review article computational challenge of fractional...

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Review ArticleComputational Challenge of Fractional Differential Equationsand the Potential Solutions A Survey

Chunye Gong123 Weimin Bao12 Guojian Tang1 Yuewen Jiang4 and Jie Liu3

1College of Aerospace Science and Engineering National University of Defense Technology Changsha 410073 China2Science and Technology on Space Physics Laboratory Beijing 100076 China3School of Computer Science National University of Defense Technology Changsha 410073 China4Department of Engineering Science University of Oxford Oxford OX2 0ES UK

Correspondence should be addressed to Chunye Gong gongchunyegmailcom

Received 13 June 2014 Revised 6 August 2014 Accepted 9 September 2014

Academic Editor Guido Maione

Copyright copy 2015 Chunye Gong 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 present a survey of fractional differential equations and in particular of the computational cost for their numerical solutionsfrom the view of computer science The computational complexities of time fractional space fractional and space-time fractionalequations are O(N2M) O(NM2) and O(NM(M + N)) compared with O(MN) for the classical partial differential equations withfinite difference methods where M N are the number of space grid points and time steps The potential solutions for thischallenge include but are not limited to parallel computing memory access optimization (fractional precomputing operator)shortmemory principle fast Fourier transform (FFT) based solutions alternating direction implicitmethodmultigridmethod andpreconditioner technologyThe relationships of these solutions for both space fractional derivative and time fractional derivative arediscussed The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcomethis challenge and some attention should be paid to the fractional killer applications high performance iteration methods highorder schemes and Monte Carlo methods Since the computation of fractional equations with high dimension and variable orderis even heavier the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in thearea of fractional calculus

1 Introduction

The idea of fractional is natural If 120597119906120597119909 and 12059721199061205971199092 exist1205971511990612059711990915maybe exists too Fractional equations can be used

to describe some physical phenomena more accurately thanthe classical integer order differential equation [1] Fractionaldifferential equations (FDEs) provide a powerful instrumentfor the description of memory and hereditary properties ofdifferent substances The fractional diffusion equations playan important role in dynamical systems of semiconductorresearch hydrogeology bioinformatics finance [2] and otherresearch areas [3ndash6] Rajeev and Kushwaha [7] presented amathematical model describing the time fractional anoma-lous diffusion process of a generalized Stefan problem whichis a limiting case of a shoreline problem Space fractionaladvection-diffusion equations arise when velocity variationsare heavy tailed and describe particle motion that account for

variation in the flow field over the entire system [8] FDEsmay be divided into two fundamental types time fractionaldifferential equations and space fractional differential equa-tions For the fractional ordinary equations and fractionalorder control systems are also studied [9 10] The stability offractional order control systems attracts many attentions [1112] For example Laguerre continued fraction expansion ofthe Tustin fractional discrete-time operator was investigatedby Maione [13]

Some analytical methods were proposed for fractionaldifferential equation [14 15] Saha Ray [16] presented theanalytical solutions of the space fractional diffusion equa-tions by two-step Adomian decomposition method Momaniand Odibat [17] gave a comparison between the homotopyperturbation method and the variational iteration methodfor linear fractional partial differential equations By usinginitial conditions the explicit solutions of the equations have

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 258265 13 pageshttpdxdoiorg1011552015258265

2 Mathematical Problems in Engineering

been presented in the closed form and then their solutionshave been represented graphically Becausemost of fractionalproblems cannot be solved analyticallymore andmoreworksfocus on their numerical solutionsThere aremany numericalsolutions proposed for fractional equations [18] such asfinite difference method (FDM) [18] FDM is intuitive tounderstand and easy to learn for inexperienced researcherfrom the areas rather than mathematics So this surveyfocuses on FDM for fractional equations

For the numerical solutions of different differential equa-tions the area ofmathematics paysmuch attention to approx-imating the equation more accurately and faster (accuracyand speed) The area of computer science mainly focuses onthe runtime (speed) and code reuse (software)Thenumericalmethods (mathematic area) have eternal value and mayexist along with the existence of human culture [19ndash21] Theimplementations (computer area) are closer to the humansociety and the real applications but vary quickly alongwith the computer architecture [22ndash28] The computationalcost of the numerical solutions for fractional equations ismuch heavier than that for the traditional integer orderequations In the near future the fractional problems withhigh dimension long time iterations and huge grid pointswill need to be solved These problems are real challenge fortodayrsquos computer technologies and algorithms

2 Fractional Differential Equations

21 Origins In 1695 LrsquoHopital wrote to Leibniz asking himabout a particular notation he had used in his publications forthe nth-derivative of the linear function 119891(119909) = 119909 119863119899119909119863119909119899[29 30] LrsquoHopitalrsquos posed the question to Leibniz whatwouldthe result be if 119899 = 12 Leibnizrsquos response was ldquoAn apparentparadox from which one day useful consequences will bedrawnrdquo In these words fractional calculus was born [31]Later Fourier Euler and Laplace dabbled with fractionalcalculus [32]

22 A Short Summary of FDE There are mainly three kindsof FDEs

(1) time fractional [33 34](2) space fractional [35](3) space-time fractional [36]

One of the main challenges in fractional differential equationis the nonlocality of the fractional operator In the case of atime fractional derivative one needs to store all the historywhereas in the case of a space fractional derivative one needsto deal with almost-dense matrices

The partial differential equations (PDEs) mainly havethree categories parabolic hyperbolic and elliptic There arecorresponding FDEs to deal with the fractional conditionsThere are various FDEs listed below

(1) the parabolic widely studied fractional diffusionequation [37 38]

(2) the hyperbolic Telegraph equation [39]

(3) the elliptic fractional Laplace equation [40](4) fractional Black-Scholes equation in computational

finance [41](5) fractional wave equation for sound light and water

waves [42](6) fractional Fokker-Planck equations describing the

time evolution of the probability density function ofthe velocity of a particle [43]

(7) fractional Euler equations and fractional Navier-Stokes equations [44] for fluid dynamics [45]

(8) fractional kinetic equations for motion of objects[46]

(9) the perfect fractionalMaxwellrsquos equations for electro-dynamics [47]

(10) fractional Boltzmann equations for particle transport[48]

We can suppose that if there is a PDE there will bea FDE The reason is that replacing the integer derivativewith fractional derivative and solving the FDE with differentnumerical methods is not a hard job

The numerical schemes [49] for these FDEs are listedbelow

(1) finite difference method (FDM) [18 50 51](2) finite element method [52ndash54](3) finite volume method [55ndash57](4) Adomian decomposition method [58](5) Fourier transform method [59](6) spectral method [60](7) meshless method [61ndash63](8) exponential difference method [64 65](9) Monte Carlo method [66]

TheMonte Carlo (MC)method which uses repeated randomsampling to obtain numerical results belongs to undeter-mined computational methods

23 AHot Topic in Recent Years Machado et al [29] collected20 special issues on fractional calculus In 2011 and 2012 thereare about 7 special issues on this topicThere are more specialissues in the nearby four years (2011ndash2014) than those of 1999to 2010 [29] And there are additional 20 special issues inthe years of 2013 and 2014 So the researches on fractionalcalculus can be regarded as a collective revelry

3 Computational Challenge

31 A Classical Partial Differential Equation (PDE)

311 Heat EquationDiffusion Equation The heat equationis a parabolic PDE which describes the distribution of heat(or variation in temperature) in a given region over time

Mathematical Problems in Engineering 3

shown in (1) This equation is also known as the diffusionequation

120597119906 (119909 119905)

120597119905= 119889 (119909)

1205972119906 (119909 119905)

1205971199092+ 119891 (119909 119905)

119906 (119909 0) = 120601 (119909) 119909 isin [119909119871 119909119877]

119906 (119909119871 119905) = 120593 (119905) 119905 isin [0 119879]

119906 (119909119877 119905) = 120595 (119905) 119905 isin [0 119879]

(1)

where the 119889(119909) is the diffusion constant

312 Numerical Method Define 119905119899= 119899120591 119909

119894= 119894ℎ for 0 le 119899 le

119873 0 le 119894 le 119872 where119872 and 119873 are positive integer and 120591 =119879119873 ℎ = (119909

119877minus 119909119871)119872 are time step size and space step size

respectively Define 119906119899119894119891119899119894119889119894as the numerical approximation

to 119906(119909119894 119905119899) 119891(119909

119894 119905119899) and 119889(119909

119894) Using a forward difference at

time 119905119899and a second-order central difference for the space

derivative at position 119909119894 we get the explicit finite difference

approximation for the one-dimensional heat equation

119906119899+1

119894minus 119906119899

119894

120591= 119889119894

119906119899

119894+1minus 2119906119899

119894+ 119906119899

119894minus1

ℎ2+ 119891119899

119894 (2)

The 119906119899+1119894

can be obtained by this way

119906119899+1

119894= 1198861119906119899

119894+1+ 1198862119906119899

119894+ 1198863119906119899

119894minus1+ 120591119891119899

119894 (3)

where 1198861= 1198863= minus119889119894120591ℎ2 and 119886

2= 1 minus 2119889

119894120591ℎ2

If we use the backward difference at time 119905119899+1

and asecond-order central difference for the space derivative atposition 119909

119894we get the explicit finite difference approximation

for (1)

119906119899+1

119894minus 119906119899

119894

120591= 119889119894

119906119899+1

119894+1minus 2119906119899+1

119894+ 119906119899+1

119894minus1

ℎ2+ 119891119899+1

119894 (4)

The 119906119899+1119894


1198861119906119899+1

119894+1+ 1198862119906119899+1

119894+ 1198863119906119899+1

119894minus1= 119906119899

119894+ 120591119891119899

119894 (5)

where 1198861= 1198863= 119889119894120591ℎ2 and 119886

2= 1 + 2119889

119894120591ℎ2

The classical PDE is used to compare the fractionalequations For convenience there are several hypotheses forthis paper

(1) We just focus on the explicit FDM of (3) becausethese kinds of comparisons between fractional equa-tions and classical equations are straightforward Thecomparison between implicit FDE of (5) and implicitschemes of fractional equations ismore complex [67]

(2) The computational cost of diffusion coefficient 119889119894and

source119891119899119894(or119891119899+1119894

) is ignored because they only needcompute once And they are different with differentequations and make the comparison complicated

313 Computational Cost Each grid point of time step 119905119899+1

needs 4 multiplications and 3 additionsThere are119872minus1 gridpoints in each time step So each time step needs 7(119872 minus 1)

arithmetic logic operations There are about 119873 time stepsSo the total computational cost is about 7119873(119872 minus 1) Thecomputational cost will vary linearly along the number oftime steps and grid points

32 Time Fractional Diffusion Equation

321 Numerical Method Liu et al [63] developed an implicitradial basis function (RBF) meshless approach for timefractional diffusion equations and found that the presentedmeshless formulation is very effective for modeling andsimulation of fractional differential equations Murillo andYuste developed an explicit difference method for solvingfractional diffusion with Caputo form [68]

120597120572119906 (119909 119905)

120597120572119905=1205972119906 (119909 119905)

1205971199092(0 lt 120572 lt 1)

119906 (119909 0) = 119892 (119909) 119909 isin (0 119909119877)

119906 (0 119905) = 119906 (119909119877 119905) = 0

(6)

on a finite domain 0 lt 119909 lt 119909119877and 0 le 119905 le 119879

The explicit finite difference approximation for (6) isdescribed as follows [69]

120591minus120572

119899+1

sum

119894=0

119908119894(119906119899+1minus119894

119895minus 1199060

119895) =

1

ℎ2(119906119899

119895+1minus 2119906119899

119895+ 119906119899

119895minus1) (7)

where 119908119894is the normalized Grunwald weight defined with

119908119894= (minus1)

119894(120572

119894) (8)

Define 120583 = 120591120572ℎ2 119887119899= sum119899minus1

119894=0119908119894 and 119880119899 = (119906

119899

1 119906119899

2

119906119899

119872minus1)119879 we can get

119880119899+1

= 119887119899+11198800minus

119899+1

sum

119894=1

119908119894119880119899+1minus119894

+ 120583119860119880119899 (9)

where matrix 119860 is a tridiagonal matrix defined by

119860119872minus1times119872minus1

=(

minus2 1

1 minus2 1

∙ ∙

∙ ∙ 1

1 minus2

) (10)

