2 119873

(21)

The concentrations of the sodium ions in the intercellularphase and intracellular phase are determined at different stepsof time and space and their weighted mean concentration atthe blood capillaries can also be obtained

3 Results and Discussion

The thickness of the intestinal wall 119871 is taken to be 214 times10minus4m [1] which is measured from a cross-section of the rat

intestinal wall using an ocular micrometer fitted to a simplemicroscopeMuller [23] reported inmorphometric studies ofrat gastricmucosa that the epithelial cells occupied 74whilethe remaining 26was occupied by lamina propria Hence achoice of 026 ismade for 120574

1and 074 for 120574

2 arbitrarily as their

values are not available in the literatures According to Xu andZhao [24] the permeability coefficient 119875 of sodium ions istaken to be 161 times 10minus5 sminus1 Varadharajan and Jayaraman [7]had studied that the numerical value of 119860 depended on thepotential difference between the serosa and the mucosa andcomparing with the experimental results of Lauterbach [25]Varadharajan obtained the value of119860 to be 0 le 119860 le 1 and theoptimum value of 119860 to be around 04 so 119860 is taken to be 04in our studies

31 The Anomalous Diffusion Coefficient 119863120572of Sodium Ions

According to the Stokes-Einstein formula 119863 = 1198961198796120587120583119903where 119896 is Boltzmannrsquos constant (14 times 10

minus23 JKminus1) 119879 isthe temperature (sim310K) 120583 is the viscosity of intercellularfluid (sim0001 Pas) and 119903 is the radius of the water molecule(sim045 nm) we can obtain the diffusivity in water119863 = 512 times

10minus10m2 sminus1 [1] Nevertheless it can also be considered as a

reasonable approximation for the diffusion coefficient 119863120572of

sodium ions Here we change this parameter to 03 07 113 and 17 times of the value of diffusivity119863


0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04

03 times of D07 times of D1 times of D

13 times of D17 times of D

C1

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x



120572 = 16 A = 04

(b)

Figure 2 (a) is 1198621 plotted against 119909 for different values of 119863

120572when 120572 = 16 (b) is 119872119862 plotted against 119909 for different values of 119863

120572when

120572 = 16

Figure 2 individual graph demonstrates the variation ofconcentration with respect to 119909 for different values of 119863

120572

when 120572 = 16 During the course of diffusion we findthat the concentrations decrease in an exponential way andthe curves tend to change very little when the values of 119863

120572

vary not too much which indicates that the distribution ofsodium ions is more uniform therefore we can consider anyof these values as a reasonable approximation for the diffusioncoefficient 119863

120572of sodium ions By amplifying the figures

we also observe that smaller 119863120572increases the amplitude of

119872119862 which indicates that larger 119863120572increases the speed of

the absorption especially at the blood capillaries a possibleexplanation is that faster movement of sodium ions makes iteasier for diffusion

32 The Order 120572 of Fractional Derivative Here based on theabove analysis we take 119863

120572to be 038 times 10

minus5 cm2 sminus1 whichis about 07 times of the diffusivity 119863 just as used in theliterature [18] Substituting the respective values of 119863

120572and

119871 we can obtain that the actual time is about 2mintimes 119905Figure 3 demonstrates that the variation of concentration

with respect to 119909 at 119905 = 20min for different values of 120572Obviously we can observe that the concentrations decrease inan exponential way and the curves become smoother as thevalue of 120572 becomes larger which indicate that the diffusionin the intercellular phase and the absorption especially at theblood capillaries become quicker as the value of 120572 becomessmaller This fact demonstrates that the diffusion of sodiumions is anomalous superdiffusion

