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Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 129404 5 pageshttpdxdoiorg1011552013129404

Research ArticleA Weighted Average Finite Difference Method for the FractionalConvection-Diffusion Equation

Lijuan Su and Pei Cheng

School of Mathematical Sciences Anhui University Hefei Anhui 230601 China

Correspondence should be addressed to Lijuan Su sulijuan8362163com

Received 18 January 2013 Revised 8 April 2013 Accepted 3 June 2013

Academic Editor R de la Llave

Copyright copy 2013 L Su and P Cheng 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

A weighted average finite difference method for solving the two-sided space-fractional convection-diffusion equation is givenwhich is an extension of the weighted average method for ordinary convection-diffusion equations Stability consistency andconvergence of the new method are analyzed A simple and accurate stability criterion valid for this method arbitrary weightedfactor and arbitrary fractional derivative is given Some numerical examples with known exact solutions are provided

1 Introduction

Thehistory of the fractional derivatives and integrals can dateback to the 17th century However only after 124 years laterLacroix first put forward a result of the simplest fractional cal-culus Nowadays the fractional derivatives and integrals havemany important applications in various fields of physics [1ndash3] finance [4 5] hydrology [6] engineering [7] mathematics[8] science and so forth

Anomalous diffusion is perhaps themost frequently stud-ied complex problem Classical (integer-order) partial differ-ential equation of diffusion and wave has been extended tothe equation with fractional time andor space by means offractional operator [9] Furthermore it has been extended tothe problems of every kind of nonlinear fractional differentialequations and to present the solutions to the problems ofinitial and boundary values subject to above equations isanother rapidly developing field of fractional operator appli-cations In general all of these equations have importantbackground of practice applications such as dispersion infractals and porous media [10] semiconductor turbulenceand condensed matter physics

As a special case of anomalous diffusion the two-sidedspace-fractional convection-diffusion equation for the force-free case is usually written in the following way [11]

120597119906 (119909 119905)

120597119905= minus119881 (119909 119905)

120597119906 (119909 119905)

120597119909+ 119863+

(119909 119905)120597120572119906 (119909 119905)

120597+119909120572

+ 119863minus

(119909 119905)120597120572119906 (119909 119905)

120597minus119909120572

+ 119904 (119909 119905)

119871 le 119909 le 119877 0 lt 119905 le 119879

119906 (119909 119905 = 0) = 1199060

(119909) 119871 le 119909 le 119877

119906 (119871 119905) = 119906 (119877 119905) = 0 0 le 119905 le 119879

(1)

where 119881(119909 119905) gt 0 is the drift of the process that is the meanadvective velocity 120572 is the order of fractional differentiation119863+(119909 119905) = (1 + 120573)119863(119909 119905)2 119863

minus(119909 119905) = (1 minus 120573)119863(119909 119905)2 119863(119909

119905) gt 0 is the coefficient of dispersion and minus1 le 120573 le 1

indicates the relative weight of forward versus backwardtransition probability The function 119906

0(119909) is the initial con-

dition the boundary conditions are zero Dirichlet boundaryconditions and the function 119904(119909 119905) is a sourcesink termThe120597120572119906(119909 119905)120597

+119909120572 and 120597

120572119906(119909 119905)120597

minus119909120572 in (1) are the Riemann-

Liouville fractional derivatives Equation (1) is a special caseof the space-fractional Fokker-Planck equation which moreadequately describes the movement of solute in an aquiferthan the traditional second-order Fokker-Planck equation

The left-sided (+) and the right-sided (minus) fractional deriv-atives in (1) are the Riemann-Liouville fractional derivativesof order 120572 of a function 119891(119909) for 119909 isin [119871 119877] defined by [12]

120597120572119891 (119909)

120597+119909120572

=1

Γ (119899 minus 120572)

119889119899

119889119909119899int

119909

119871

119891 (120585) 119889120585

(119909 minus 120585)120572minus119899+1

2 Advances in Mathematical Physics

120597120572119891 (119909)

120597minus119909120572

=(minus1)119899

Γ (119899 minus 120572)

119889119899

119889119909119899int

119877

119909

119891 (120585) 119889120585

(120585 minus 119909)120572minus119899+1

(2)

where 119899 minus 1 lt 120572 le 119899 (119899 is an integer) and Γ(sdot) is the Gammafunction

In some cases there are somemethods to solve fractionalpartial differential equations and get the analytical solutions[12] such as Fourier transform methods Laplace transformmethods Mellin transform methods the method of imagesand the method of separation of variables In this paper theexact solution of (1) can be obtained by Fourier transformmethodsHowever as in the cases of integer-order differentialequations there are only very few cases of fractional partialdifferential equations in which the closed-form analyticalsolutions are available Therefore numerical means have tobe used in general