322 Computational Cost In order to get 119880119899+1 the right-sided computation of (9) should be performed There aremainly one tridiagonal matrix-vector multiplication manyconstant-vector multiplications and many vector-vectoradditions in the right-sided computation

(1) The tridiagonal matrix-vector multiplication is 119860119880119899and a new vector 119880119899+1 = 119860119880119899 is got


(2) The constant-vector multiplications are 119881= 119887119899+11198800

11988110158401015840= 1205831198811 119881119894 = 119908

119894119880119899+1minus119894

(3) The vector-vector additions are 119880119899+1 = 119880119899+1

+ 119881+

11988110158401015840minus sum119899+1

119894=1119881119894

There are about 5(119872 minus 1) operations for tridiagonalmatrix-vector multiplication For time step 119905

119899+1needs (119899 +

8)(119872 minus 1) arithmetic logic operations with 119899 = 1 rarr 119873 Sothe total computational cost for (9) is about

8119873 (119872 minus 1) + (1 + 2 + sdot sdot sdot + 119873 minus 1) (119872 minus 1)

= 8119873 (119872 minus 1) +119873 (119873 minus 1) (119872 minus 1)

2

= 119873(119873

2+ 75) (119872 minus 1)

(11)

The computational cost varies linearly along the numberof grid points but squares with the number of time steps

33 Space Fractional Diffusion Equation

331 Numerical Method A classical numerical scheme forthe space fractional diffusion equation is the second-orderfractional Crank-Nicolson method proposed by Tadjeran etal [38 70] where the Richardson extrapolation techniqueis used to the first order shift Grunwald formula for spacefractional derivative Tadjeran et al [70] presented a practicalnumerical method in time and space to solve a class of initial-boundary value with variable coefficients on a finite domainfor

120597119906 (119909 119905)

120597119905= 119889 (119909)

120597120572119906 (119909 119905)

120597119909120572+ 119902 (119909 119905) (12)

on a finite domain 119909119871lt 119909 lt 119909

119877with 1 lt 120572 lt 2 and 0 le 119905 le 119879

The case of 1 lt 120572 lt 2models a super-diffusive flow in whicha cloud of diffusing particles spreads at a faster rate than theclassical diffusion model predictions [71 72]

The explicit finite difference approximation for (12) is

119906119899+1

119894minus 119906119899

119894

120591=119889119894

ℎ120572(

119894+1

sum

119896=0

119892120572119896119906119899

119894minus119896+1) + 119902

119899

119894 (13)

where 119906119899119894= 119906(119909

119894 119905119899) 119889119894= 119889(119909

119894) and 119902119899

119894= 119902(119909

119894 119905119899) The 119892

120572119896is

the normalized Grunwald weight [70 73]

119892120572119896

=Γ (119896 minus 120572)

Γ (minus120572) Γ (119896 + 1) (14)

These normalized weights only depend on the order 120572 andthe index 119896 The resulting equation can be explicitly solvedfor 119906119899+1119894

to give

119906119899+1

119894= 119906119899

119894+119889119894120591

ℎ120572(

119894+1

sum

119896=0

119892120572119896119906119899

119894minus119896+1) + 120591119902

119899

119894 (15)

332 Computational Cost The 119894th grid point of time step 119905119899+1

needs 119894 + 6 multiplication addition and division There are119872minus 1 grid points in each time step So each time step needs(119872 minus 1)6 + (1 + 2 + sdot sdot sdot + 119872 minus 1) = (119872 minus 1)((1198722) + 6)

arithmetic logic operations There are about119873 time steps Sothe total computational cost is about119873(119872 minus 1)((1198722) + 6)

34 Riesz Space Fractional Diffusion Equation

341 Numerical Method Shen et al [74] investigated theGreen function and a discrete random walk model for Rieszfractional advection-dispersion equation on infinite domainwith an initial condition and also presented implicit andexplicit finite differences to this problem on a finite domainCelik andDuman [75] applied theCrank-Nicolsonmethod toa fractional diffusion equation which has the Riesz fractionalderivative and obtained that the method is unconditionallystable and convergent The Riesz space fractional reaction-diffusion equation [76 77] is

120597119906 (119909 119905)

120597119905= minus119906 (119909 119905) +

119909119863120572

0119906 (119909 119905) + 119902 (119909 119905)

119906 (119909 0) = 119892 (119909) 119909 isin (119909119871 119909119877)

119906 (119909119871 119905) = 119906 (119909

119877 119905) = 0

(16)

with 1 lt 120572 le 2 and 0 le 119905 le 119879 Both 119906(119909 119905) and 119892(119909) are realvalued and sufficiently well-behaved function

119909119863120572

0119906(119909 119905) is

the Riesz space fractional derivativeWith adopting an Euler approximation in time the

explicit difference approximation can be got

119906119899+1

119894minus 119906119899

119894

120591= minus119906119899

119894minus

1

119888ℎ120572

times [

119894+1

sum

119896=0

119892119896119906119899

119894+1minus119896+

119872minus119894+1

sum

119896=0

119892119896119906119899

119894minus1+119896] + 119891

119899

119894

(17)

where 1198920= 1 119892

119896= (minus1)

119896120572 (120572 minus 1) sdot sdot sdot (120572 minus 119896 + 1) 119896 119896 = 1 2

3 is the normalized Grunwald weight 119888 = 12 cos(1205721205872)and 119891119899

119894= 119902(119909

119894 119905119899)

Equation (17) results in a linear system of equations

119880119899+1

= 119860119880119899+ 119876119899 (18)

where 119880119899 = (1199061198991 119906119899

2 119906

119899

119872minus1)119879 119876119899 = (120591119891119899

1 120591119891119899

2 120591119891

119899

119872minus1)119879

with 119902119899119894= 120591119891119899

119894(1 le 119894 lt 119872 minus 1) and 119860 = (119886

119894119895)(119872minus1)times(119872minus1)

is amatrix of coefficient 119860 is defined by

119886119894119895=

minus120591

119888ℎ120572119892119894+1minus119895

for 1 le 119895 lt 119894 minus 1

minus120591

119888ℎ120572(1198920+ 1198922) for 119895 = 119894 minus 1

(1 minus 120591) minus 2120591

119888ℎ1205721198921 for 119895 = 119894

minus120591

119888ℎ120572(1198920+ 1198922) for 119895 = 119894 + 1

minus120591

119888ℎ120572119892119895minus119894+1

for 119894 + 1 lt 119895 lt 119872 minus 1

(19)


342 Computational Cost From (18) the 119872 minus 1 results areproduced by 119860119880119899 for each time step The inner product ofvectors V1 and V2 with size119898 has119898multiplications and119898minus1

additions 119860119880119899 for each time step needs (119872 minus 1)(2119872 minus 3)

operations Assuming 120591119891119899119894is pre-performed each time step

needs (119872minus1)(2119872minus2) arithmetic logic operationsThere areabout 119873 time steps So the total computational cost is about119873(119872minus1)(2119872minus 2) The computational cost will vary linearlyalong the number of time steps but square with the numberof grid points From analytical view the computation of (18)is about four times heavier than that of (15)

35 Space-Time Riesz-Caputo Fractional Convection-DiffusionEquation The fractional advection-diffusion (dispersion)equation has been applied to many problems Fractionaladvection-dispersion equations are used in groundwaterhydrology to model the transport of passive tracers carriedout by fluid flow in a porous medium [78] Shen et al[79] presented an explicit difference approximation andan implicit difference approximation for the space-timeRiesz-Caputo fractional advection-diffusion equation withinitial and boundary conditions in a finite domain Theyproved that the implicit difference approximation is uncon-ditionally stable and convergent but the explicit differenceapproximation is conditionally stable and convergent Liuet al [80] proposed an implicit difference method and anexplicit difference method to solve the space-time fractionaladvection-diffusion equation and discussed the stability andconvergence of the method

351 Numerical Method We consider the following space-time Riesz-Caputo fractional advection-diffusion equation(STRCFADE) studied by Shen et al [79] This equation isobtained by replacing the space-derivative in the advection-diffusion equation with a generalized derivative of order 120573

1

1205732with 0 lt 120573

1lt 1 1 lt 120573

2lt 2 and time-derivative with a

generalized derivative of order 120572with 0 lt 120572 lt 1 We consider

0119863120572

119905119906 (119909 119905) = 119861

1

1205971205731119906 (119909 119905)

120597 |119909|1205731

+ 1198612

1205971205732119906 (119909 119905)

120597 |119909|1205732

+ 119891 (119909 119905)

119906 (119909 0) = 120601 (119909) 119909 isin [0 119909119877]

119906 (0 119905) = 119906 (119909119877 119905) = 0 119905 isin [0 119879]

(20)

on a finite domain 0 le 119909 le 119909119877and 0 le 119905 le 119879 The coef-

ficients 1198611and 119861

2are both positive constants and represent

the diffusion (dispersion) coefficient and the average fluidvelocity The 1205971205731119906(119909 119905)120597|119909|1205731 and 1205971205732119906(119909 119905)120597|119909|1205732 are Rieszspace fractional derivatives of order 120573

1and 120573

2 respectively

Then we can get the explicit finite differential approximationfor (20) [79]

119906119899+1

119894=

119899minus1

sum

119895=0

(119887119895minus 119887119895+1) 119906119899minus119895

119894+ 1198871198991199060

119894+ 11986111205831(

119873minus119894

sum

119896=minus119894

1205961205731

119896119906119899

119894+119896)

+ 11986121205832(

119873minus119894

sum

119896=minus119894

1205961205732

119896119906119899

119894+119896) + 120583

0120593119899

119894 119894 = 1 119873 minus 1

119906119899+1

0= 119906119899+1

119873= 0

1199060

119894= 120601119894

(21)

where 1205830= 120591120572Γ(2 minus 120572) 120583

1= 1205830ℎ1205731 and 120583

2= 1205830ℎ1205732 More

information can be referred to in [79]

352 Computational Cost From Section 342 we knowthat the computational cost of 119861

11205831(sum119873minus119894

119896=minus1198941205961205731

119896119906119899

119894+119896) +

11986121205832(sum119873minus119894

119896=minus1198941205961205732

119896119906119899

119894+119896) + 120583

0120593119899

119894of 119873 time steps is about

119873(119872 minus 1)(2119872 minus 2) operations From Section 322 we knowthat the computational cost of sum119899minus1

119895=0(119887119895minus 119887119895+1)119906119899minus119895

119894+ 1198871198991199060

119894

of 119873 time steps with 119899 ranging from 1 to 119873 is about119873(1198732)(119872 minus 1) So the total computational cost is about119873(119872 minus 1)(2119872 minus 2 + (1198732)) The computational cost variesquadratically with the number of time steps or the numberof grid points

For fixed 119872 = 4097 the comparison between the com-putational costs of the numerical solution among classicalPDE time fractional space fractional Riesz space fractionaland space-time Riesz-Caputo fractional equations is shownin Figure 1 For fixed 119873 = 2048 the computational cost isshown in Figure 2

36 More Challenges There are more computational chal-lenges listed below

(1) high dimensional problems [55 81]

(2) implicit schemes [67 82]

(3) high order schemes [33 83]

(4) variable order problems [84]

(5) huge memory space requirement

The high dimensional problems are more computationexpensive [55 81 85 86] For example the two-dimensionaltime fractional diffusion equation (2D-TFDE) [67 87] is

0119863120572

119905119906 (119909 119910 119905) = 119886 (119909 119910 119905)

1205972119906 (119909 119910 119905)

1205971199092

+ 119886 (119909 119910 119905)1205972119906 (119909 119910 119905)

1205971199102+ 119891 (119909 119910 119905)

119906 (119909 119910 0) = 120601 (119909 119910) (119909 119910) isin Ω

119906 (119909 119910 119905)1003816100381610038161003816120597Ω = 0 119905 isin [0 119879]

(22)

where Ω = (119909 119910) | 0 le 119909 le 1198711 0 le 119910 le 119871

2 119886(119909 119910 119905) gt 0

and 119887(119909 119910 119905) gt 0


40

35

30

25

20

N

ClassicalTime fractionalSpace fractional

Riesz space fractionalSpace-time Riesz-Caputo fractional

215214213212211210

Valu

e (lo

g 2)