Figure 4 demonstrates that the variation of concentra-tion with respect to 119909 at different times when 120572 = 16119860 = 04 During the course of diffusion we find that theconcentrations decrease in an exponential way By amplifyingthe figures we obviously observe that the decay tends to besmoother and smootherwhen time increases which indicatesthe distribution of sodium ions is more uniform Figure 4(a)shows that at earlier times most of the sodium ions areabsorbed by the cells and for later times they tend to passtowards the serosa Meanwhile we observe that most ofthe obsorption takes place at 119909 lt 02 from Figure 4(b)which can be explained as the distance at which the bloodcapillaries lie This is quite reasonable since in rats themucosal epithelium is about 014 of the total wall thickness[1] All these phenomena are connective with the resultsof Varadharajan and Jayaraman [7] which indicate that theanomalous diffusion is appropriate for describing the uptakeof sodium ions

Figure 5(a) is 119872119862 plotted against 119905 at different 119909 when120572 = 19 119860 = 04 We find that at a particular distance119872119862 increases with time but at a farther distance the fluxis lower because the major absorption is made available atthe distance where the blood capillaries lie Figure 5(b) is119872Cplotted against 119905 at 119909 = 015 for different values of 120572 We findthat at a particular distance 119909 = 015 119872119862 is lower when 120572is smaller which indicates that the diffusion is quicker whenthe value of 120572 is smaller and this leads to the lower value of119872119862

Figure 6(a) is1198621 plotted against 119905 at 119909 = 015 for different

values of 120572 We find that at a particular distance 119909 = 015


0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

C1

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(b)

Figure 3 (a) is 1198621 plotted against 119909 at 119905 = 20min for different values of 120572 (b) is119872119862 plotted against 119909 at 119905 = 20min for different values of 120572

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04

C1

t = 2mint = 20min

t = 40mint = 76min

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 16 A = 04

t = 2mint = 20min

t = 40mint = 76min

(b)

Figure 4 (a) is 1198621 plotted against 119909 at different times when 120572 = 16 119860 = 04 (b) is119872119862 plotted against 119909 at different times when 120572 = 16

119860 = 04

1198621is lower when 120572 is smaller which indicates that the

diffusion is quicker when the value of 120572 is smaller and thisleads to the lower value of119862

1 Figure 6(b) is119862

1plotted against

119905 at different 119909 when 120572 = 196 119860 = 04 We find thatabout 23 sodium ion absorption is achieved at a distance

of 01ndash015 when we choose 120572 = 196 and it is in goodagreement with the experimental results of Lauterbach [25]which indicate that the concentration of Na in the cell waterapproaches 23 of the initial concentration of the incubationmedium after the addition to the luminal side


0 05 1 15 2 25 3 35 40

002

004

006

008

01

012

014

016

018

02

MC

x = 01

x = 015

x = 05

t

120572 = 19 A = 04

(a) (b)

Figure 5 (a) is119872119862 plotted against 119905 at different 119909 when 120572 = 19 119860 = 04 (b) is119872119862 plotted against 119905 at 119909 = 015 for different values of 120572

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

C1

t

x = 015

120572 = 19

120572 = 17

120572 = 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

C1

t

x = 01

x = 015

x = 05

120572 = 196

(b)

Figure 6 (a) is 1198621 plotted against 119905 at 119909 = 015 for different values of 120572 (b) is 119862

1 plotted against 119905 at different 119909 when 120572 = 196 119860 = 04

4 Conclusions

In summary in this paper we have derived a fractionalanomalous diffusion model for sodium ion transport in theintestinal wall using space fractional Fickrsquos law Appropriatepartial differential equations describing the variation withtime of concentrations of sodium ions in both the intersti-tial phase and the intracellular phase across the intestinalwall are obtained using Riemann-Liouville space-fractionalderivatives and are solved by finite difference methods Thenumerical simulations have been discussed and numeri-cal results are presented graphically for various values of

different parameters It demonstrates that fractional anoma-lous diffusion model is appropriate for describing the uptakeof sodium ions across the epithelium of gastrointestinalmucosa This research also provides some new points forstudying ions transferring processes in biological systems

Acknowledgment

The project was supported by the National Natural ScienceFoundation of China (Grants nos 11072134 11102102 and91130017)