Many of researches on the numerical methods for solv-ing fractional partial differential equations have been pro-posed for example L2 or L2C methods [13] standard orshifted Grunwald-Letnikov formulae [14] convolution for-mulae [15] homotopy perturbationmethod and so forth Forexample Langlands and Henry [16] use L1 scheme form [8]to discretize the Riemann-Liouville fractional time derivativeof order between 1 and 2 Yuste [17] considered a Grunwald-Letnikov approximation for the Riemann-Liouville timefractional derivative and used a weighted average for thesecond-order space derivative Lin and Xu [18] proposedthe method based on a finite difference scheme in timeLegendrersquos spectral method in space and so on

In this paper based on shifted Grunwald-Letnikov for-mula we consider a fractional weighted average (FWA)finite difference method which is very close to the classicalWAmethods for ordinary (nonfractional) partial differentialequations The FWA method has some better propertiesthan the fractional explicit and full implicit methods [19]such as higher-order accuracy in time step when weightingcoefficient 120582 = 12

The rest of this paper is organized as follows In Section 2the FWA finite difference method is developed The stabilityand convergence of themethod are proved in Section 3 Somenumerical examples are given in Section 4 Finally we drawour conclusions in Section 5

2 Fractional Weighted Average Methods

To present the new finite difference method we give somenotations Δ119905 is the time step Δ119909 is the spatial step thecoordinates of the mesh points are 119909

119895= 119871 + 119895Δ119909 119895 = 0 1 2

119873 119873 = (119877 minus 119871)Δ119909 and 119905119898

= 119898Δ119905 119898 = 0 1 2 119872119872 = 119879Δ119905 and the values of the solution 119906(119909 119905) at thesegrid points are 119906(119909

119895 119905119898

) equiv 119906119898

119895≃ 119880119898

119895 where we denote by

119880119898

119895the numerical estimate of the exact value of 119906(119909 119905) at

the point (119909119895 119905119898

) Define 119881119898+12

119895= 119881(119909

119895 119905119898+12

) 119863119898+12

119895=

119863(119909119895 119905119898+12

) and 119904119898+12

119895= 119904(119909119895 119905119898+12

)

The centered time difference scheme is [20]

120597119906

120597119905

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898+12)=

119906119898+1

119895minus 119906119898

119895

Δ119905+ 119874(Δ119905)

2

(3)

and the backward space difference scheme is

120597119906

120597119909

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)=

119906119898

119895minus 119906119898

119895minus1

Δ119909+ 119874 (Δ119909) (4)

According to the shifted Grunwald-Letnikov definition[8] the definition (2) can be written as

120597120572119906 (119909)

120597+119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)

=1

ℎ120572

119895+1

sum

119896=0

119892(120572)

119896119906119898

119895minus119896+1+ 119874 (ℎ)

120597120572119906 (119909)

120597minus119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)

=1

ℎ120572

119873minus119895+1

sum

119896=0

119892(120572)

119896119906119898

119895+119896minus1+ 119874 (ℎ)

(5)

Here 119892(120572)

119896= (minus1)

119896

(120572

119896) can be evaluated recursively

119892(120572)

0= 1 119892

(120572)

119896= (1 minus

120572 + 1

119896) 119892(120572)

119896minus1 (6)

In the weighted average method (1) can be evaluated atthe intermediate point of the grid (119909

119895 119905119898+12

) by the followingformula

120597119906 (119909 119905)

120597119905

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898+12)

= minus119881 (119909119895 119905119898+12

) [120582120597119906 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)

+ (1 minus 120582)120597119906 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898+1)]

+ 119863+

(119909119895 119905119898+12

) [120582120597120572119906 (119909 119905)

120597+119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)

+ (1 minus 120582)120597120572119906 (119909 119905)

120597+119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898+1)

]

+ 119863minus

(119909119895 119905119898+12

) [120582120597120572119906 (119909 119905)

120597+119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)

+ (1 minus 120582)120597120572119906 (119909 119905)

120597+119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898+1)

]

+ 119904 (119909 119905) |(119909119895 119905119898+12)

(7)

where 0 le 120582 le 1 is the weighting coefficient

Advances in Mathematical Physics 3

Applying (3)sim(5) to (7) letting ℎ = Δ119909 and neglectingthe truncation error we get the FWA difference scheme

119880119898+1

119895+ (1 minus 120582) [minus 119903

119898+12

119895119880119898+1

119895minus1+ 119903119898+12

119895119880119898+1

119895

minus 120585119898+12

119895

119895+1

sum

119896=0

119892(120572)

119896119880119898+1

119895minus119896+1

minus 120578119898+12

119895

119873minus119895+1

sum

119896=0

119892(120572)

119896119880119898+1

119895+119896minus1]

= 119880119898

119895minus 120582 [minus 119903

119898+12

119895119880119898

119895minus1+ 119903119898+12

119895119880119898

119895

minus 120585119898+12

119895

119895+1

sum

119896=0

119892(120572)

119896119880119898

119895minus119896+1

minus120578119898+12

119895

119873minus119895+1

sum

119896=0

119892(120572)