Figure 1 Comparison of the computational cost with different gridpoints for fixed 119872 = 4097 among classical time fractional spacefractional Riesz space fractional and space-time Riesz-Caputofractional equations

40

35

30

25

20



215214213212211210

Valu

e (lo

g 2)

Mminus 1

Figure 2 Comparison of the computational cost with different timestep for fixed 119873 = 2048 among classical time fractional spacefractional Riesz space fractional and space-time Riesz-Caputofractional equations

The approximating scheme is [87]

119906119899+1

119894119895minus 119906119899

119894119895+

119899

sum

119904=1

(119887119904) (119906119899+1minus119904

119894119895minus 119906119899minus119904

119894119895)

= 1205831Γ (2 minus 120572) 119886

119899

119894119895(119906119899

119894+1119895minus 2119906119899

119894119895+ 119906119899

119894minus1119895)

+ 1205832Γ (2 minus 120572) 119887

119899

119894119895(119906119899

119894119895+1minus 2119906119899

119894119895+ 119906119899

119894119895minus1)

+ 120591120572Γ (2 minus 120572) 119891

119899

119894119895

(23)

where 1205831= 120591120572ℎ2

119909and 120583

2= 120591120572ℎ2

119910 The ℎ

119909and ℎ

119910are

the step size along 119883 and 119884 directions The computationalcomplexity is about 119874(11987321198722) which is much bigger than

the computational complexity of one-dimensional problemswhen the number of grid points is bigger enough

The direct Gauss elimination for implicit scheme ofFDE is not convenient Solving unsteady FDE with implicitschemes relies on the iteration method at each time step Thevariable order problems [84 88] have complex coefficientswhich need more arithmetic logic operationsThe high orderschemes [33 38 70] need more arithmetic logic operationstoo

The memory usage belongs to the computing resourcesSo the hugememory requirement of FDE is a kind of compu-tational challenge in the broader sense This is especially truefor time fractional problems Ignoring the memory usage ofthe coefficients and source terms the one-dimensional equa-tion (9) needs 8(119872minus1)119873 bytes ofmemory space and the two-dimensional equation (23) needs 81198722119873 bytes of memoryspace For three-dimensional problems the memory usage is81198723119873 at least It needs 15625 PB (1 PB = 10245 bytes) mem-

ory space with119872 = 10240119873 = 2048 for three-dimensionalproblemsMaybe only themost powerful supercomputer [89]can satisfy the huge memory requirement

4 Potential Solutions

There are several ways which can be used to overcome thecomputational challenge of FDEs

41 Parallel Computing Large scale applications and algo-rithms in science and engineering such as neutron transport[90ndash92] light transport [93] computational fluid dynamics[94 95]molecular dynamics [96] and computational financeand different numerical methods [97] rely on parallel com-puting [98 99]

Gong et al [77] present a parallel algorithm for Rieszspace fractional diffusion equation based on MPI parallelprogramming model at the first time The parallel algo-rithm is as accurate as the serial algorithm and achieves7939 parallel efficiency compared with 8 processes ona distributed memory cluster system The parallel implicititerative algorithm was studied for two-dimensional timefractional problem and a task distribution model is shownin Figure 3 [67] 119872

119909 119872119910 119875119909 119875119910stand for the discrete grid

points parallel processes along119883 119884 directionsDomain decomposition method is regarded as the basic

mathematical background for many parallel applications[100ndash102] A domain decomposition algorithm for time frac-tional reaction-diffusion equation with implicit finite differ-encemethodwas proposed [103]The domain decompositionalgorithm keeps the same parallelism as Jacobi iteration butneeds much fewer iterations

Diethelm [104] implemented the fractional version ofthe second-order Adams-Bashforth-Moulton method on aparallel computer and discussed the precise nature of the par-allelization concept Parallel computing has already appearedin some studies on FDEs but until today their power forapproximating fractional derivatives and solving FDEs hasnot been fully recognized


P0My

(0 Py)P0Myminus1

(1 Py)

PMx minus1 My

(Px Py)

PMx My

(1 0)(0 0) (Px 0)

middot middot middot middot middot middot

middot middot middot


PMxMyminus1PMxminus1Myminus1

PMxminus10 PMx0P00 P01

Figure 3 The two-dimensional task distribution model for 2D-TFDE

42 Memory Access Optimization (Fractional Precomput-ing Operator) Memory access optimization is generallyregarded as a technology of computer architecture It isalso useful from the view of applications One feature ofthe modern computer architecture is multilayer memoryFor example the traditional CPU has fast cache and slowmain memory The new graphics processing unit (GPU) hasfast shared memory and slow memory Reusing the data inshared memory is a key point to improve the performanceof GPU applications [105ndash109] A very useful optimization ispresented for fractional derivative [69 110]We can name thisfractional precomputing operator which will be a very basicand useful optimizing technology for the implementationof fractional algorithms and applications The example offractional precomputing operator can refer to Algorithms 4and 5 in reference [69]

OnCPUplatform an optimization of the sum of constantvector multiplication is presented and 2-time speedup can begot for both serial and parallel algorithm for the time frac-tional equation [69] The key technology is reusing the datain cache through loop unrolling Zhang et al [110] presentedcode optimization for fractional Adams-Bashforth-Moultonmethod by loop fusion and loop unrolling

On GPU platform Liu et al [111] presented an optimizedCUDA kernel for the numerical solution of time frac-tional equation and 12ndash27-time performance improvementis achieved

43 Short Memory Principle The short memory principle[3] means the unknown grid points only rely on the recentpast (in time) or near neighbors (in space) This principlehas been proved to be an easy and powerful way forvarious kinds of fractional differential equations [112 113]The short memory principle is also called fixed memoryprinciple or logarithmic memory principle The idea of shortmemory principle is simple Taking time fractional equationas an example in Section 321 the value of 119908

119894becomes

smaller while 119894 is increasing In (9) the accumulation of

sum119899+1

119894=1119908119894119880119899+1minus119894 is replaced withsum119899+1

119894=min1119899+1minus119870 119908119894119880119899+1minus119894119870 is a

positive integer whichmeans if 119899 is very big the computationof the accumulation is fixed

Based on short memory principle some principles withbetter balance of computation speed and computation accu-racy are presented The logarithmic memory principle isdeveloped with a good approximation to the true solutionat reasonable computational cost [114] Another principle isequal-weight memory principle in which an equal-weight isapplied to all past data in history and the result is reservedinstead of being discarded [115] The equal-weight memoryprinciple is an interesting and useful approximation methodto fractional derivative

44 FFT Based Solution Because of the nonlocal propertyof fractional differential operators the numerical methodshave full coefficient matrices which require computationalcost of 119874(1198723) for implicit scheme per time step where119872 is the number of grid points Ford and Simpson [114]developed a faster scheme for the calculation of fractionalintegrals A reduction in the amount of computational workcan be achieved by using a graded mesh thereby makingthe 119874(1198732)method to a 119874(119873 log119873)method The underlyingidea is based on the fact that the fractional integral possessesa fundamental scaling property that can be exploited in anatural way [114]

Wang et al [116 117] developed a fast finite differ-ence method for fractional diffusion equations which onlyrequires computational cost of 119874(119873log

2119873) while retaining

the same accuracy and approximation property as the regularfinite difference method Numerical experiments show thatthe fast method has a significant reduction of CPU time [86]The fast method should have a banded coefficient matrixinstead of the full matrix The properties of Toeplitz andcirculant matrices fast Fourier transform (FFT) and inverseFFT are used to reduce the computational cost The methodis also called superfast-preconditioned iterative method forsteady-state space fractional diffusion equations [118]

An efficient iteration method for Toeplitz-plus-band tri-angular systems which may produced by fractional ordi-nary differential equations was developed Some methodssuch as matrix splitting FFT compress memory storageand adjustable matrix bandwidth are used in the presentedsolution The interesting technologies are the adjustablematrix bandwidth and solving fractional ordinary differentialequations with iteration method The experimental resultsshow that the presented efficient iteration method is 425times faster than the regular solution [10]

45 Alternating Direction Implicit Method The alternatingdirection implicit (ADI) method is a finite difference methodfor solving traditional PDEs The approximation methodsfor fractional equations result in a very complicated set ofequations inmultiple dimensions which are hard to solve [6781 85] So the ADImethod is developed for high dimensionalproblems [33 117] The advantage of the fractional ADImethod is that the equations that have to be solved in eachstep have a simpler structure The time fractional problems


can be solved efficientlywith the tridiagonalmatrix algorithm[33] The space fractional problems can use the FFT toaccelerate the computation [117]

46 Multigrid Method The multigrid method is usuallyexploited for solving ill-conditioned systems The main ideaof multigrid is to accelerate the convergence of a basiciterative method by global correction form accomplishedby solving a coarse problem Pang and Sun [119] proposeda multigrid method to solve the initial-boundary valueproblem of a fractional diffusion equation The experimentalresults show that the multigrid + FFT method runs hundredtimes faster than Gaussian elimination method and theconjugate gradient normal residual (CGNR) method A fullV-cycle multigrid method is proposed for the stationaryfractional advection dispersion equation [120] and ten-timeperformance improvement is achieved

47 Preconditioning Technology Preconditioning is typicallyrelated to reducing a condition number of the problem withiterative methods It shows that both the average number ofiterations and the CPU time by the PCGNR (preconditionerCGNR) method with circulant preconditioners are muchless than those by the CGNR method and less than thatby the multigrid method [121] The circulant preconditioner[121] banded preconditioner [122] fast Poisson precondi-tioners [123] and preconditioned conjugate gradient squaredmethod plus FFT [118] are developed for different FDEs

48 Relationships among These Potential Solutions Thepotential solutions for the computational challenge of FDEare investigated above Many people will be curious aboutthese relationships canwe combine thesemethods to developa fastest solution for FDEs The answer is still unknownThe performance of these solutions varies from differentFDE applications Their relationships are shown in Table 1The PC MAO SMP FFT ADI MGM and PT standfor parallel computing memory access optimization shortmemory principle FFT based solution alternating directionimplicit method multigrid method and preconditioningtechnologyThe score means the degrees of the two solutionsare harmonious Higher score means using two solutions canachieve better performance

For time fractional derivative [124] only memory accessoptimization [69 111] and short memory principle [3] areuseful Here the time fractional derivative is different fromtime fractional problems For example the one- and two-dimensional FDEs are parallelized by Gong et al [67 69]These parallel algorithms are based on the partition of spacenot time

5 Future Directions

51 Fractional Killer Applications Killer application is a kindof application that is so necessary or desirable that it provesthe core value of some larger technology A killer applicationis something like Project Apollo in space technology theIBM 360 in personal computer industry and the IPhone in

Table 1 Relationships among the potential solutions for spacefractional derivative

PC MAO SMP FFT ADI MGM FTPC mdash 3 3 2a 3 2a 2a

MAO mdash 3 1b 3 3 3SMP mdash 1c 3 3 3FFT mdash 3 3 3ADI mdash 3 3MGM mdash 2d

PT mdashaThe parallel efficiency of FFT multigrid preconditioner is limitingbIt is hard to use MAO within FFTcBecause of the Toeplitz structure FFT cannot use short memory principledThe philosophy of multigrid and preconditioning technology are differentBut there are some multigrid preconditioners

the smart phone industry The fractional research still lacksthese kinds of killer applications It needs fractional applica-tions to solve scientific or engineering problems in physicalworld such as the fractional flowcontrol of hypersonicvehicle not only the academical problems The fractionalkiller application should be proved that it is more useful thanthe traditional classical application Solving real problemsin physical world will build an economic foundation forfractional researches