References

[1] N Karmakar and G Jayaraman ldquoLinear diffusion of leadin the intestinal wall A Theoretical Studyrdquo IMA Journal ofMathematics Applied in Medicine and Biology vol 5 no 1 pp33ndash43 1988

[2] WM Armstrong ldquoCellular mechanisms of ion transport in thesmall intestinerdquo Physiology of the Gastrointestinal Tract 2 pp1251ndash1266 1987

[3] S G Schultz ldquoPrinciples of electrophysiology and their appli-cation to epithelial tissues gastrointestinal physiologyrdquo in Gas-trointestinal Physiology pp 87ndash91 Butterworth London UK1974

[4] S G Schultz and P F Curran Handbook of Physiology vol 3Section 6 1968

[5] M S Fadali J W Steadman and R G Jacquot ldquoA modelof water absorption in human intestinerdquo Biomedical SciencesInstrumentation vol 16 pp 49ndash53 1980

[6] B A Hills ldquoLinear bulk diffusion into heterogeneous tissuerdquoTheBulletin of Mathematical Biophysics vol 30 no 1 pp 47ndash591968

[7] M Varadharajan and G Jayaraman ldquoSodium ion transportin the intestinal wall a mathematical modelrdquo IMA Journal ofMathematical Medicine and Biology vol 11 no 3 pp 193ndash2051994

[8] D Baleanu A K Golmankhaneh R Nigmatullin and AK Golmankhaneh ldquoFractional Newtonian mechanicsrdquo CentralEuropean Journal of Physics vol 8 no 1 pp 120ndash125 2010

[9] J Singh P K Gupta and K N Rai ldquoSolution of fractionalbioheat equations by finite difference method and HPMrdquoMathematical and Computer Modelling vol 54 no 9-10 pp2316ndash2325 2011

[10] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000

[11] R Metzler and J Klafter ldquoThe restaurant at the end of the ran-domwalk recent developments in the description of anomaloustransport by fractional dynamicsrdquo Journal of Physics A vol 37no 31 pp R161ndashR208 2004

[12] R L Magin S C Boregowda and C Deodhar ldquoModeling ofpulsating peripheral bio-Heat transferusing fractional calculusand constructal theoryrdquo International Journal of Design Natureand Ecodynamics vol 1 no 18 pp 18ndash33 2007

[13] X Y Jiang and H T Qi ldquoThermal wave model of bioheattransfer with modified Riemann-Liouville fractional deriva-tiverdquo Journal of Physics vol 45 no 48 10 pages 2012

[14] W C Tan C Fu C Fu W Xie and H Cheng ldquoAn anomaloussubdiffusion model for calcium spark in cardiac myocytesrdquoApplied Physics Letters vol 91 no 18 3 pages 2007

[15] R G LarsonThe Structure and Rheology of Complex Fluids vol2 Oxford University Press New York NY USA 1999

[16] R L Magin ldquoFractional calculus models of complex dynamicsin biological tissuesrdquo Computers amp Mathematics with Applica-tions vol 59 no 5 pp 1586ndash1593 2010

[17] X Y Jiang M Y Xu and H T Qi ldquoThe fractional diffusionmodel with an absorption term and modified Fickrsquos law fornon-local transport processesrdquo Nonlinear Analysis Real WorldApplications vol 11 no 1 pp 262ndash269 2010

[18] R I Macey Membrane Physiology Plenum New York NYUSA 2nd edition 1980

[19] D E Goldman ldquoPotential impedence and rectification inmembranesrdquo The Journal of General Physiology vol 27 no 1pp 37ndash60 1943

[20] K S Cole ldquoElectrodiffusion models for the membrane of squidgiant axonrdquo Physiological Reviews vol 45 pp 340ndash379 1965

[21] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004

[22] C Li and F Zeng ldquoFinite difference methods for fractionaldifferential equationsrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 22 no 4 2012

[23] O Muller ldquoThe quantitation of the rat gastric mucosa by mor-phometric methodsrdquo Scandinavian Journal of GastroenterologySupplement vol 101 no 1 pp 1ndash6 1984