119896119880119898

119895+119896minus1] + Δ119905119904

119898+12

119895

119895 = 1 2 119873 minus 1 119898 = 0 1 2 119872 minus 1

(8)

where 119903119898+12

119895= 119881119898+12

119895Δ119905ℎ 120585

119898+12

119895= 119863119898+12

+119895Δ119905ℎ120572 120578119898+12

119895=

119863119898+12

minus119895Δ119905ℎ120572 and the initial values are calculated by 119880

(0)

119895=

1199060(119909119895) 119895 = 1 2 119873 minus 1 Generally the quantity 119903

119898+12

119895is

called the Courant (or CFL) number the 120585119898+12

119895and 120578

119898+12

119895

are associated with the diffusion coefficientsObviously the scheme is explicit when 120582 = 1 and the

scheme is fully implicit when 120582 = 0 particularly when120582 = 12 the FWA scheme is called the fractional Crank-Nicholson (FCN) scheme

3 Stability and Accuracy Analysis

In this section we study the stability of the FWAmethod anddiscuss the truncating error According to our analysis wecan get a conclusion which is similar to the result of classicalWAmethods In fact the following theorem can be viewed asa generalization of these stability conditions for classical WAmethods [20]

Lemma 1 The coefficients 119892(120572)

119896given in (6) with 1 lt 120572 le 2

satisfy the following properties

119892(120572)

0= 1 119892

(120572)

1= minus120572 lt 0

1 ge 119892(120572)

2ge 119892(120572)

3ge sdot sdot sdot ge 0

infin

sum

119896=0

119892(120572)

119896= 0

119898

sum

119896=0

119892(120572)

119896le 0 (119898 ge 1)

(9)

Theorem 2 When 0 le 120582 le 12 the FWA (8) is uncondition-ally stable based on the shifted Grunwald approximation (5) tothe fractional equation (1) with 1 lt 120572 le 2 When 12 lt 120582 le 1the FWA (8) is conditionally stable if

120572Δ119905119863maxℎ120572

+Δ119905119881max

ℎle

1

2120582 minus 1 (10)

where 119881max = max119871le119909le1198770le119905le119879

119881(119909 119905) and 119863max =

max119871le119909le1198770le119905le119879

119863(119909 119905)

Proof The FWA scheme (8) can be rewritten as [119868 + (1 minus

120582)119860]119880119898+1

= (119868 minus 120582119860)119880119898 119898 = 0 1 2 119872 minus 1 here

119880119898

= [119880119898

0 119880119898

1 119880119898

2 119880

119898

119873]119879 119860 = (119886

119894119895) 119894 119895 = 0 1 2 119873

The matrix entries 119886119894119895

for 119894 = 1 2 119873 minus 1 and 119895 =

0 1 119873 are defined by

119886119894119895

=

119903119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894) 119892(120572)

1 119895 = 119894

minus119903119898+12

119894minus 120585119898+12

119894119892(120572)

2minus 120578119898+12

119894119892(120572)

0 119895 = 119894 minus 1

minus120585119898+12

119894119892(120572)

0minus 120578119898+12

119894119892(120572)

2 119895 = 119894 + 1

minus120585119898+12

119894119892(120572)

119894minus119895+1 119895 lt 119894 minus 1

minus120578119898+12

119894119892(120572)

119895minus119894+1 119895 gt 119894 + 1

(11)

while 1198860119895

= 119886119873119895

= 0 for 119895 = 0 1 119873According to Lemma 1 and the Gerschgorin theorem

the eigenvalues of the matrix 119860 (noted 120596119894) are in the disks

centered at 119886119894119894

= 119903119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894)119892(120572)

1 with radius

119877119894=

119873

sum

119895=0119895 = 119894

10038161003816100381610038161003816119886119894119895

10038161003816100381610038161003816

= 119903119898+12

119894+ 120585119898+12

119894

119894+1

sum

119896=0119896 = 1

119892(120572)

119896+ 120578119898+12

119894

119873minus119894+1

sum

119896=0119896 = 1

119892(120572)

119896

le 119903119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894) 119892(120572)

1

(12)

Therefore we have

0 le 120596119894le 2 [119903

119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894) 119892(120572)

1]

= 2 [119863119898+12

119894120572Δ119905

ℎ120572+

119881119898+12

119894Δ119905

ℎ]

(13)

Next note that 120596119894is an eigenvalue of 119860 if and only if (1 minus

120582120596119894)(1+(1minus120582)120596

119894) is an eigenvalue of the matrix (119868minus120582119860)[119868+

(1 minus 120582)119860]minus1 Because of 0 le 120596

119894and 0 le 120582 le 1 we get

1 minus 120582120596119894

1 + (1 minus 120582) 120596119894

= 1 minus120596119894

1 + (1 minus 120582) 120596119894

le 1 (14)

In addition (1 minus 120582120596i)(1 + (1 minus 120582)120596i) ge minus1 as long as(2120582 minus 1)120596