52 Parallel Computing The technologies of parallel comput-ing should be regarded as a basic method to overcome thecomputational challenge for FDEs There are three potentialsolutions for the computational challenge of FDEs The shortmemory principle is an experimentalmethod which is usefulin real fractional applicationsThe119874(119873 log119873)methods [117]used the property of Toeplitz matrices and FFT technologyParallel computing is a foundational technology for scientificand engineering computation Fortunately the shortmemoryprinciple and119874(119873 log119873)methods are compatiblewith paral-lel computing So it is interesting to develop algorithmswhichis faster than119874(119873 log119873)methodsThenumericalmethods ofspace fractional equations with global dependence are muchharder to be parallelized than that of the time fractionalequations So the task distribution model load balance of theparallel algorithm for space fractional equations should bepaid attention to

53 High Performance Iteration Methods Different kind ofnumerical methods is very easily found for FDEs The directmethods such as Gauss elimination are not suitable for largescale fractional applications The iterative methods for thesenumerical methods are not fully studied and very few workscan be found [118 125] We think that different iterativemethods such as Jacobimethod andGauss-Seidel andKrylovsubspace method are effective for fractional linear systemswhich are produced from FDEs Does the coefficient matrixof FDEs have some other special properties or not Theanswer is still unknown The convergence and stability ofthese iterative methods should be proved as well


54 High Order Schemes for Fractional Derivatives Thetraditional partial derivatives have many high order schemesFor time fractional equation the high order schemes fortraditional integer derivative are not hard to build But forfractional derivatives the high order schemes are underdeveloping [70 126ndash128] High order schemes will be usedin the numerical solutions of FDEs where high accuracy isrequired in the presence of shocks or discontinuities

55 Monte Carlo Method The Monte Carlo method hasadvantage in solving nonlinear high dimensional complexgeometry problems In order to get the approximation of asmall domain the determined methods such as FDM mustresolve the total definition domainwith boundary conditionsThe Monte Carlo method only focuses on the small domainThis property is very useful if we are only interested inthis small domain The Monte Carlo method is easy tobe parallelized and needs much less memory space thandetermined methods The fractional equations are a kindof nonlinear problem and their high dimensional problemsare very computation intensive So Monte Carlo method forFDEs needs to be studied in the future Because of the highnonlinear and nonlocal property of FDEs the sampling effi-ciency will be the key point ofMonte Carlomethod for FDEs

6 Conclusions

In this paper we give a comprehensive review of FDEs and itscomputational challenge This kind of challenge will becomean incoming problem for the computer industry if the realfractional problems need to be approximated We revieweda wide range of computational costs that come from differentkinds of fractional equationsWhile we have collected severalpotential solutions on this challenge we believe that the long-term legacy of solutionswill allow the real world scientific andengineering applications come true

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research work is supported in part by the NationalNatural Science Foundation of China under Grants no61402039 no 61303265 and no 60970033 in part by Spe-cialized Research Fund for the Doctoral Program of HigherEducation underGrant no 20114307110001 and in part by 973Program of China under Grants no 2014CB430205 and no61312701001 The authors would like to thank the anonymousreviewers for their helpful comments as well

References

[1] R Klages G Radons and I Sokolov Anomalous TransportFoundations and Applications Wiley-VCH Weinheim Ger-many 2008

[2] R R Nigmatullin T Omay and D Baleanu ldquoOn fractionalfiltering versus conventional filtering in economicsrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 15no 4 pp 979ndash986 2010

[3] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[4] G Maione and A Punzi ldquoCombining differential evolutionand particle swarm optimization to tune and realize fractional-order controllersrdquo Mathematical and Computer Modelling ofDynamical Systems vol 19 no 3 pp 277ndash299 2013

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

[6] C Farges M Moze and J Sabatier ldquoPseudo-state feedbackstabilization of commensurate fractional order systemsrdquo Auto-matica vol 46 no 10 pp 1730ndash1734 2010

[7] Rajeev andM Kushwaha ldquoHomotopy perturbationmethod fora limit case Stefan problem governed by fractional diffusionequationrdquo Applied Mathematical Modelling vol 37 no 5 pp3589ndash3599 2013

[8] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvectiondispersion equations for modeling transport at theEarth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009

[9] G Maione ldquoHigh-speed digital realizations of fractional oper-ators in the delta domainrdquo IEEE Transactions on AutomaticControl vol 56 no 3 pp 697ndash702 2011

[10] C Gong W Bao G Tang C Min and J Liu ldquoAn efficientiteration method for toeplitz -plus-band triangular systemsgenerated from fractional ordinary differential equationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 194249 5pages 2014

[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011

[12] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[13] G Maione ldquoOn the Laguerre rational approximation to frac-tional discrete derivative and integral operatorsrdquo IEEE Transac-tions on Automatic Control vol 58 no 6 pp 1579ndash1585 2013

[14] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solutionsfor the multi-term time-fractional diffusion-wavediffusionequations in a finite domainrdquo Computers and Mathematics withApplications vol 64 no 10 pp 3377ndash3388 2012

[15] M Alipour and D Baleanu ldquoApproximate analytical solutionfor nonlinear system of fractional differential equations by BPsoperational matricesrdquo Advances in Mathematical Physics vol2013 Article ID 954015 9 pages 2013

[16] S Saha Ray ldquoAnalytical solution for the space fractional dif-fusion equation by two-step adomian decomposition methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1295ndash1306 2009

[17] S Momani and Z Odibat ldquoComparison between the homotopyperturbation method and the variational iteration method forlinear fractional partial differential equationsrdquo Computers andMathematics with Applications vol 54 no 7-8 pp 910ndash9192007

[18] S Chen F Liu and V Anh ldquoA novel implicit finite differencemethod for the one-dimensional fractional percolation equa-tionrdquo Numerical Algorithms vol 56 no 4 pp 517ndash535 2011


[19] J Yan G-M Tan and N-H Sun ldquoOptimizing parallel Snsweeps on unstructured grids for multi-core clustersrdquo Journalof Computer Science and Technology vol 28 no 4 pp 657ndash6702013

[20] H Yang and X-C Cai ldquoParallel fully implicit two-gridmethodsfor distributed control of unsteady incompressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 72 no 1pp 1ndash21 2013

[21] X Liu T Gu X Hang and Z Sheng ldquoA parallel version ofQMRCGSTAB method for large linear systems in distributedparallel environmentsrdquo Applied Mathematics and Computationvol 172 no 2 pp 744ndash752 2006

[22] D Ming ldquoA parallel Attainable Region construction methodsuitable for implementation on a graphics processing unit(GPU)rdquo Computers and Chemical Engineering vol 67 pp 103ndash120 2014

[23] X Liao L Xiao C Yang and Y Lu ldquoMilkyway-2 supercom-puter system and applicationrdquo Frontiers of Computer Sciencevol 8 no 3 pp 345ndash356 2014

[24] W Xu Y Lu Q Li et al ldquoHybrid hierarchy storage system inMilkyWay-2 supercomputerrdquo Frontiers of Computer Science vol8 no 3 pp 367ndash377 2014

[25] F Aquotte and A F da Silva ldquoPSIM a modular particle systemon graphics processing unitrdquo IEEE Latin America Transactionsvol 12 no 2 pp 321ndash329 2014

[26] S J Pennycook S D Hammond S A Jarvis and G RMudalige ldquoPerformance analysis of a hybrid MPICUDAimplementation of the NASLU benchmarkrdquo ACM SIGMET-RICS Performance Evaluation Review vol 38 no 4 pp 23ndash292011

[27] D Mu P Chen and L Wang ldquoAccelerating the discontinuousGalerkin method for seismic wave propagation simulationsusing the graphic processing unit (GPU)-single-GPU imple-mentationrdquo Computers and Geosciences vol 51 pp 282ndash2922013

[28] S Mauger G C de Verdiere L Berge and S Skupin ldquoGPUaccelerated fully space and time resolved numerical simulationsof self-focusing laser beams in SBS-active mediardquo Journal ofComputational Physics vol 235 pp 606ndash625 2013

[29] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[30] J A Tenreiro Machado V Kiryakova and F Mainardi ldquoAposter about the recent history of fractional calculusrdquo FractionalCalculus amp Applied Analysis vol 13 no 3 pp 329ndash334 2010

[31] A Loverro ldquoFractional calculus history definitions and appli-cations for the engineerrdquo Tech Rep 46556 Department ofAerospace and Mechanical Engineering Notre Dame IndUSA 2004

[32] K Nishimoto An Essence of Nishimotorsquos Fractional Calculus(Calculus in the 21st Century) Integrations and Differentiationsof Arbitrary Order Descartes Press Company 1991

[33] M Cui ldquoCompact alternating direction implicit method fortwo-dimensional time fractional diffusion equationrdquo Journal ofComputational Physics vol 231 no 6 pp 2621ndash2633 2012

[34] W Zhang X Cai and S Holm ldquoTime-fractional heat equationsand negative absolute temperaturesrdquoComputers andMathemat-ics with Applications vol 67 no 1 pp 164ndash171 2014

[35] Q Yang F Liu and I Turner ldquoNumerical methods for frac-tional partial differential equations with Riesz space fractionalderivativesrdquo Applied Mathematical Modelling Simulation and

Computation for Engineering and Environmental Systems vol34 no 1 pp 200ndash218 2010

[36] J Zhao J Xiao and Y Xu ldquoA finite element method forthe multiterm time-space Riesz fractional advection-diffusionequations in finite domainrdquo Abstract and Applied Analysis vol2013 Article ID 868035 15 pages 2013

[37] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Compu-tational Physics vol 225 no 2 pp 1533ndash1552 2007

[38] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[39] N J Ford J Xiao and Y Yan ldquoStability of a numerical methodfor a space-time-fractional telegraph equationrdquo ComputationalMethods in Applied Mathematics vol 12 no 3 pp 273ndash2882012

[40] T Chang ldquoBoundary integral operator for the fractional laplaceequation in a bounded lipschitz domainrdquo Integral Equations andOperator Theory vol 72 no 3 pp 345ndash361 2012

[41] G Jumarie ldquoDerivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time Applica-tion to Mertonrsquos optimal portfoliordquo Computers amp Mathematicswith Applications vol 59 no 3 pp 1142ndash1164 2010

[42] P Straka M M Meerschaert R J McGough and Y ZhouldquoFractional wave equations with attenuationrdquo Fractional Calcu-lus and Applied Analysis vol 16 no 1 pp 262ndash272 2013

[43] W Deng ldquoFinite element method for the space and timefractional Fokker-Planck equationrdquo SIAM Journal onNumericalAnalysis vol 47 no 1 pp 204ndash226 2008

[44] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[45] M A Herzallah and D Baleanu ldquoFractional Euler-Lagrangeequations revisitedrdquoNonlinear Dynamics vol 69 no 3 pp 977ndash982 2012

[46] A I Saichev andGM Zaslavsky ldquoFractional kinetic equationssolutions and applicationsrdquo Chaos vol 7 no 4 pp 753ndash7641997

[47] T Wenchang P Wenxiao and X Mingyu ldquoA note on unsteadyflows of a viscoelastic fluid with the fractional Maxwell modelbetween two parallel platesrdquo International Journal of Non-LinearMechanics vol 38 no 5 pp 645ndash650 2003

[48] R A El-Nabulsi ldquoThe fractional Boltzmann transport equa-tionrdquoComputers andMathematics with Applications vol 62 no3 pp 1568ndash1575 2011

[49] K Diethelm ldquoAn investigation of some nonclassical methodsfor the numerical approximation of Caputo-type fractionalderivativesrdquo Numerical Algorithms vol 47 no 4 pp 361ndash3902008

[50] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[51] M Cui ldquoCompact finite difference method for the fractionaldiffusion equationrdquo Journal of Computational Physics vol 228no 20 pp 7792ndash7804 2009

[52] N J Ford J Xiao and Y Yan ldquoA finite element method for timefractional partial differential equationsrdquoFractional Calculus andApplied Analysis vol 14 no 3 pp 454ndash474 2011


[53] X Zhang J Liu J Wen B Tang and Y He ldquoAnalysis for one-dimensional time-fractional Tricomi-type equations by LDGmethodsrdquoNumerical Algorithms vol 63 no 1 pp 143ndash164 2013

[54] K Mustapha and W McLean ldquoUniform convergence for adiscontinuous Galerkin time-stepping method applied to afractional diffusion equationrdquo IMA Journal of Numerical Anal-ysis vol 32 no 3 pp 906ndash925 2012