[24] M Y Xu and G L Zhao ldquoNonisotonic reabsorption of waterand steady state distribution of sodiumion in renal descendinglimb of Henlerdquo Acta Mechanica Sinica vol 1 no 3 1986

[25] F O Lauterbach Intestinal Transport University Park Press1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


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


Lumen

MucosaEpithelium

Blood capillary

Cellular phase

Intercellular phase

Serosa

x = 0

x = L1

x = L

Figure 1 Schematic diagram of the intestinal wall

in which a less permeable phase was distributed as particlesof irregular profile the overall uptake of solute by a parallel-faced section of tissue could be expressed as the sum of aninfinite number of exponential terms The advantage of thismodel was that it represented the histology of cellular tissuein the most realistic manner Karmakar and Jayaraman [1]presented a linear diffusion model of the intestinal wall todescribe the uptake of lead ions across the epithelium ofgastrointestinal mucosa and their subsequent diffusion in theunderlying blood capillariesThemodel studied the variationof concentration with time in the extracellular phase andthe cellular phase and the mean concentration available forabsorption by the blood capillaries as a function of time andit also reported the determination of membrane permeabil-ity for lead through theoretical analysis Varadharajan andJayaraman [7] presented a theoretical approach to study theuptake of sodium ions across the gastrointestinal mucosaand the concentrations at which they were taken up into theunderlying blood capillaries The model took into accountboth the diffusion under concentration gradient and poten-tial gradient and active transport which was ATPase enzymemediated and appropriate partial differential equations forthe two mechanisms of transport had been derived and weresolved by iterative methods

Recently fractional calculus has been a subject of world-wide attention due to its surprisingly broad range of applica-tions in physics chemistry engineering economics biologyand so forth [8ndash11] In particular fractional calculus is akey tool to study anomalous diffusion in transport processeswhich implies a fractional Fickrsquos law for the flux that accountsfor spatial and temporal nonlocality Many literatures haveshown that the power-law behavior is a hallmark of manybiological phenomena observed at different scales and atvarious levels of organization and that a rheological behaviorthat conforms to the power law can be described by usingmethods of fractional calculus [12ndash16] For example Maginet al [12] describe the formulation of the bioheat transfer inone dimension in terms of the fractional order differentiationwith respect to time His study demonstrates that fractionalcalculus can provide a unified approach to examine theperiodic heat transfer in peripheral tissue regions In thispaper according to Maginrsquos idea based on the previousanalysis and fractional calculus theory we consider a frac-tional anomalous diffusion model to describe the uptakeof sodium ions across the epithelium of gastrointestinalmucosa using fractional Fickrsquos law In Section 2 we presentthe fractional anomalous diffusion model and appropriate

partial differential equations describing the variation withtime of concentrations of sodium ions in both the interstitialphase and the intracellular phase across the intestinal wall areobtained using Riemann-Liouville space-fractional deriva-tive and are solved by numerical computation In Section 3numerical results are presented graphically for various valuesof different parameters In Section 4 we have presented ourconclusions

2 Materials and Methods

The fractional anomalous diffusion model considers a two-phase structure of the intestinal wall in which the epitheliumis treated as a thin layer The apical plasma membrane isadjacent to the lumen of the intestine and at the origin of aone-dimensional coordinate system The rest of the cellularelements form a uniformly distributed array of identical cellsIn between are the intercellular spaces which correspond tointerstitial phase (Figure 1)

21 Fractional Fickrsquos Law Fickrsquos law is extensively adopted asa model for standard diffusion processes For example thesimplest reaction diffusion model in spherical coordinatescan be expressed as

120597119862 (119903 119905)

120597119905= minus

1

1199032

120597 (1199032119869 (119903 119905))

120597119903+ 119891 (119903 119905) (1)

where 119862(119903 119905) is the concentration of solute (with radialsymmetry) 119891(119903 119905) represents reaction kinetics and 119869(119903 119905)

is dispersive flux Generally Fickrsquos law is used in normaldiffusion for dispersive flux based on empirical observations