119894le 2

Hence when 0 le 120582 le 12 we can find that minus1 le (1 minus

120582120596119894)(1 + (1 minus 120582)120596

119894) le 1 always holds that is |(1 minus 120582120596

119894)(1 +

(1 minus 120582)120596119894)| le 1 Then the FWA (8) is unconditionally stable

On the other hand when 12 lt 120582 le 1 from (13) and120596119894le 2(2120582 minus 1) we get the limited condition (120572Δ119905119863maxℎ

120572) +

(Δ119905119881maxℎ) le 1(2120582 minus 1) where 119881max = max119871le119909le1198770le119905le119879

119881(119909 119905)

and 119863max = max119871le119909le1198770le119905le119879

119863(119909 119905) Therefore the FWA (8) isconditionally stable


minus5 0 5 10 150

02

04

06

08

1

12

x

n = 1000

n = 100

n = 4000

u(xt)

Figure 1 Numerical solutions of (1) by means of the FWA methodfor 120582 = 09 Δ119909 = 140 and 119878 = 12 The numerical solutions areshown after 100 (dots) 1000 (stars) and 4000 (circles) time stepsThe lines correspond to the exact solutions

Let

119878 =120572Δ119905119863max

ℎ120572+

Δ119905119881maxℎ

(15)

the stability limit 119878timesis 119878times

= 1(2120582 minus 1)In addition taking into account (3)sim(5) for arbitrary Δ119909

and Δ119905 we derive that this method is consistent with a localtruncation error 119874(Δ119909 + Δ119905) except for the FCN methodwhose accuracy is of (Δ119905)

2 with respect to the time step [21]Therefore according to Laxrsquos equivalence theorem the FWAmethod converges at the same rate too

Remark 3 Instead of (4) if forward space difference schemeis used Theorem 2 still holds and its proof does not changebasically However if centered space difference scheme isused we cannot obtain the same conclusion as Theorem 2

4 Numerical Simulations

In this section we apply the FWA scheme (8) to solve the two-sided space-fractional convection-diffusion equation (1) with120573 = 0 119881(119909 119905) = 119881 119863(119909 119905) = 119863 and 119904(119909 119905) = 0 the initialcondition is

1199060

(119909) =20

120587int

+infin

0

cos [(119909 minus 01119881) 120585] 11989001119863 cos(1205871205722)120585120572

119889120585 (16)

In this case the analytical solution of (1) solved by theFourier transform methods is [12]119906 (119909 119905)

=20

120587int

+infin

0

cos [(119909 minus 119881 (119905 + 01)) 120585] 119890119863(119905+01) cos(1205871205722)120585120572

119889120585

(17)

In the following numerical experiments the data arechosen as follow 120572 = 19 119863 = 2 119881 = 2 119879 = 25 119871 = minus5 and119877 = 15

minus5 0 5 10 15minus8

minus6

minus4

minus2

0

2

4

6

8

x

Error

times105

Figure 2 The same as Figure 1 but for 119878 = 13 The errors betweennumerical solution and exact solution after 1000 time steps areshown by line

minus5 0 5 10 150

01

02

03

04

05

06

07

x

n = 100

n = 10

n = 50

u(xt)

Figure 3 Numerical solutions of (1) by means of the FCN methodfor Δ119909 = 140 and 119878 = 100 The numerical solutions are shown after10 (dots) 50 (stars) and 100 (circles) time stepsThe lines correspondto the exact solutions

The numerical solutions are obtained from the FWAscheme (8) discussed above with different 120582 Δ119905 119905 119878 and ℎFrom (15) the values of Δ119905 for fixed 119878 and Δ119909 = ℎ are

Δ119905 =119878

((120572119863maxℎ120572) + (119881maxℎ))=

119878

((120572119863ℎ120572) + (119881ℎ))

(18)

The computational results are shown in Figures 1 2 and3 Figures 1 and 2 show two different cases where the FWAmethod is stable and unstable according to the theoreticalpredictions ofTheorem 2 Figure 1 shows numerical solutionsobtained by the FWA method (8) with 120582 = 09 Δ119909 = 140and small 119878 = 12 after 100 1000 and 4000 time stepsThe numerical solutions compare well to the exact solutions


which proves that the FWAmethod is stable At the momentwe gain the very small time step Δ119905 = 28 times 10

minus4 calculatedfrom (18) Figure 2 has the same assumptions as Figure 1 butfor 119878 = 13 after 1000 time steps and the large errors betweennumerical solutions and exact solutions obviously prove thatthe FWA method is unstable In the both figures because of120582 = 09 the stability limit is 119878

times= 1(2120582 minus 1) = 125

Next we consider the special case of 120582 = 12 under theassumption that the FWAmethod becomes the FCNmethodFigure 3 shows numerical solutions obtained by the FCNmethod with Δ119909 = 140 and large 119878 = 100 after 10 50 and100 time steps Meanwhile we can gain the large time stepΔ119905 = 23 times 10

minus2 calculated from (18) which is much largerthan Δ119905 = 28 times 10

minus4 in Figure 1 The numerical solutionsapproximate well to the exact solutions and the FCNmethodis always stable so it allows the large time steps to be used