[55] Q Yang I Turner F Liu andM Ilic ldquoNovel numericalmethodsfor solving the time-space fractional diffusion equation in twodimensionsrdquo SIAM Journal on Scientific Computing vol 33 no3 pp 1159ndash1180 2011

[56] H Hejazi T Moroney and F Liu ldquoA finite volume methodfor solving the two-sided time-space fractional advection-dispersion equationrdquo Central European Journal of Physics vol11 no 10 pp 1275ndash1283 2013

[57] H Hejazi T Moroney and F Liu ldquoStability and convergenceof a finite volume method for the space fractional advection-dispersion equationrdquo Journal of Computational and AppliedMathematics vol 255 pp 684ndash697 2014

[58] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

[59] B Tang C Jiang and H Zhu ldquoFractional fourier transform forconfluent hypergeometric beamsrdquoPhysics Letters A vol 376 no38-39 pp 2627ndash2631 2012

[60] A R Carella and C A Dorao ldquoLeast-squares spectral methodfor the solution of a fractional advection-dispersion equationrdquoJournal of Computational Physics vol 232 pp 33ndash45 2013

[61] A Shirzadi L Ling and S Abbasbandy ldquoMeshless simulationsof the two-dimensional fractional-time convection-diffusion-reaction equationsrdquo Engineering Analysis with Boundary Ele-ments vol 36 no 11 pp 1522ndash1527 2012

[62] Q Liu F Liu I Turner V Anh and Y T Gu ldquoA RBF meshlessapproach for modeling a fractal mobileimmobile transportmodelrdquo Applied Mathematics and Computation vol 226 pp336ndash347 2014

[63] Q Liu Y T Gu P Zhuang F Liu and Y F Nie ldquoAn implicitRBFmeshless approach for time fractional diffusion equationsrdquoComputational Mechanics vol 48 no 1 pp 1ndash12 2011

[64] R Garrappa ldquoA family of Adams exponential integrators forfractional linear systemsrdquo Computers and Mathematics withApplications vol 66 no 5 pp 717ndash727 2013

[65] R Garrappa and M Popolizio ldquoGeneralized exponential timedifferencingmethods for fractional order problemsrdquoComputersamp Mathematics with Applications vol 62 no 3 pp 876ndash8902011

[66] D Fulger E Scalas and G Germano ldquoMonte Carlo simu-lation of uncoupled continuous-time random walks yieldinga stochastic solution of the space-time fractional diffusionequationrdquo Physical Review E vol 77 no 2 Article ID 0211222008

[67] C Gong W Bao G Tang Y Jiang and J Liu ldquoA parallelalgorithm for the two-dimensional time fractional diffusionequation with implicit difference methodrdquoThe Scientific WorldJournal vol 2014 Article ID 219580 8 pages 2014

[68] J Q Murillo and S B Yuste ldquoAn explicit difference methodfor solving fractional diffusion and diffusion-wave equationsin the caputo formrdquo Journal of Computational and NonlinearDynamics vol 6 no 2 Article ID 021014 2011

[69] C Gong W Bao G Tang B Yang and J Liu ldquoAn efficient par-allel solution forCaputo fractional reaction-diffusion equationrdquoJournal of Supercomputing vol 68 no 3 pp 1521ndash1537 2014

[70] C Tadjeran M M Meerschaert and H P Scheffler ldquoA second-order accurate numerical approximation for the fractional dif-fusion equationrdquo Journal of Computational Physics vol 213 no1 pp 205ndash213 2006 httpdxdoiorg101016jjcp200508008

[71] MMMeerschaert D A BensonH-P Scheffler and P Becker-Kern ldquoGoverning equations and solutions of anomalous ran-dom walk limitsrdquo Physical Review E vol 66 no 6 Article ID060102 2002

[72] R Metzler and J Klafter ldquoThe restaurant at the end of therandomwalk recent developments in the description of anoma-lous transport by fractional dynamicsrdquo Journal of Physics AMathematical and General vol 37 no 31 pp R161ndashR208 2004

[73] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[74] S Shen F Liu V Anh and I Turner ldquoThe fundamental solutionand numerical solution of the Riesz fractional advection-dispersion equationrdquo IMA Journal of Applied Mathematics vol73 no 6 pp 850ndash872 2008

[75] C Celik andM Duman ldquoCrank-Nicolson method for the frac-tional diffusion equation with the Riesz fractional derivativerdquoJournal of Computational Physics vol 231 no 4 pp 1743ndash17502012

[76] J Chen and F Liu ldquoAnalysis of stability and convergence ofnumerical approximation for the Riesz fractional reaction-dispersion equationrdquo Journal of Xiamen University vol 45 no4 pp 466ndash469 2006 (Chinese)

[77] C Gong W Bao and G Tang ldquoA parallel algorithm for theRiesz fractional reaction-diffusion equation with explicit finitedifference methodrdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 654ndash669 2013

[78] A Yildirim andH Kocak ldquoHomotopy perturbationmethod forsolving the space-time fractional advection-dispersion equa-tionrdquo Advances in Water Resources vol 32 no 12 pp 1711ndash17162009

[79] S Shen F Liu and V Anh ldquoNumerical approximations andsolution techniques for the space-time Riesz-Caputo fractionaladvection-diffusion equationrdquo Numerical Algorithms vol 56no 3 pp 383ndash403 2011

[80] F Liu P Zhuang V Anh I Turner and K Burrage ldquoStabilityand convergence of the difference methods for the space-timefractional advection-diffusion equationrdquo Applied Mathematicsand Computation vol 191 no 1 pp 12ndash20 2007

[81] C-M Chen and F Liu ldquoA numerical approximationmethod forsolving a three-dimensional space Galilei invariant fractionaladvection-diffusion equationrdquo Journal of Applied Mathematicsand Computing vol 30 no 1-2 pp 219ndash236 2009

[82] K Mustapha ldquoAn implicit finite-difference time-steppingmethod for a sub-diffusion equation with spatial discretizationby finite elementsrdquo IMA Journal of Numerical Analysis vol 31no 2 pp 719ndash739 2011

[83] J Huang N Nie and Y Tang ldquoA second order finite difference-spectral method for space fractional diffusion equationsrdquo Sci-ence China Mathematics vol 57 no 6 pp 1303ndash1317 2014

[84] P Zhuang F Liu VAnh and I Turner ldquoNumericalmethods forthe variable-order fractional advection-diffusion equation witha nonlinear source termrdquo SIAM Journal on Numerical Analysisvol 47 no 3 pp 1760ndash1781 2009

[85] M Cui ldquoConvergence analysis of high-order compact alternat-ing direction implicit schemes for the two-dimensional timefractional diffusion equationrdquo Numerical Algorithms vol 62no 3 pp 383ndash409 2013


[86] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012

[87] P Zhuang and F Liu ldquoFinite difference approximation fortwo-dimensional time fractional diffusion equationrdquo Journal ofAlgorithms amp Computational Technology vol 1 no 1 pp 1ndash152007

[88] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[89] X Liao ldquoMilkyWay-2 back to the world Top 1rdquo Frontiers ofComputer Science vol 8 no 3 pp 343ndash344 2014

[90] S J Pennycook S D Hammond G R Mudalige S A Wrightand S A Jarvis ldquoOn the acceleration of wavefront applicationsusing distributed many-core architecturesrdquo Computer Journalvol 55 no 2 pp 138ndash153 2012

[91] Q Xu G-L Yu K Wang and J-L Sun ldquoResearch on gpu-accelerated algorithm in 3d finite difference neutro n diffusioncalculationmethodrdquoNuclear Science andTechniques vol 25 no1 Article ID 010501 2014

[92] M Rosa J S Warsa and M Perks ldquoA cellwise block-gauss-seidel iterative method for multigroup S

119873transport on a

hybrid parallel computer architecturerdquo Nuclear Science andEngineering vol 174 no 3 pp 209ndash226 2013

[93] A K Jha M A Kupinski H H Barrett E Clarkson andJ H Hartman ldquoThree-dimensional Neumann-series approachto model light transport in nonuniform mediardquo Journal of theOptical Society of America A vol 29 no 9 pp 1885ndash1899 2012

[94] C Xu X Deng L Zhang et al ldquoParallelizing a high-order CFDsoftware for 3D multi-block structural grids on the TianHe-1Asupercomputerrdquo in Supercomputing vol 7905 of Lecture Notesin Computer Science Springer Berlin Germany 2013

[95] Y-X Wang L-L Zhang W Liu et al ldquoEfficient parallel imple-mentation of large scale 3D structured grid CFD applicationson the Tianhe-1A supercomputerrdquo Computers amp Fluids vol 80no 1 pp 244ndash250 2013

[96] QWu C Yang T Tang and L Xiao ldquoExploiting hierarchy par-allelism for molecular dynamics on a petascale heterogeneoussystemrdquo Journal of Parallel and Distributed Computing vol 73no 12 pp 1592ndash1604 2013

[97] F Chen and J Shen ldquoA GPU parallelized spectral method forelliptic equations in rectangular domainsrdquo Journal of Computa-tional Physics vol 250 pp 555ndash564 2013

[98] Z Mo A Zhang X Cao et al ldquoJASMIN a parallel softwareinfrastructure for scientific computingrdquo Frontiers of ComputerScience in China vol 4 no 4 pp 480ndash488 2010

[99] C Xu X Deng L Zhang et al ldquoCollaborating CPU andGPU for large scale high-order CFD for simulations withcomplex grids on the tianhe-1A supercom puterrdquo Journal ofComputational Physics vol 278 pp 275ndash297 2014

[100] R Chen Y Wu Z Yan Y Zhao and X-C Cai ldquoA paralleldomain decomposition method for 3D unsteady incompress-ible flows at high reynolds numberrdquo Journal of Scientific Com-puting vol 58 no 2 pp 275ndash289 2014

[101] H Yang C Yang and X-C Cai ldquoParallel domain decomposi-tion methods with mixed order discretization for fully implicitsolution of tracer transport problems on the cubed-sphererdquoJournal of Scientific Computing 2014

[102] C Yang J Cao and X C Cai ldquoA fully implicit domaindecomposition algorithm for shallow water equations on thecubed-sphererdquo SIAM Journal on Scientific Computing vol 32no 1 pp 418ndash438 2010

[103] C Gong W Bao G Tang Y Jiang and J Liu ldquoA domaindecomposition method for time fractional reaction-diffusionequationrdquo The Scientific World Journal vol 2014 Article ID681707 5 pages 2014

[104] K Diethelm ldquoAn efficient parallel algorithm for the numericalsolution of fractional differential equationsrdquo Fractional Calculusand Applied Analysis vol 14 no 3 pp 475ndash490 2011

[105] C Gong J Liu L Chi H Huang J Fang and Z Gong ldquoGPUaccelerated simulations of 3D deterministic particle transportusing discrete ordinates methodrdquo Journal of ComputationalPhysics vol 230 no 15 pp 6010ndash6022 2011

[106] C Gong J Liu Z Gong J Qin and J Xie ldquoOptimizingSweep3D for graphic processor unitrdquo in Algorithms and Archi-tectures for Parallel Processing C-H Hsu L Yang J Park andS-S Yeo Eds vol 6081 of Lecture Notes in Computer Sciencepp 416ndash426 Springer Berlin Germany 2010

[107] X Gan C Liu Z Wang et al ldquoAccelerating GOR algorithmusing CUDArdquo Applied Mathematics and Information Sciencesvol 7 no 2 pp 563ndash567 2013

[108] C Gong J Liu HHuang and Z Gong ldquoParticle transport withunstructured grid onGPUrdquoComputer Physics Communicationsvol 183 no 3 pp 588ndash593 2012

[109] X Gan Z Wang L Shen and Q Zhu ldquoData layout pruning onGPUrdquo Applied Mathematics amp Information Sciences vol 5 no2 pp 129ndash138 2011

[110] W Zhang W Wei and X Cai ldquoPerformance modeling ofserial and parallel implementations of the fractional Adams-Bashforth-Moulton methodrdquo Fractional Calculus and AppliedAnalysis vol 17 no 3 pp 617ndash637 2014

[111] J Liu C Gong W Bao G Tang and Y Jiang ldquoSolvingthe Caputo fractional reaction-diffusion equation on GPUrdquoDiscrete Dynamics in Nature and Society vol 2014 Article ID820162 2014