119869 (119903 119905) = minus119863120597119862 (119903 119905)

120597119903 (2)

where119863 is the diffusion coefficientHowever requiring separation of scales it is not suitable

for describing nonlocal transport process In order to studythe anomalous diffusion the fractional Fickrsquos law has beenproposed [17] where the gradient of the solute concentrationin the empirical flux equation is replaced by a fractional-orderderivative

119869 (119903 119905) = minus1198631205971minus120582

1205971199051minus120582(120597120572minus1

119862 (119903 119905)

120597119903120572minus1) (3)

where 0 lt 120582 le 1 1 lt 120572 le 2 and 119863 is the anomalousdiffusion coefficient 1205971minus1205821205971199051minus120582 and 120597

120572120597119903120572 are Riemann-

Liouville operators which are defined as follows

1205971minus120582

119862 (119903 119905)

1205971199051minus120582=

1

Γ (120582)

120597

120597119905int

119905

0

119862 (119903 120591)

(119905 minus 120591)1minus120582

119889120591

120597120572119862 (119903 119905)

120597119903120572=

1

Γ (2 minus 120572)

1205972

1205971199032int

119903

0

119862 (120591 119905)

(119903 minus 120591)120572minus1

119889120591

(4)

where 0 lt 120582 le 1 1 lt 120572 le 2 We name it as the time-spacefractional Fickrsquos law [17] Some special cases of this equation




119869 = minus119863(120597119862

120597119909+119885119865

119877119879

119862120597120595

120597119909) (5)



119869 = minus119863120572(120597120572minus1

119862

120597119909120572minus1+119885119865

119877119879

119862120597120595

120597119909) (6)




1205971198621015840

1

120597119905= minusnabla sdot 119869 + 119875 (119862

1015840

2minus 1198621015840

1) (7)

where11986210158401and1198621015840



1205971198621015840

1

120597119905= 119863120572[1205971205721198621015840

1

120597119909120572+119885119865

119877119879(1205971198621015840

1

120597119909

120597120595

120597119909+1205972120595

12059711990921198621015840

1)]

+ 119875 (1198621015840

2minus 1198621015840

1)

(8)


1205971198621015840

2

120597119905= 119875 (119862

1015840

1minus 1198621015840

2) (9)




1205971198621015840

1

120597119905= 119863120572(1205971205721198621015840

1

120597119909120572+119885119865

119877119879

1205971198621015840

1

1205971199091198601015840) + 119875 (119862

1015840

2minus 1198621015840

1) (10)

that is

1205971198621015840

1

120597119905= 119863120572(1205971205721198621015840

1

120597119909120572+ 119860

1205971198621015840

1

120597119909) + 119875 (119862

1015840

2minus 1198621015840

1) (11)

where 119860 = 1198601015840(119885119865119877119879)




1015840

1and 119862

1015840



119872119862 =12057411198621(1198711) + 12057421198622(1198711)

1205741+ 1205742

(12)



119905lowast=119905119863120572

1198712 119909

lowast=119909

119871 120573 =

1198751198712

119863120572

1198621=1198621015840

1

1198621015840119871

1198622=1198621015840

2

1198621015840119871

119871lowast

1=1198711

119871

(13)


1205971198621

120597119905=

1

119871120572minus2

1205971205721198621

120597119909120572+ 119860

1205971198621

120597119909+ 120573 (119862

2minus 1198621)

1205971198622

120597119905= 120573 (119862

1minus 1198622)

(14)







1198621(0 119905) = 1 119862

1(1 119905) = 0 (15)


1and 119862

2


1198621(119909 0) = 0 119862

2(119909 0) = 0 (16)


119895 119905119896) with 119909

119895= 119895ℎ 119895 = 0 1 119873 and 119905

119896=


1


1(119909119895 119905119896) the solution of



2at




denoted by 1198621198962119895


1120597119905 and 120597119862



1205971198621(119909119895 119905119896)