5 Conclusions

Based on the shifted Grunwald approximation to the frac-tional derivative we propose the FWA method in this paperwhich can be viewed as a generalization of the classical WAmethods for ordinary diffusion equations [17]The stability ofthe FWAmethod depends on weighting parameter 120582 and itsaccuracy is of order 119874(Δ119909 + Δ119905) except for the FCNmethodwhose accuracy with respect to the time step is of (Δ119905)

2 (see[21])

Obviously the FCN method is much better and moreconvenient than the fractional explicit and fully implicitmethods because it is not only unconditionally stable but alsoof second-order accuracy in time

Acknowledgments

This research was supported by the National Natural ScienceFoundations of China (Grants nos 11126179 and 11226247)the 211 Project of Anhui University (nos 02303319 and12333010266) the Scientific Research Award for ExcellentMiddle-Aged andYoung Scientists of ShandongProvince (noBS2010HZ012) and the Nature Science Foundation of AnhuiProvincial (no 1308085QA15) The authors acknowledge theanonymous reviewers for their helpful comments

References

[1] M de la Sen ldquoPositivity and stability of the solutions ofCaputo fractional linear time-invariant systems of any orderwith internal point delaysrdquo Abstract and Applied Analysis vol2011 Article ID 161246 25 pages 2011

[2] H Yang ldquoExistence of mild solutions for a class of fractionalevolution equations with compact analytic semigrouprdquoAbstractandAppliedAnalysis vol 2012 Article ID 903518 15 pages 2012

[3] A Ashyralyev ldquoA note on fractional derivatives and fractionalpowers of operatorsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 232ndash236 2009

[4] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A vol 314 no 1ndash4 pp 749ndash755 2002

[5] L Sabatelli S Keating J Dudley and P Richmond ldquoWaitingtime distributions in financial marketsrdquo The European PhysicalJournal B vol 27 no 2 pp 273ndash275 2002

[6] B Baeumer D A Benson M M Meerschaert and S WWheatcraft ldquoSubordinated advection-dispersion equation forcontaminant transportrdquo Water Resources Research vol 37 no6 pp 1543ndash1550 2001

[7] R LMagin Fractional Calculus in Bioengineering Begell HousePublishers New York NY USA 2006

[8] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[9] F Mainardi ldquoFractional relaxation-oscillation and fractionaldiffusion-wave phenomenardquo Chaos Solitons and Fractals vol7 no 9 pp 1461ndash1477 1996

[10] B I Henry and S L Wearne ldquoFractional reaction-diffusionrdquoPhysica A vol 276 no 3-4 pp 448ndash455 2000

[11] D A Benson S W Wheatcraft and M M Meerschaert ldquoThefractional-order governing equation of Levy motionrdquo WaterResources Research vol 36 no 6 pp 1413ndash1423 2000

[12] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[13] F Liu V Anh and I Turner ldquoNumerical solution of the spacefractional Fokker-Planck equationrdquo Journal of Computationaland Applied Mathematics vol 166 no 1 pp 209ndash219 2004

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

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

[16] T A M Langlands and B I Henry ldquoThe accuracy and stabilityof an implicit solution method for the fractional diffusionequationrdquo Journal of Computational Physics vol 205 no 2 pp719ndash736 2005

[17] S B Yuste ldquoWeighted average finite differencemethods for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 216 no 1 pp 264ndash274 2006

[18] 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

[19] M M Meerschaert and C Tadjeran ldquoFinite difference approxi-mations for two-sided space-fractional partial differential equa-tionsrdquoApplied Numerical Mathematics vol 56 no 1 pp 80ndash902006

[20] K W Morton and D F Mayers Numerical Solution of PartialDifferential Equations CambridgeUniversity Press CambridgeUK 1994

[21] L Su W Wang and Z Yang ldquoFinite difference approximationsfor the fractional advection-diffusion equationrdquo Physics LettersA vol 373 pp 4405ndash4408 2009

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


120597120572119891 (119909)

120597minus119909120572

=(minus1)119899

Γ (119899 minus 120572)

119889119899

119889119909119899int

119877

119909

119891 (120585) 119889120585

(120585 minus 119909)120572minus119899+1

(2)

where 119899 minus 1 lt 120572 le 119899 (119899 is an integer) and Γ(sdot) is the Gammafunction

In some cases there are somemethods to solve fractionalpartial differential equations and get the analytical solutions[12] such as Fourier transform methods Laplace transformmethods Mellin transform methods the method of imagesand the method of separation of variables In this paper theexact solution of (1) can be obtained by Fourier transformmethodsHowever as in the cases of integer-order differentialequations there are only very few cases of fractional partialdifferential equations in which the closed-form analyticalsolutions are available Therefore numerical means have tobe used in general

Many of researches on the numerical methods for solv-ing fractional partial differential equations have been pro-posed for example L2 or L2C methods [13] standard orshifted Grunwald-Letnikov formulae [14] convolution for-mulae [15] homotopy perturbationmethod and so forth Forexample Langlands and Henry [16] use L1 scheme form [8]to discretize the Riemann-Liouville fractional time derivativeof order between 1 and 2 Yuste [17] considered a Grunwald-Letnikov approximation for the Riemann-Liouville timefractional derivative and used a weighted average for thesecond-order space derivative Lin and Xu [18] proposedthe method based on a finite difference scheme in timeLegendrersquos spectral method in space and so on

