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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 696179, 15 pages
http://dx.doi.org/10.1155/2012/696179
Research Article

On the Numerical Solution of Fractional Parabolic Partial Differential Equations with the Dirichlet Condition

1Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey
2Department of Mathematics, ITTU, Ashgabat, Turkmenistan, Turkey
3Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey

Received 13 April 2012; Revised 14 June 2012; Accepted 12 July 2012

Academic Editor: Seenith Sivasundaram

Copyright © 2012 Allaberen Ashyralyev and Zafer Cakir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The first and second order of accuracy stable difference schemes for the numerical solution of the mixed problem for the fractional parabolic equation are presented. Stability and almost coercive stability estimates for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of one-dimensional fractional parabolic partial differential equations.

1. Introduction

It is known that various problems in fluid mechanics (dynamics, elasticity) and other areas of physics lead to fractional partial differential equations. Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers (see, e.g., [128] and the references therein).

The role played by stability inequalities (well posedness) in the study of boundary value problems for parabolic partial differential equations is well known (see, e.g., [2934]). In the present paper, the mixed boundary value problem for the fractional parabolic equation 𝜕𝑢(𝑡,𝑥)𝜕𝑡+𝐷𝑡1/2𝑢(𝑡,𝑥)𝑚𝑝=1𝑎𝑝(𝑥)𝑢𝑥𝑝𝑥𝑝𝑥=𝑓(𝑡,𝑥),𝑥=1,,𝑥𝑚Ω,0<𝑡<𝑇,𝑢(𝑡,𝑥)=0,𝑥𝑆,𝑢(0,𝑥)=0,𝑥Ω(1.1) is considered. Here 𝐷𝑡1/2=𝐷1/20+ is the standard Riemann-Liouville's derivative of order 1/2 and Ω is the open cube in the 𝑚-dimensional Euclidean space 𝑚𝑥𝑥Ω𝑥=1,,𝑥𝑚;0<𝑥𝑗<1,1𝑗𝑚(1.2) with boundary 𝑆,Ω=Ω𝑆,𝑎𝑝(𝑥)(𝑥Ω) and 𝑓(𝑡,𝑥)(𝑡(0,𝑇),𝑥Ω) are given smooth functions and 𝑎𝑝(𝑥)𝑎>0.

The first and second order of accuracy in 𝑡 and second orders of accuracy in space variables difference schemes for the approximate solution of problem (1.1) are presented. The stability and almost coercive stability estimates for the solution of these difference schemes are established. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of one-dimensional fractional parabolic partial differential equations.

2. Difference Schemes and Stability Estimates

The discretization of problem (1.1) is carried out in two steps. In the first step, let us define the grid space Ω=𝑥=𝑥𝑝=1𝑝1,,𝑚𝑝𝑚𝑝,𝑝=1,,𝑝𝑚,0𝑝𝑗𝑀𝑗,𝑗𝑀𝑗,Ω=1,𝑗=1,,𝑚=ΩΩ,𝑆=Ω𝑆.(2.1) We introduce the Hilbert space 𝐿2=𝐿2(Ω) of the grid function 𝜑(𝑥)={𝜑(1𝑗1,,𝑚𝑗𝑚)} defined on Ω, equipped with the norm 𝜑𝐿2(Ω)=𝑥Ω||𝜑(||𝑥)21𝑚1/2.(2.2) To the differential operator 𝐴𝑥 generated by problem (1.1), we assign the difference operator 𝐴𝑥 by the formula 𝐴𝑥𝑢=𝑚𝑝=1𝑎𝑝(𝑥)𝑢𝑥𝑝𝑥𝑝,𝑗𝑝(2.3) acting in the space of grid functions 𝑢(𝑥), satisfying the conditions 𝑢(𝑥)=0 for all 𝑥𝑆. It is known that 𝐴𝑥 is a self-adjoint positive definite operator in 𝐿2(Ω). Here, 𝜑𝑥𝑝,𝑗𝑝=1𝑝𝜑1𝑗1,,𝑗𝑗𝑗+1,,𝑚𝑗𝑚𝜑1𝑗1,,𝑗𝑗𝑗,,𝑚𝑗𝑚,𝜑𝑥𝑝,𝑗𝑝=1𝑝𝜑1𝑗1,,𝑗𝑗𝑗,,𝑚𝑗𝑚𝜑1𝑗1,,𝑗𝑗𝑗1,,𝑚𝑗𝑚.(2.4) With the help of 𝐴𝑥, we arrive at the initial boundary value problem 𝑑𝑣(𝑡,𝑥)𝑑𝑡+𝐷𝑡1/2𝑣(𝑡,𝑥)+𝐴𝑥𝑣(𝑡,𝑥)=𝑓(𝑡,𝑥),0<𝑡<𝑇,𝑥Ω,𝑣(0,𝑥)=0,𝑥Ω(2.5) for a finite system of ordinary fractional differential equations.