)

120591

1205971198622(119909119895 119905119896)


)

120591

(17)



1205971198621(119909119895 119905119896minus1

)

120597119909asymp1198621(119909119895+1 119905119896minus1

) minus 1198621(119909119895 119905119896minus1

)

ℎ (18)


1205971205721198621(119909119895 119905119896minus1

)

120597119909120572asymp ℎminus120572

119895+1

sum

119897=0

120596(120572)

1198971198621(119909119895minus (119897 minus 1) ℎ 119905

119896minus1) (19)

where 120596(120572)0

= 1 120596(120572)119896




119862119896

1119895minus 119862119896minus1

1119895

120591=

ℎminus120572

119871120572minus2

119895+1

sum

119897=0

120596(120572)

119897119862119896minus1

1119895minus119897+1

+ 119860119862119896minus1

1119895+1minus 119862119896minus1

1119895

ℎ+ 120573 (119862

119896minus1

2119895minus 119862119896minus1

1119895)

119896 = 1 2 119872 119895 = 1 2 119873 minus 1

119862119896

2119895minus 119862119896minus1

2119895

120591= 120573 (119862

119896minus1

1119895minus 119862119896minus1

2119895)

119896 = 1 2 119872 119895 = 0 1 2 119873 minus 1119873

(20)


119862119896

10= 1 119862

119896

1119873= 0 119896 = 0 1 119872

1198620

1119895= 0 119895 = 1 2 119873

1198620

2119895= 0 119895 = 0 1 2 119873

(21)





1and 074 for 120574










0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04



C1

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x



120572 = 16 A = 04

(b)



120572when

120572 = 16


120572


120572







120572to be 038 times 10


120572and








0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

C1

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(b)


0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04

C1

t = 2mint = 20min

t = 40mint = 76min

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 16 A = 04

t = 2mint = 20min

t = 40mint = 76min

(b)


119860 = 04




1plotted against




0 05 1 15 2 25 3 35 40

002

004

006

008

01

012

014

016

018

02

MC

x = 01

x = 015

x = 05

t

120572 = 19 A = 04

(a) (b)


0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

C1

t

x = 015

120572 = 19

120572 = 17

120572 = 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

C1

t

x = 01

x = 015

x = 05

120572 = 196

(b)



4 Conclusions



Acknowledgment



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119869 = minus119863(120597119862

120597119909+119885119865

119877119879

119862120597120595

120597119909) (5)



119869 = minus119863120572(120597120572minus1

119862

120597119909120572minus1+119885119865

119877119879

119862120597120595

120597119909) (6)




1205971198621015840

1

120597119905= minusnabla sdot 119869 + 119875 (119862

1015840

2minus 1198621015840

1) (7)

where11986210158401and1198621015840



1205971198621015840

1

120597119905= 119863120572[1205971205721198621015840

1

120597119909120572+119885119865

119877119879(1205971198621015840

1

120597119909

120597120595

120597119909+1205972120595

12059711990921198621015840

1)]

+ 119875 (1198621015840

2minus 1198621015840

1)

(8)


1205971198621015840

2

120597119905= 119875 (119862

1015840

1minus 1198621015840

2) (9)




1205971198621015840

1

120597119905= 119863120572(1205971205721198621015840

1

120597119909120572+119885119865

119877119879

1205971198621015840

1

1205971199091198601015840) + 119875 (119862

1015840

2minus 1198621015840

1) (10)

that is

1205971198621015840

1

120597119905= 119863120572(1205971205721198621015840

1

120597119909120572+ 119860

1205971198621015840

1

120597119909) + 119875 (119862

1015840

2minus 1198621015840

1) (11)

where 119860 = 1198601015840(119885119865119877119879)




1015840

1and 119862

1015840



119872119862 =12057411198621(1198711) + 12057421198622(1198711)

1205741+ 1205742

(12)