In this paper based on shifted Grunwald-Letnikov for-mula we consider a fractional weighted average (FWA)finite difference method which is very close to the classicalWAmethods for ordinary (nonfractional) partial differentialequations The FWA method has some better propertiesthan the fractional explicit and full implicit methods [19]such as higher-order accuracy in time step when weightingcoefficient 120582 = 12

The rest of this paper is organized as follows In Section 2the FWA finite difference method is developed The stabilityand convergence of themethod are proved in Section 3 Somenumerical examples are given in Section 4 Finally we drawour conclusions in Section 5

2 Fractional Weighted Average Methods

To present the new finite difference method we give somenotations Δ119905 is the time step Δ119909 is the spatial step thecoordinates of the mesh points are 119909

119895= 119871 + 119895Δ119909 119895 = 0 1 2

119873 119873 = (119877 minus 119871)Δ119909 and 119905119898

= 119898Δ119905 119898 = 0 1 2 119872119872 = 119879Δ119905 and the values of the solution 119906(119909 119905) at thesegrid points are 119906(119909

119895 119905119898

) equiv 119906119898

119895≃ 119880119898

119895 where we denote by

119880119898

119895the numerical estimate of the exact value of 119906(119909 119905) at

the point (119909119895 119905119898

) Define 119881119898+12

119895= 119881(119909

119895 119905119898+12

) 119863119898+12

119895=

119863(119909119895 119905119898+12

) and 119904119898+12

119895= 119904(119909119895 119905119898+12

)

The centered time difference scheme is [20]

120597119906

120597119905

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898+12)=

119906119898+1

119895minus 119906119898

119895

Δ119905+ 119874(Δ119905)

2

(3)

and the backward space difference scheme is

120597119906

120597119909

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)=

119906119898

119895minus 119906119898

119895minus1

Δ119909+ 119874 (Δ119909) (4)

According to the shifted Grunwald-Letnikov definition[8] the definition (2) can be written as

120597120572119906 (119909)

120597+119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)

=1

ℎ120572

119895+1

sum

119896=0

119892(120572)

119896119906119898

119895minus119896+1+ 119874 (ℎ)

120597120572119906 (119909)

120597minus119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)

=1

ℎ120572

119873minus119895+1

sum

119896=0

119892(120572)

119896119906119898

119895+119896minus1+ 119874 (ℎ)

(5)

Here 119892(120572)

119896= (minus1)

119896

(120572

119896) can be evaluated recursively

119892(120572)

0= 1 119892

(120572)

119896= (1 minus

120572 + 1

119896) 119892(120572)

119896minus1 (6)

In the weighted average method (1) can be evaluated atthe intermediate point of the grid (119909

119895 119905119898+12

) by the followingformula

120597119906 (119909 119905)

120597119905

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898+12)

= minus119881 (119909119895 119905119898+12

) [120582120597119906 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)

+ (1 minus 120582)120597119906 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898+1)]

+ 119863+

(119909119895 119905119898+12

) [120582120597120572119906 (119909 119905)

120597+119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)

+ (1 minus 120582)120597120572119906 (119909 119905)

120597+119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898+1)

]

+ 119863minus

(119909119895 119905119898+12

) [120582120597120572119906 (119909 119905)

120597+119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898)

+ (1 minus 120582)120597120572119906 (119909 119905)

120597+119909120572

10038161003816100381610038161003816100381610038161003816(119909119895 119905119898+1)

]

+ 119904 (119909 119905) |(119909119895 119905119898+12)

(7)

where 0 le 120582 le 1 is the weighting coefficient



119880119898+1

119895+ (1 minus 120582) [minus 119903

119898+12

119895119880119898+1

119895minus1+ 119903119898+12

119895119880119898+1

119895

minus 120585119898+12

119895

119895+1

sum

119896=0

119892(120572)

119896119880119898+1

119895minus119896+1

minus 120578119898+12

119895

119873minus119895+1

sum

119896=0

119892(120572)

119896119880119898+1

119895+119896minus1]

= 119880119898

119895minus 120582 [minus 119903

119898+12

119895119880119898

119895minus1+ 119903119898+12

119895119880119898

119895

minus 120585119898+12

119895

119895+1

sum

119896=0

119892(120572)

119896119880119898

119895minus119896+1

minus120578119898+12

119895

119873minus119895+1

sum

119896=0

119892(120572)

119896119880119898

119895+119896minus1] + Δ119905119904

119898+12

119895

119895 = 1 2 119873 minus 1 119898 = 0 1 2 119872 minus 1

(8)

where 119903119898+12

119895= 119881119898+12

119895Δ119905ℎ 120585

119898+12

119895= 119863119898+12

+119895Δ119905ℎ120572 120578119898+12

119895=

119863119898+12


(0)