[112] Y Xu and Z He ldquoThe short memory principle for solvingAbel differential equation of fractional orderrdquo Computers andMathematics with Applications vol 62 no 12 pp 4796ndash48052011

[113] W Deng ldquoShort memory principle and a predictor-correctorapproach for fractional differential equationsrdquo Journal of Com-putational and AppliedMathematics vol 206 no 1 pp 174ndash1882007

[114] N J Ford and A C Simpson ldquoThe numerical solution of frac-tional differential equations speed versus accuracyrdquo NumericalAlgorithms vol 26 no 4 pp 333ndash346 2001

[115] H-L Cao X Li Z-H Deng and Y Qin ldquoControl-orientedfast numerical approaches of fractional-order modelsrdquo ControlTheory and Applications vol 28 no 5 pp 715ndash721 2011(Chinese)

[116] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifference method for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010

[117] H Wang and K Wang ldquoAn O(Nlog2N) alternating-directionfinite difference method for two-dimensional fractional diffu-sion equationsrdquo Journal of Computational Physics vol 230 no21 pp 7830ndash7839 2011

[118] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013


[119] H-K Pang and H-W Sun ldquoMultigrid method for fractionaldiffusion equationsrdquo Journal of Computational Physics vol 231no 2 pp 693ndash703 2012

[120] Z Zhou and H Wu ldquoFinite element multigrid method forthe boundary value problem of fractional advection dispersionequationrdquo Journal of Applied Mathematics vol 2013 Article ID385403 8 pages 2013

[121] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013

[122] TMoroney andQ Yang ldquoA banded preconditioner for the two-sided nonlinear space-fractional diffusion equationrdquo Comput-ers and Mathematics with Applications vol 66 no 5 pp 659ndash667 2013

[123] T Moroney and Q Yang ldquoEfficient solution of two-sided non-linear space-fractional diffusion equations using fast poissonpreconditionersrdquo Journal of Computational Physics vol 246 pp304ndash317 2013

[124] J A Tenreiro Machado ldquoFractional derivatives probabilityinterpretation and frequency response of rational approxima-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 9-10 pp 3492ndash3497 2009

[125] F-R Lin S-W Yang and X-Q Jin ldquoPreconditioned iterativemethods for fractional diffusion equationrdquo Journal of Compu-tational Physics vol 256 pp 109ndash117 2014

[126] W Deng S Du and Y Wu ldquoHigh order finite differenceWENO schemes for fractional differential equationsrdquo AppliedMathematics Letters vol 26 no 3 pp 362ndash366 2013

[127] H Zhou W Tian and W Deng ldquoQuasi-compact finite differ-ence schemes for space fractional diffusion equationsrdquo Journalof Scientific Computing vol 56 no 1 pp 45ndash66 2013

[128] C Lubich ldquoDiscretized fractional calculusrdquo SIAM Journal onMathematical Analysis vol 17 no 3 pp 704ndash719 1986

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


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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


been presented in the closed form and then their solutionshave been represented graphically Becausemost of fractionalproblems cannot be solved analyticallymore andmoreworksfocus on their numerical solutionsThere aremany numericalsolutions proposed for fractional equations [18] such asfinite difference method (FDM) [18] FDM is intuitive tounderstand and easy to learn for inexperienced researcherfrom the areas rather than mathematics So this surveyfocuses on FDM for fractional equations

For the numerical solutions of different differential equa-tions the area ofmathematics paysmuch attention to approx-imating the equation more accurately and faster (accuracyand speed) The area of computer science mainly focuses onthe runtime (speed) and code reuse (software)Thenumericalmethods (mathematic area) have eternal value and mayexist along with the existence of human culture [19ndash21] Theimplementations (computer area) are closer to the humansociety and the real applications but vary quickly alongwith the computer architecture [22ndash28] The computationalcost of the numerical solutions for fractional equations ismuch heavier than that for the traditional integer orderequations In the near future the fractional problems withhigh dimension long time iterations and huge grid pointswill need to be solved These problems are real challenge fortodayrsquos computer technologies and algorithms

2 Fractional Differential Equations

21 Origins In 1695 LrsquoHopital wrote to Leibniz asking himabout a particular notation he had used in his publications forthe nth-derivative of the linear function 119891(119909) = 119909 119863119899119909119863119909119899[29 30] LrsquoHopitalrsquos posed the question to Leibniz whatwouldthe result be if 119899 = 12 Leibnizrsquos response was ldquoAn apparentparadox from which one day useful consequences will bedrawnrdquo In these words fractional calculus was born [31]Later Fourier Euler and Laplace dabbled with fractionalcalculus [32]

22 A Short Summary of FDE There are mainly three kindsof FDEs

(1) time fractional [33 34](2) space fractional [35](3) space-time fractional [36]

One of the main challenges in fractional differential equationis the nonlocality of the fractional operator In the case of atime fractional derivative one needs to store all the historywhereas in the case of a space fractional derivative one needsto deal with almost-dense matrices

The partial differential equations (PDEs) mainly havethree categories parabolic hyperbolic and elliptic There arecorresponding FDEs to deal with the fractional conditionsThere are various FDEs listed below

(1) the parabolic widely studied fractional diffusionequation [37 38]

(2) the hyperbolic Telegraph equation [39]

(3) the elliptic fractional Laplace equation [40](4) fractional Black-Scholes equation in computational

finance [41](5) fractional wave equation for sound light and water

waves [42](6) fractional Fokker-Planck equations describing the

time evolution of the probability density function ofthe velocity of a particle [43]

(7) fractional Euler equations and fractional Navier-Stokes equations [44] for fluid dynamics [45]

(8) fractional kinetic equations for motion of objects[46]

(9) the perfect fractionalMaxwellrsquos equations for electro-dynamics [47]

(10) fractional Boltzmann equations for particle transport[48]

We can suppose that if there is a PDE there will bea FDE The reason is that replacing the integer derivativewith fractional derivative and solving the FDE with differentnumerical methods is not a hard job

The numerical schemes [49] for these FDEs are listedbelow

(1) finite difference method (FDM) [18 50 51](2) finite element method [52ndash54](3) finite volume method [55ndash57](4) Adomian decomposition method [58](5) Fourier transform method [59](6) spectral method [60](7) meshless method [61ndash63](8) exponential difference method [64 65](9) Monte Carlo method [66]

TheMonte Carlo (MC)method which uses repeated randomsampling to obtain numerical results belongs to undeter-mined computational methods

23 AHot Topic in Recent Years Machado et al [29] collected20 special issues on fractional calculus In 2011 and 2012 thereare about 7 special issues on this topicThere are more specialissues in the nearby four years (2011ndash2014) than those of 1999to 2010 [29] And there are additional 20 special issues inthe years of 2013 and 2014 So the researches on fractionalcalculus can be regarded as a collective revelry

3 Computational Challenge

31 A Classical Partial Differential Equation (PDE)

311 Heat EquationDiffusion Equation The heat equationis a parabolic PDE which describes the distribution of heat(or variation in temperature) in a given region over time



120597119906 (119909 119905)

120597119905= 119889 (119909)

1205972119906 (119909 119905)

1205971199092+ 119891 (119909 119905)

119906 (119909 0) = 120601 (119909) 119909 isin [119909119871 119909119877]

119906 (119909119871 119905) = 120593 (119905) 119905 isin [0 119879]

119906 (119909119877 119905) = 120595 (119905) 119905 isin [0 119879]

(1)



119894= 119894ℎ for 0 le 119899 le




to 119906(119909119894 119905119899) 119891(119909

119894 119905119899) and 119889(119909





119906119899+1

119894minus 119906119899

119894

120591= 119889119894

119906119899

119894+1minus 2119906119899

119894+ 119906119899

119894minus1

ℎ2+ 119891119899

119894 (2)

The 119906119899+1119894


119906119899+1

119894= 1198861119906119899

119894+1+ 1198862119906119899

119894+ 1198863119906119899

119894minus1+ 120591119891119899

119894 (3)

where 1198861= 1198863= minus119889119894120591ℎ2 and 119886

2= 1 minus 2119889

119894120591ℎ2




for (1)

119906119899+1

119894minus 119906119899

119894

120591= 119889119894

119906119899+1

119894+1minus 2119906119899+1

119894+ 119906119899+1

119894minus1

ℎ2+ 119891119899+1

119894 (4)

The 119906119899+1119894


1198861119906119899+1

119894+1+ 1198862119906119899+1

119894+ 1198863119906119899+1

119894minus1= 119906119899

119894+ 120591119891119899

119894 (5)

where 1198861= 1198863= 119889119894120591ℎ2 and 119886

2= 1 + 2119889

119894120591ℎ2




source119891119899119894(or119891119899+1119894







120597120572119906 (119909 119905)

120597120572119905=1205972119906 (119909 119905)

1205971199092(0 lt 120572 lt 1)

119906 (119909 0) = 119892 (119909) 119909 isin (0 119909119877)

119906 (0 119905) = 119906 (119909119877 119905) = 0

(6)



120591minus120572

119899+1

sum

119894=0

119908119894(119906119899+1minus119894

119895minus 1199060

119895) =

1

ℎ2(119906119899

119895+1minus 2119906119899

119895+ 119906119899

119895minus1) (7)


119908119894= (minus1)

119894(120572

119894) (8)


119894=0119908119894 and 119880119899 = (119906

119899

1 119906119899

2

119906119899


119880119899+1

= 119887119899+11198800minus

119899+1

sum

119894=1

119908119894119880119899+1minus119894

+ 120583119860119880119899 (9)



=(

minus2 1

1 minus2 1

∙ ∙

∙ ∙ 1

1 minus2

) (10)





11988110158401015840= 1205831198811 119881119894 = 119908

119894119880119899+1minus119894


+ 119881+

11988110158401015840minus sum119899+1

119894=1119881119894


119899+1needs (119899 +




2

= 119873(119873

2+ 75) (119872 minus 1)

(11)




120597119906 (119909 119905)

120597119905= 119889 (119909)

120597120572119906 (119909 119905)

120597119909120572+ 119902 (119909 119905) (12)





119906119899+1

119894minus 119906119899

119894

120591=119889119894

ℎ120572(

119894+1

sum

119896=0

119892120572119896119906119899

119894minus119896+1) + 119902

119899

119894 (13)

where 119906119899119894= 119906(119909

119894 119905119899) 119889119894= 119889(119909

119894) and 119902119899

119894= 119902(119909

119894 119905119899) The 119892

120572119896is


119892120572119896

=Γ (119896 minus 120572)

Γ (minus120572) Γ (119896 + 1) (14)


to give

119906119899+1

119894= 119906119899

119894+119889119894120591

ℎ120572(

119894+1

sum

119896=0

119892120572119896119906119899

119894minus119896+1) + 120591119902

119899

119894 (15)






120597119906 (119909 119905)

120597119905= minus119906 (119909 119905) +

119909119863120572

0119906 (119909 119905) + 119902 (119909 119905)

119906 (119909 0) = 119892 (119909) 119909 isin (119909119871 119909119877)

119906 (119909119871 119905) = 119906 (119909

119877 119905) = 0

(16)


119909119863120572

0119906(119909 119905) is



119906119899+1

119894minus 119906119899

119894

120591= minus119906119899

119894minus

1

119888ℎ120572

times [

119894+1

sum

119896=0

119892119896119906119899

119894+1minus119896+

119872minus119894+1

sum

119896=0

119892119896119906119899

119894minus1+119896] + 119891

119899

119894

(17)

where 1198920= 1 119892

119896= (minus1)



119894= 119902(119909

119894 119905119899)


119880119899+1

= 119860119880119899+ 119876119899 (18)

where 119880119899 = (1199061198991 119906119899

2 119906

119899

119872minus1)119879 119876119899 = (120591119891119899

1 120591119891119899

2 120591119891

119899

119872minus1)119879

with 119902119899119894= 120591119891119899

119894(1 le 119894 lt 119872 minus 1) and 119860 = (119886



119886119894119895=

minus120591

119888ℎ120572119892119894+1minus119895

for 1 le 119895 lt 119894 minus 1

minus120591

119888ℎ120572(1198920+ 1198922) for 119895 = 119894 minus 1

(1 minus 120591) minus 2120591

119888ℎ1205721198921 for 119895 = 119894

minus120591

119888ℎ120572(1198920+ 1198922) for 119895 = 119894 + 1

minus120591

119888ℎ120572119892119895minus119894+1

for 119894 + 1 lt 119895 lt 119872 minus 1

(19)