119905lowast=119905119863120572

1198712 119909

lowast=119909

119871 120573 =

1198751198712

119863120572

1198621=1198621015840

1

1198621015840119871

1198622=1198621015840

2

1198621015840119871

119871lowast

1=1198711

119871

(13)


1205971198621

120597119905=

1

119871120572minus2

1205971205721198621

120597119909120572+ 119860

1205971198621

120597119909+ 120573 (119862

2minus 1198621)

1205971198622

120597119905= 120573 (119862

1minus 1198622)

(14)







1198621(0 119905) = 1 119862

1(1 119905) = 0 (15)


1and 119862

2


1198621(119909 0) = 0 119862

2(119909 0) = 0 (16)


119895 119905119896) with 119909

119895= 119895ℎ 119895 = 0 1 119873 and 119905

119896=


1


1(119909119895 119905119896) the solution of



2at




denoted by 1198621198962119895


1120597119905 and 120597119862



1205971198621(119909119895 119905119896)


)

120591

1205971198622(119909119895 119905119896)


)

120591

(17)



1205971198621(119909119895 119905119896minus1

)

120597119909asymp1198621(119909119895+1 119905119896minus1

) minus 1198621(119909119895 119905119896minus1

)

ℎ (18)


1205971205721198621(119909119895 119905119896minus1

)

120597119909120572asymp ℎminus120572

119895+1

sum

119897=0

120596(120572)

1198971198621(119909119895minus (119897 minus 1) ℎ 119905

119896minus1) (19)

where 120596(120572)0

= 1 120596(120572)119896




119862119896

1119895minus 119862119896minus1

1119895

120591=

ℎminus120572

119871120572minus2

119895+1

sum

119897=0

120596(120572)

119897119862119896minus1

1119895minus119897+1

+ 119860119862119896minus1

1119895+1minus 119862119896minus1

1119895

ℎ+ 120573 (119862

119896minus1

2119895minus 119862119896minus1

1119895)

119896 = 1 2 119872 119895 = 1 2 119873 minus 1

119862119896

2119895minus 119862119896minus1

2119895

120591= 120573 (119862

119896minus1

1119895minus 119862119896minus1

2119895)

119896 = 1 2 119872 119895 = 0 1 2 119873 minus 1119873

(20)


119862119896

10= 1 119862

119896

1119873= 0 119896 = 0 1 119872

1198620

1119895= 0 119895 = 1 2 119873

1198620

2119895= 0 119895 = 0 1 2 119873

(21)





1and 074 for 120574










0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04



C1

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x



120572 = 16 A = 04

(b)



120572when

120572 = 16


120572


120572







120572to be 038 times 10


120572and








0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

C1

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(b)


0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04

C1

t = 2mint = 20min

t = 40mint = 76min

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 16 A = 04

t = 2mint = 20min

t = 40mint = 76min

(b)


119860 = 04




1plotted against




0 05 1 15 2 25 3 35 40

002

004

006

008

01

012

014

016

018

02

MC

x = 01

x = 015

x = 05

t

120572 = 19 A = 04

(a) (b)


0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

C1

t

x = 015

120572 = 19

120572 = 17

120572 = 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

C1

t

x = 01

x = 015

x = 05

120572 = 196

(b)



4 Conclusions



Acknowledgment



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198621(0 119905) = 1 119862

1(1 119905) = 0 (15)


1and 119862

2


1198621(119909 0) = 0 119862

2(119909 0) = 0 (16)


119895 119905119896) with 119909

119895= 119895ℎ 119895 = 0 1 119873 and 119905

119896=


1


1(119909119895 119905119896) the solution of



2at




denoted by 1198621198962119895


1120597119905 and 120597119862



1205971198621(119909119895 119905119896)


)

120591

1205971198622(119909119895 119905119896)


)

120591

(17)



1205971198621(119909119895 119905119896minus1

)

120597119909asymp1198621(119909119895+1 119905119896minus1

) minus 1198621(119909119895 119905119896minus1

)

ℎ (18)


1205971205721198621(119909119895 119905119896minus1

)