119895=


119898+12

119895is


119895and 120578

119898+12

119895








119892(120572)

0= 1 119892

(120572)

1= minus120572 lt 0

1 ge 119892(120572)

2ge 119892(120572)


infin

sum

119896=0

119892(120572)

119896= 0

119898

sum

119896=0

119892(120572)

119896le 0 (119898 ge 1)

(9)


120572Δ119905119863maxℎ120572

+Δ119905119881max

ℎle

1

2120582 minus 1 (10)


119881(119909 119905) and 119863max =

max119871le119909le1198770le119905le119879

119863(119909 119905)


120582)119860]119880119898+1

= (119868 minus 120582119860)119880119898 119898 = 0 1 2 119872 minus 1 here

119880119898

= [119880119898

0 119880119898

1 119880119898

2 119880

119898

119873]119879 119860 = (119886

119894119895) 119894 119895 = 0 1 2 119873


for 119894 = 1 2 119873 minus 1 and 119895 =


119886119894119895

=

119903119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894) 119892(120572)

1 119895 = 119894

minus119903119898+12

119894minus 120585119898+12

119894119892(120572)

2minus 120578119898+12

119894119892(120572)

0 119895 = 119894 minus 1

minus120585119898+12

119894119892(120572)

0minus 120578119898+12

119894119892(120572)

2 119895 = 119894 + 1

minus120585119898+12

119894119892(120572)

119894minus119895+1 119895 lt 119894 minus 1

minus120578119898+12

119894119892(120572)

119895minus119894+1 119895 gt 119894 + 1

(11)

while 1198860119895

= 119886119873119895



centered at 119886119894119894

= 119903119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894)119892(120572)

1 with radius

119877119894=

119873

sum

119895=0119895 = 119894

10038161003816100381610038161003816119886119894119895

10038161003816100381610038161003816

= 119903119898+12

119894+ 120585119898+12

119894

119894+1

sum

119896=0119896 = 1

119892(120572)

119896+ 120578119898+12

119894

119873minus119894+1

sum

119896=0119896 = 1

119892(120572)

119896

le 119903119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894) 119892(120572)

1

(12)

Therefore we have

0 le 120596119894le 2 [119903

119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894) 119892(120572)

1]

= 2 [119863119898+12

119894120572Δ119905

ℎ120572+

119881119898+12

119894Δ119905

ℎ]

(13)


120582120596119894)(1+(1minus120582)120596



119894and 0 le 120582 le 1 we get

1 minus 120582120596119894

1 + (1 minus 120582) 120596119894

= 1 minus120596119894

1 + (1 minus 120582) 120596119894

le 1 (14)


119894le 2


120582120596119894)(1 + (1 minus 120582)120596


119894)(1 +



120572) +


119881(119909 119905)




minus5 0 5 10 150

02

04

06

08

1

12

x

n = 1000

n = 100

n = 4000

u(xt)


Let

119878 =120572Δ119905119863max

ℎ120572+

Δ119905119881maxℎ

(15)








1199060

(119909) =20

120587int

+infin

0

cos [(119909 minus 01119881) 120585] 11989001119863 cos(1205871205722)120585120572

119889120585 (16)


=20

120587int

+infin

0

cos [(119909 minus 119881 (119905 + 01)) 120585] 119890119863(119905+01) cos(1205871205722)120585120572

119889120585

(17)



minus6

minus4

minus2

0

2

4

6

8

x

Error

times105


minus5 0 5 10 150

01

02

03

04

05

06

07

x

n = 100

n = 10

n = 50

u(xt)



Δ119905 =119878

((120572119863maxℎ120572) + (119881maxℎ))=

119878

((120572119863ℎ120572) + (119881ℎ))

(18)





times= 1(2120582 minus 1) = 125




5 Conclusions


2 (see[21])


Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880119898+1

119895+ (1 minus 120582) [minus 119903

119898+12

119895119880119898+1

119895minus1+ 119903119898+12

119895119880119898+1

119895

minus 120585119898+12

119895

119895+1

sum

119896=0

119892(120572)

119896119880119898+1

119895minus119896+1

minus 120578119898+12

119895

119873minus119895+1

sum

119896=0

119892(120572)

119896119880119898+1

119895+119896minus1]

= 119880119898

119895minus 120582 [minus 119903

119898+12

119895119880119898

119895minus1+ 119903119898+12

119895119880119898

119895

minus 120585119898+12

119895

119895+1

sum

119896=0

119892(120572)

119896119880119898

119895minus119896+1

minus120578119898+12

119895

119873minus119895+1

sum

119896=0

119892(120572)

119896119880119898

119895+119896minus1] + Δ119905119904

119898+12

119895

119895 = 1 2 119873 minus 1 119898 = 0 1 2 119872 minus 1

(8)

where 119903119898+12

119895= 119881119898+12

119895Δ119905ℎ 120585

119898+12

119895= 119863119898+12

+119895Δ119905ℎ120572 120578119898+12

119895=

119863119898+12


(0)