1

1205732with 0 lt 120573

1lt 1 1 lt 120573



0119863120572

119905119906 (119909 119905) = 119861

1

1205971205731119906 (119909 119905)

120597 |119909|1205731

+ 1198612

1205971205732119906 (119909 119905)

120597 |119909|1205732

+ 119891 (119909 119905)

119906 (119909 0) = 120601 (119909) 119909 isin [0 119909119877]

119906 (0 119905) = 119906 (119909119877 119905) = 0 119905 isin [0 119879]

(20)





1and 120573

2 respectively


119906119899+1

119894=

119899minus1

sum

119895=0

(119887119895minus 119887119895+1) 119906119899minus119895

119894+ 1198871198991199060

119894+ 11986111205831(

119873minus119894

sum

119896=minus119894

1205961205731

119896119906119899

119894+119896)

+ 11986121205832(

119873minus119894

sum

119896=minus119894

1205961205732

119896119906119899

119894+119896) + 120583

0120593119899

119894 119894 = 1 119873 minus 1

119906119899+1

0= 119906119899+1

119873= 0

1199060

119894= 120601119894

(21)

where 1205830= 120591120572Γ(2 minus 120572) 120583

1= 1205830ℎ1205731 and 120583

2= 1205830ℎ1205732 More



11205831(sum119873minus119894

119896=minus1198941205961205731

119896119906119899

119894+119896) +

11986121205832(sum119873minus119894

119896=minus1198941205961205732

119896119906119899

119894+119896) + 120583

0120593119899



119895=0(119887119895minus 119887119895+1)119906119899minus119895

119894+ 1198871198991199060

119894










0119863120572

119905119906 (119909 119910 119905) = 119886 (119909 119910 119905)

1205972119906 (119909 119910 119905)

1205971199092

+ 119886 (119909 119910 119905)1205972119906 (119909 119910 119905)

1205971199102+ 119891 (119909 119910 119905)

119906 (119909 119910 0) = 120601 (119909 119910) (119909 119910) isin Ω

119906 (119909 119910 119905)1003816100381610038161003816120597Ω = 0 119905 isin [0 119879]

(22)

where Ω = (119909 119910) | 0 le 119909 le 1198711 0 le 119910 le 119871

2 119886(119909 119910 119905) gt 0

and 119887(119909 119910 119905) gt 0


40

35

30

25

20

N



215214213212211210

Valu

e (lo

g 2)


40

35

30

25

20



215214213212211210

Valu

e (lo

g 2)

Mminus 1



119906119899+1

119894119895minus 119906119899

119894119895+

119899

sum

119904=1

(119887119904) (119906119899+1minus119904

119894119895minus 119906119899minus119904

119894119895)

= 1205831Γ (2 minus 120572) 119886

119899

119894119895(119906119899

119894+1119895minus 2119906119899

119894119895+ 119906119899

119894minus1119895)

+ 1205832Γ (2 minus 120572) 119887

119899

119894119895(119906119899

119894119895+1minus 2119906119899

119894119895+ 119906119899

119894119895minus1)

+ 120591120572Γ (2 minus 120572) 119891

119899

119894119895

(23)

where 1205831= 120591120572ℎ2

119909and 120583

2= 120591120572ℎ2

119910 The ℎ

119909and ℎ

119910are















P0My

(0 Py)P0Myminus1

(1 Py)

PMx minus1 My

(Px Py)

PMx My

(1 0)(0 0) (Px 0)











119894becomes


sum119899+1

















5 Future Directions












6 Conclusions




Acknowledgments


References















































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119906 (119909 119905)

120597119905= 119889 (119909)

1205972119906 (119909 119905)

1205971199092+ 119891 (119909 119905)

119906 (119909 0) = 120601 (119909) 119909 isin [119909119871 119909119877]

119906 (119909119871 119905) = 120593 (119905) 119905 isin [0 119879]

119906 (119909119877 119905) = 120595 (119905) 119905 isin [0 119879]

(1)



119894= 119894ℎ for 0 le 119899 le




to 119906(119909119894 119905119899) 119891(119909

119894 119905119899) and 119889(119909





119906119899+1

119894minus 119906119899

119894

120591= 119889119894

119906119899

119894+1minus 2119906119899

119894+ 119906119899

119894minus1

ℎ2+ 119891119899

119894 (2)

The 119906119899+1119894


119906119899+1

119894= 1198861119906119899

119894+1+ 1198862119906119899

119894+ 1198863119906119899

119894minus1+ 120591119891119899

119894 (3)

where 1198861= 1198863= minus119889119894120591ℎ2 and 119886

2= 1 minus 2119889

119894120591ℎ2




for (1)

119906119899+1

119894minus 119906119899

119894

120591= 119889119894

119906119899+1

119894+1minus 2119906119899+1

119894+ 119906119899+1

119894minus1

ℎ2+ 119891119899+1

119894 (4)

The 119906119899+1119894


1198861119906119899+1

119894+1+ 1198862119906119899+1

119894+ 1198863119906119899+1

119894minus1= 119906119899

119894+ 120591119891119899

119894 (5)

where 1198861= 1198863= 119889119894120591ℎ2 and 119886

2= 1 + 2119889

119894120591ℎ2




source119891119899119894(or119891119899+1119894







120597120572119906 (119909 119905)

120597120572119905=1205972119906 (119909 119905)

1205971199092(0 lt 120572 lt 1)

119906 (119909 0) = 119892 (119909) 119909 isin (0 119909119877)

119906 (0 119905) = 119906 (119909119877 119905) = 0

(6)



120591minus120572

119899+1

sum

119894=0

119908119894(119906119899+1minus119894

119895minus 1199060

119895) =

1

ℎ2(119906119899

119895+1minus 2119906119899

119895+ 119906119899

119895minus1) (7)


119908119894= (minus1)

119894(120572

119894) (8)


119894=0119908119894 and 119880119899 = (119906

119899

1 119906119899

2

119906119899


119880119899+1

= 119887119899+11198800minus

119899+1

sum

119894=1

119908119894119880119899+1minus119894

+ 120583119860119880119899 (9)



=(

minus2 1

1 minus2 1

∙ ∙

∙ ∙ 1

1 minus2

) (10)





11988110158401015840= 1205831198811 119881119894 = 119908

119894119880119899+1minus119894


+ 119881+

11988110158401015840minus sum119899+1

119894=1119881119894


119899+1needs (119899 +




2

= 119873(119873

2+ 75) (119872 minus 1)

(11)




120597119906 (119909 119905)

120597119905= 119889 (119909)

120597120572119906 (119909 119905)

120597119909120572+ 119902 (119909 119905) (12)





119906119899+1

119894minus 119906119899

119894

120591=119889119894

ℎ120572(

119894+1

sum

119896=0

119892120572119896119906119899

119894minus119896+1) + 119902

119899

119894 (13)

where 119906119899119894= 119906(119909

119894 119905119899) 119889119894= 119889(119909

119894) and 119902119899

119894= 119902(119909

119894 119905119899) The 119892

120572119896is


119892120572119896

=Γ (119896 minus 120572)

Γ (minus120572) Γ (119896 + 1) (14)


to give

119906119899+1

119894= 119906119899

119894+119889119894120591

ℎ120572(

119894+1

sum

119896=0

119892120572119896119906119899

119894minus119896+1) + 120591119902

119899

119894 (15)






120597119906 (119909 119905)

120597119905= minus119906 (119909 119905) +

119909119863120572

0119906 (119909 119905) + 119902 (119909 119905)

119906 (119909 0) = 119892 (119909) 119909 isin (119909119871 119909119877)

119906 (119909119871 119905) = 119906 (119909

119877 119905) = 0

(16)


119909119863120572

0119906(119909 119905) is



119906119899+1

119894minus 119906119899

119894

120591= minus119906119899

119894minus

1

119888ℎ120572

times [

119894+1

sum

119896=0

119892119896119906119899

119894+1minus119896+

119872minus119894+1

sum

119896=0

119892119896119906119899

119894minus1+119896] + 119891

119899

119894

(17)

where 1198920= 1 119892

119896= (minus1)



119894= 119902(119909

119894 119905119899)


119880119899+1

= 119860119880119899+ 119876119899 (18)

where 119880119899 = (1199061198991 119906119899

2 119906

119899

119872minus1)119879 119876119899 = (120591119891119899

1 120591119891119899

2 120591119891

119899

119872minus1)119879

with 119902119899119894= 120591119891119899

119894(1 le 119894 lt 119872 minus 1) and 119860 = (119886



119886119894119895=

minus120591

119888ℎ120572119892119894+1minus119895

for 1 le 119895 lt 119894 minus 1

minus120591

119888ℎ120572(1198920+ 1198922) for 119895 = 119894 minus 1

(1 minus 120591) minus 2120591

119888ℎ1205721198921 for 119895 = 119894

minus120591

119888ℎ120572(1198920+ 1198922) for 119895 = 119894 + 1

minus120591

119888ℎ120572119892119895minus119894+1

for 119894 + 1 lt 119895 lt 119872 minus 1

(19)








1

1205732with 0 lt 120573

1lt 1 1 lt 120573



0119863120572

119905119906 (119909 119905) = 119861

1

1205971205731119906 (119909 119905)

120597 |119909|1205731

+ 1198612

1205971205732119906 (119909 119905)

120597 |119909|1205732

+ 119891 (119909 119905)

119906 (119909 0) = 120601 (119909) 119909 isin [0 119909119877]

119906 (0 119905) = 119906 (119909119877 119905) = 0 119905 isin [0 119879]

(20)





1and 120573

2 respectively


119906119899+1

119894=

119899minus1

sum

119895=0

(119887119895minus 119887119895+1) 119906119899minus119895

119894+ 1198871198991199060

119894+ 11986111205831(

119873minus119894

sum

119896=minus119894

1205961205731

119896119906119899

119894+119896)

+ 11986121205832(

119873minus119894

sum

119896=minus119894

1205961205732

119896119906119899

119894+119896) + 120583

0120593119899

119894 119894 = 1 119873 minus 1

119906119899+1

0= 119906119899+1

119873= 0

1199060

119894= 120601119894

(21)

where 1205830= 120591120572Γ(2 minus 120572) 120583

1= 1205830ℎ1205731 and 120583

2= 1205830ℎ1205732 More



11205831(sum119873minus119894

119896=minus1198941205961205731

119896119906119899

119894+119896) +

11986121205832(sum119873minus119894

119896=minus1198941205961205732

119896119906119899

119894+119896) + 120583

0120593119899



119895=0(119887119895minus 119887119895+1)119906119899minus119895

119894+ 1198871198991199060

119894










0119863120572

119905119906 (119909 119910 119905) = 119886 (119909 119910 119905)

1205972119906 (119909 119910 119905)

1205971199092

+ 119886 (119909 119910 119905)1205972119906 (119909 119910 119905)

1205971199102+ 119891 (119909 119910 119905)

119906 (119909 119910 0) = 120601 (119909 119910) (119909 119910) isin Ω

119906 (119909 119910 119905)1003816100381610038161003816120597Ω = 0 119905 isin [0 119879]

(22)

where Ω = (119909 119910) | 0 le 119909 le 1198711 0 le 119910 le 119871

2 119886(119909 119910 119905) gt 0

and 119887(119909 119910 119905) gt 0


40

35

30

25

20

N



215214213212211210

Valu

e (lo

g 2)


40

35

30

25

20



215214213212211210

Valu

e (lo

g 2)

Mminus 1



119906119899+1

119894119895minus 119906119899

119894119895+

119899

sum

119904=1

(119887119904) (119906119899+1minus119904

119894119895minus 119906119899minus119904

119894119895)

= 1205831Γ (2 minus 120572) 119886

119899

119894119895(119906119899

119894+1119895minus 2119906119899

119894119895+ 119906119899

119894minus1119895)