120597119909120572asymp ℎminus120572

119895+1

sum

119897=0

120596(120572)

1198971198621(119909119895minus (119897 minus 1) ℎ 119905

119896minus1) (19)

where 120596(120572)0

= 1 120596(120572)119896




119862119896

1119895minus 119862119896minus1

1119895

120591=

ℎminus120572

119871120572minus2

119895+1

sum

119897=0

120596(120572)

119897119862119896minus1

1119895minus119897+1

+ 119860119862119896minus1

1119895+1minus 119862119896minus1

1119895

ℎ+ 120573 (119862

119896minus1

2119895minus 119862119896minus1

1119895)

119896 = 1 2 119872 119895 = 1 2 119873 minus 1

119862119896

2119895minus 119862119896minus1

2119895

120591= 120573 (119862

119896minus1

1119895minus 119862119896minus1

2119895)

119896 = 1 2 119872 119895 = 0 1 2 119873 minus 1119873

(20)


119862119896

10= 1 119862

119896

1119873= 0 119896 = 0 1 119872

1198620

1119895= 0 119895 = 1 2 119873

1198620

2119895= 0 119895 = 0 1 2 119873

(21)





1and 074 for 120574










0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04



C1

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x



120572 = 16 A = 04

(b)



120572when

120572 = 16


120572


120572







120572to be 038 times 10


120572and








0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

C1

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(b)


0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04

C1

t = 2mint = 20min

t = 40mint = 76min

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 16 A = 04

t = 2mint = 20min

t = 40mint = 76min

(b)


119860 = 04




1plotted against




0 05 1 15 2 25 3 35 40

002

004

006

008

01

012

014

016

018

02

MC

x = 01

x = 015

x = 05

t

120572 = 19 A = 04

(a) (b)


0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

C1

t

x = 015

120572 = 19

120572 = 17

120572 = 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

C1

t

x = 01

x = 015

x = 05

120572 = 196

(b)



4 Conclusions



Acknowledgment



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04



C1

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x



120572 = 16 A = 04

(b)



120572when

120572 = 16


120572


120572







120572to be 038 times 10


120572and








0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

C1

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(b)


0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04

C1

t = 2mint = 20min

t = 40mint = 76min

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 16 A = 04

t = 2mint = 20min

t = 40mint = 76min

(b)


119860 = 04




1plotted against




0 05 1 15 2 25 3 35 40

002

004

006

008

01

012

014

016

018

02

MC

x = 01

x = 015

x = 05

t

120572 = 19 A = 04

(a) (b)


0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

C1

t

x = 015

120572 = 19

120572 = 17

120572 = 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

C1

t

x = 01

x = 015

x = 05

120572 = 196

(b)



4 Conclusions



Acknowledgment



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

C1

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 13

120572 = 15

120572 = 17

t = 20min A = 04

(b)


0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

x

120572 = 16 A = 04

C1

t = 2mint = 20min

t = 40mint = 76min

(a)

0

005

01

015

02

025

03

035

MC

0 02 04 06 08 1

x

120572 = 16 A = 04

t = 2mint = 20min

t = 40mint = 76min

(b)


119860 = 04




1plotted against




0 05 1 15 2 25 3 35 40

002

004

006

008

01

012

014

016

018

02

MC

x = 01

x = 015

x = 05

t

120572 = 19 A = 04

(a) (b)


0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

C1

t

x = 015

120572 = 19

120572 = 17

120572 = 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

C1

t

x = 01

x = 015

x = 05

120572 = 196

(b)



4 Conclusions



Acknowledgment



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 25 3 35 40

002

004

006

008

01

012

014

016

018

02

MC

x = 01

x = 015

x = 05

t

120572 = 19 A = 04

(a) (b)


0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

C1

t

x = 015

120572 = 19

120572 = 17

120572 = 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

C1

t

x = 01

x = 015

x = 05

120572 = 196

(b)



4 Conclusions



Acknowledgment



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 a fractional anomalous diffusion...

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