119895=


119898+12

119895is


119895and 120578

119898+12

119895








119892(120572)

0= 1 119892

(120572)

1= minus120572 lt 0

1 ge 119892(120572)

2ge 119892(120572)


infin

sum

119896=0

119892(120572)

119896= 0

119898

sum

119896=0

119892(120572)

119896le 0 (119898 ge 1)

(9)


120572Δ119905119863maxℎ120572

+Δ119905119881max

ℎle

1

2120582 minus 1 (10)


119881(119909 119905) and 119863max =

max119871le119909le1198770le119905le119879

119863(119909 119905)


120582)119860]119880119898+1

= (119868 minus 120582119860)119880119898 119898 = 0 1 2 119872 minus 1 here

119880119898

= [119880119898

0 119880119898

1 119880119898

2 119880

119898

119873]119879 119860 = (119886

119894119895) 119894 119895 = 0 1 2 119873


for 119894 = 1 2 119873 minus 1 and 119895 =


119886119894119895

=

119903119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894) 119892(120572)

1 119895 = 119894

minus119903119898+12

119894minus 120585119898+12

119894119892(120572)

2minus 120578119898+12

119894119892(120572)

0 119895 = 119894 minus 1

minus120585119898+12

119894119892(120572)

0minus 120578119898+12

119894119892(120572)

2 119895 = 119894 + 1

minus120585119898+12

119894119892(120572)

119894minus119895+1 119895 lt 119894 minus 1

minus120578119898+12

119894119892(120572)

119895minus119894+1 119895 gt 119894 + 1

(11)

while 1198860119895

= 119886119873119895



centered at 119886119894119894

= 119903119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894)119892(120572)

1 with radius

119877119894=

119873

sum

119895=0119895 = 119894

10038161003816100381610038161003816119886119894119895

10038161003816100381610038161003816

= 119903119898+12

119894+ 120585119898+12

119894

119894+1

sum

119896=0119896 = 1

119892(120572)

119896+ 120578119898+12

119894

119873minus119894+1

sum

119896=0119896 = 1

119892(120572)

119896

le 119903119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894) 119892(120572)

1

(12)

Therefore we have

0 le 120596119894le 2 [119903

119898+12

119894minus (120585119898+12

119894+ 120578119898+12

119894) 119892(120572)

1]

= 2 [119863119898+12

119894120572Δ119905

ℎ120572+

119881119898+12

119894Δ119905

ℎ]

(13)


120582120596119894)(1+(1minus120582)120596



119894and 0 le 120582 le 1 we get

1 minus 120582120596119894

1 + (1 minus 120582) 120596119894

= 1 minus120596119894

1 + (1 minus 120582) 120596119894

le 1 (14)


119894le 2


120582120596119894)(1 + (1 minus 120582)120596


119894)(1 +



120572) +


119881(119909 119905)




minus5 0 5 10 150

02

04

06

08

1

12

x

n = 1000

n = 100

n = 4000

u(xt)


Let

119878 =120572Δ119905119863max

ℎ120572+

Δ119905119881maxℎ

(15)








1199060

(119909) =20

120587int

+infin

0

cos [(119909 minus 01119881) 120585] 11989001119863 cos(1205871205722)120585120572

119889120585 (16)


=20

120587int

+infin

0

cos [(119909 minus 119881 (119905 + 01)) 120585] 119890119863(119905+01) cos(1205871205722)120585120572

119889120585

(17)



minus6

minus4

minus2

0

2

4

6

8

x

Error

times105


minus5 0 5 10 150

01

02

03

04

05

06

07

x

n = 100

n = 10

n = 50

u(xt)



Δ119905 =119878

((120572119863maxℎ120572) + (119881maxℎ))=

119878

((120572119863ℎ120572) + (119881ℎ))

(18)





times= 1(2120582 minus 1) = 125




5 Conclusions


2 (see[21])


Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus5 0 5 10 150

02

04

06

08

1

12

x

n = 1000

n = 100

n = 4000

u(xt)


Let

119878 =120572Δ119905119863max

ℎ120572+

Δ119905119881maxℎ

(15)








1199060

(119909) =20

120587int

+infin

0

cos [(119909 minus 01119881) 120585] 11989001119863 cos(1205871205722)120585120572

119889120585 (16)


=20

120587int

+infin

0

cos [(119909 minus 119881 (119905 + 01)) 120585] 119890119863(119905+01) cos(1205871205722)120585120572

119889120585

(17)



minus6

minus4

minus2

0

2

4

6

8

x

Error

times105


minus5 0 5 10 150

01

02

03

04

05

06

07

x

n = 100

n = 10

n = 50

u(xt)



Δ119905 =119878

((120572119863maxℎ120572) + (119881maxℎ))=

119878

((120572119863ℎ120572) + (119881ℎ))

(18)





times= 1(2120582 minus 1) = 125




5 Conclusions


2 (see[21])


Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












times= 1(2120582 minus 1) = 125




5 Conclusions


2 (see[21])


Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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