+ 1205832Γ (2 minus 120572) 119887

119899

119894119895(119906119899

119894119895+1minus 2119906119899

119894119895+ 119906119899

119894119895minus1)

+ 120591120572Γ (2 minus 120572) 119891

119899

119894119895

(23)

where 1205831= 120591120572ℎ2

119909and 120583

2= 120591120572ℎ2

119910 The ℎ

119909and ℎ

119910are















P0My

(0 Py)P0Myminus1

(1 Py)

PMx minus1 My

(Px Py)

PMx My

(1 0)(0 0) (Px 0)











119894becomes


sum119899+1

















5 Future Directions












6 Conclusions




Acknowledgments


References















































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11988110158401015840= 1205831198811 119881119894 = 119908

119894119880119899+1minus119894


+ 119881+

11988110158401015840minus sum119899+1

119894=1119881119894


119899+1needs (119899 +




2

= 119873(119873

2+ 75) (119872 minus 1)

(11)




120597119906 (119909 119905)

120597119905= 119889 (119909)

120597120572119906 (119909 119905)

120597119909120572+ 119902 (119909 119905) (12)





119906119899+1

119894minus 119906119899

119894

120591=119889119894

ℎ120572(

119894+1

sum

119896=0

119892120572119896119906119899

119894minus119896+1) + 119902

119899

119894 (13)

where 119906119899119894= 119906(119909

119894 119905119899) 119889119894= 119889(119909

119894) and 119902119899

119894= 119902(119909

119894 119905119899) The 119892

120572119896is


119892120572119896

=Γ (119896 minus 120572)

Γ (minus120572) Γ (119896 + 1) (14)


to give

119906119899+1

119894= 119906119899

119894+119889119894120591

ℎ120572(

119894+1

sum

119896=0

119892120572119896119906119899

119894minus119896+1) + 120591119902

119899

119894 (15)






120597119906 (119909 119905)

120597119905= minus119906 (119909 119905) +

119909119863120572

0119906 (119909 119905) + 119902 (119909 119905)

119906 (119909 0) = 119892 (119909) 119909 isin (119909119871 119909119877)

119906 (119909119871 119905) = 119906 (119909

119877 119905) = 0

(16)


119909119863120572

0119906(119909 119905) is



119906119899+1

119894minus 119906119899

119894

120591= minus119906119899

119894minus

1

119888ℎ120572

times [

119894+1

sum

119896=0

119892119896119906119899

119894+1minus119896+

119872minus119894+1

sum

119896=0

119892119896119906119899

119894minus1+119896] + 119891

119899

119894

(17)

where 1198920= 1 119892

119896= (minus1)



119894= 119902(119909

119894 119905119899)


119880119899+1

= 119860119880119899+ 119876119899 (18)

where 119880119899 = (1199061198991 119906119899

2 119906

119899

119872minus1)119879 119876119899 = (120591119891119899

1 120591119891119899

2 120591119891

119899

119872minus1)119879

with 119902119899119894= 120591119891119899

119894(1 le 119894 lt 119872 minus 1) and 119860 = (119886



119886119894119895=

minus120591

119888ℎ120572119892119894+1minus119895

for 1 le 119895 lt 119894 minus 1

minus120591

119888ℎ120572(1198920+ 1198922) for 119895 = 119894 minus 1

(1 minus 120591) minus 2120591

119888ℎ1205721198921 for 119895 = 119894

minus120591

119888ℎ120572(1198920+ 1198922) for 119895 = 119894 + 1

minus120591

119888ℎ120572119892119895minus119894+1

for 119894 + 1 lt 119895 lt 119872 minus 1

(19)








1

1205732with 0 lt 120573

1lt 1 1 lt 120573



0119863120572

119905119906 (119909 119905) = 119861

1

1205971205731119906 (119909 119905)

120597 |119909|1205731

+ 1198612

1205971205732119906 (119909 119905)

120597 |119909|1205732

+ 119891 (119909 119905)

119906 (119909 0) = 120601 (119909) 119909 isin [0 119909119877]

119906 (0 119905) = 119906 (119909119877 119905) = 0 119905 isin [0 119879]

(20)





1and 120573

2 respectively


119906119899+1

119894=

119899minus1

sum

119895=0

(119887119895minus 119887119895+1) 119906119899minus119895

119894+ 1198871198991199060

119894+ 11986111205831(

119873minus119894

sum

119896=minus119894

1205961205731

119896119906119899

119894+119896)

+ 11986121205832(

119873minus119894

sum

119896=minus119894

1205961205732

119896119906119899

119894+119896) + 120583

0120593119899

119894 119894 = 1 119873 minus 1

119906119899+1

0= 119906119899+1

119873= 0

1199060

119894= 120601119894

(21)

where 1205830= 120591120572Γ(2 minus 120572) 120583

1= 1205830ℎ1205731 and 120583

2= 1205830ℎ1205732 More



11205831(sum119873minus119894

119896=minus1198941205961205731

119896119906119899

119894+119896) +

11986121205832(sum119873minus119894

119896=minus1198941205961205732

119896119906119899

119894+119896) + 120583

0120593119899



119895=0(119887119895minus 119887119895+1)119906119899minus119895

119894+ 1198871198991199060

119894










0119863120572

119905119906 (119909 119910 119905) = 119886 (119909 119910 119905)

1205972119906 (119909 119910 119905)

1205971199092

+ 119886 (119909 119910 119905)1205972119906 (119909 119910 119905)

1205971199102+ 119891 (119909 119910 119905)

119906 (119909 119910 0) = 120601 (119909 119910) (119909 119910) isin Ω

119906 (119909 119910 119905)1003816100381610038161003816120597Ω = 0 119905 isin [0 119879]

(22)

where Ω = (119909 119910) | 0 le 119909 le 1198711 0 le 119910 le 119871

2 119886(119909 119910 119905) gt 0

and 119887(119909 119910 119905) gt 0


40

35

30

25

20

N



215214213212211210

Valu

e (lo

g 2)


40

35

30

25

20



215214213212211210

Valu

e (lo

g 2)

Mminus 1



119906119899+1

119894119895minus 119906119899

119894119895+

119899

sum

119904=1

(119887119904) (119906119899+1minus119904

119894119895minus 119906119899minus119904

119894119895)

= 1205831Γ (2 minus 120572) 119886

119899

119894119895(119906119899

119894+1119895minus 2119906119899

119894119895+ 119906119899

119894minus1119895)

+ 1205832Γ (2 minus 120572) 119887

119899

119894119895(119906119899

119894119895+1minus 2119906119899

119894119895+ 119906119899

119894119895minus1)

+ 120591120572Γ (2 minus 120572) 119891

119899

119894119895

(23)

where 1205831= 120591120572ℎ2

119909and 120583

2= 120591120572ℎ2

119910 The ℎ

119909and ℎ

119910are















P0My

(0 Py)P0Myminus1

(1 Py)

PMx minus1 My

(Px Py)

PMx My

(1 0)(0 0) (Px 0)











119894becomes


sum119899+1

















5 Future Directions












6 Conclusions




Acknowledgments


References















































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















1

1205732with 0 lt 120573

1lt 1 1 lt 120573



0119863120572

119905119906 (119909 119905) = 119861

1

1205971205731119906 (119909 119905)

120597 |119909|1205731

+ 1198612

1205971205732119906 (119909 119905)

120597 |119909|1205732

+ 119891 (119909 119905)

119906 (119909 0) = 120601 (119909) 119909 isin [0 119909119877]

119906 (0 119905) = 119906 (119909119877 119905) = 0 119905 isin [0 119879]

(20)





1and 120573

2 respectively


119906119899+1

119894=

119899minus1

sum

119895=0

(119887119895minus 119887119895+1) 119906119899minus119895

119894+ 1198871198991199060

119894+ 11986111205831(

119873minus119894

sum

119896=minus119894

1205961205731

119896119906119899

119894+119896)

+ 11986121205832(

119873minus119894

sum

119896=minus119894

1205961205732

119896119906119899

119894+119896) + 120583

0120593119899

119894 119894 = 1 119873 minus 1

119906119899+1

0= 119906119899+1

119873= 0

1199060

119894= 120601119894

(21)

where 1205830= 120591120572Γ(2 minus 120572) 120583

1= 1205830ℎ1205731 and 120583

2= 1205830ℎ1205732 More



11205831(sum119873minus119894

119896=minus1198941205961205731

119896119906119899

119894+119896) +

11986121205832(sum119873minus119894

119896=minus1198941205961205732

119896119906119899

119894+119896) + 120583

0120593119899



119895=0(119887119895minus 119887119895+1)119906119899minus119895

119894+ 1198871198991199060

119894










0119863120572

119905119906 (119909 119910 119905) = 119886 (119909 119910 119905)

1205972119906 (119909 119910 119905)

1205971199092

+ 119886 (119909 119910 119905)1205972119906 (119909 119910 119905)

1205971199102+ 119891 (119909 119910 119905)

119906 (119909 119910 0) = 120601 (119909 119910) (119909 119910) isin Ω

119906 (119909 119910 119905)1003816100381610038161003816120597Ω = 0 119905 isin [0 119879]

(22)

where Ω = (119909 119910) | 0 le 119909 le 1198711 0 le 119910 le 119871

2 119886(119909 119910 119905) gt 0

and 119887(119909 119910 119905) gt 0


40

35

30

25

20

N



215214213212211210

Valu

e (lo

g 2)


40

35

30

25

20



215214213212211210

Valu

e (lo

g 2)

Mminus 1



119906119899+1

119894119895minus 119906119899

119894119895+

119899

sum

119904=1

(119887119904) (119906119899+1minus119904

119894119895minus 119906119899minus119904

119894119895)

= 1205831Γ (2 minus 120572) 119886

119899

119894119895(119906119899

119894+1119895minus 2119906119899

119894119895+ 119906119899

119894minus1119895)

+ 1205832Γ (2 minus 120572) 119887

119899

119894119895(119906119899

119894119895+1minus 2119906119899

119894119895+ 119906119899

119894119895minus1)

+ 120591120572Γ (2 minus 120572) 119891

119899

119894119895

(23)

where 1205831= 120591120572ℎ2

119909and 120583

2= 120591120572ℎ2

119910 The ℎ

119909and ℎ

119910are















P0My

(0 Py)P0Myminus1

(1 Py)

PMx minus1 My

(Px Py)

PMx My

(1 0)(0 0) (Px 0)











119894becomes


sum119899+1

















5 Future Directions












6 Conclusions




Acknowledgments


References















































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










40

35

30

25

20

N



215214213212211210

Valu

e (lo

g 2)


40

35

30

25

20



215214213212211210

Valu

e (lo

g 2)

Mminus 1



119906119899+1

119894119895minus 119906119899

119894119895+

119899

sum

119904=1

(119887119904) (119906119899+1minus119904

119894119895minus 119906119899minus119904

119894119895)

= 1205831Γ (2 minus 120572) 119886

119899

119894119895(119906119899

119894+1119895minus 2119906119899

119894119895+ 119906119899

119894minus1119895)

+ 1205832Γ (2 minus 120572) 119887

119899

119894119895(119906119899

119894119895+1minus 2119906119899

119894119895+ 119906119899

119894119895minus1)

+ 120591120572Γ (2 minus 120572) 119891

119899

119894119895

(23)

where 1205831= 120591120572ℎ2

119909and 120583

2= 120591120572ℎ2

119910 The ℎ

119909and ℎ

119910are















P0My

(0 Py)P0Myminus1

(1 Py)

PMx minus1 My

(Px Py)

PMx My

(1 0)(0 0) (Px 0)











119894becomes


sum119899+1

















5 Future Directions












6 Conclusions




Acknowledgments


References















































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










P0My

(0 Py)P0Myminus1

(1 Py)

PMx minus1 My

(Px Py)

PMx My

(1 0)(0 0) (Px 0)











119894becomes


sum119899+1

















5 Future Directions












6 Conclusions




Acknowledgments


References















































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















5 Future Directions












6 Conclusions




Acknowledgments


References















































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












6 Conclusions




Acknowledgments


References















































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









review article computational challenge of fractional...

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