In the second step, applying the first order of approximation formula 𝐷𝑡1/2𝑘𝑢𝑘=1𝜋𝑘𝑟=1Γ(𝑘𝑟+1/2)𝑢(𝑘𝑟)!𝑟𝑢𝑟1𝜏1/2(2.6) for 𝐷𝑡1/2𝑘𝑢𝑡𝑘=1Γ(1/2)𝑡𝑘0𝑡𝑘𝑠1/2𝑢(𝑠)𝑑𝑠(2.7) (see [35]) and using the first order of accuracy stable difference scheme for parabolic equations, one can present the first order of accuracy difference scheme with respect to 𝑡𝑢𝑘(𝑥)𝑢𝑘1(𝑥)𝜏+𝐷𝑡1/2𝑘𝑢𝑘(𝑥)+𝐴𝑥𝑢𝑘(𝑥)=𝑓𝑘(𝑥),𝑥Ω,𝑓𝑘(𝑥)=𝑓𝑡𝑘,𝑥,𝑡𝑘𝑢=𝑘𝜏,1𝑘𝑁,𝑁𝜏=𝑇,0(𝑥)=0,𝑥Ω(2.8) for the approximate solution of problem (2.5). Here Γ1𝑘𝑟+2=0𝑡𝑘𝑟+1/2𝑒𝑡𝑑𝑡.(2.9)

Moreover, applying the second order of approximation formula 𝐷𝑡1/2𝑘𝜏/2𝑢𝑘=223𝜋𝜏𝑢0+223𝜋𝜏𝑢1+2𝜏3𝜋𝑢(0),𝑘=1,6𝜋𝜏45𝑢0+25𝑢1+25𝑢26𝜏5𝜋𝑢𝑑(0),𝑘=2,𝑘1𝑚=2(𝑘𝑚)𝑏1(𝑘𝑚)+𝑏2𝑢(𝑘𝑚)𝑚2+(2𝑚2𝑘1)𝑏1(𝑘𝑚)2𝑏2𝑢(𝑘𝑚)𝑚1+(𝑘𝑚+1)𝑏1(𝑘𝑚)+𝑏2𝑢(𝑘𝑚)𝑚+𝑐𝑢𝑘24𝑢𝑘1+5𝑢𝑘,3𝑘𝑁(2.10) for 𝐷𝑡1/2𝑘𝜏/2𝑢𝑡𝑘𝜏2=1Γ(1/2)𝑡𝑘0𝜏/2𝑡𝑘𝜏2𝑠1/2𝑢(𝑠)𝑑𝑠(2.11) (see [27]) and the Crank-Nicholson difference scheme for parabolic equations, one can present the second order of accuracy difference scheme with respect to 𝑡 and to 𝑥 and 𝑢𝑘(𝑥)𝑢𝑘1(𝑥)𝜏+𝐷𝑡1/2𝑘𝑢𝑘1(𝑥)+2𝐴𝑥𝑢𝑘(𝑥)+𝑢𝑘1(𝑥)=𝑓𝑘(𝑥),𝑥Ω,𝑓𝑘𝑡(𝑥)=𝑓𝑘𝜏2,𝑥,𝑡𝑘𝑢=𝑘𝜏,1𝑘𝑁,𝑁𝜏=𝑇,0(𝑥)=0,𝑥Ω(2.12) for the approximate solution of problem (2.5). Here and in the future 2𝑑=𝜋𝜏,𝑐=26𝜋𝜏,𝑏1(𝑟)=1𝑟+21𝑟2,𝑏21(𝑟)=31𝑟+23/21𝑟23/2.(2.13)

Theorem 2.1. Let 𝜏 and ||=21++2𝑛 be sufficiently small positive numbers. Then, the solutions of difference scheme (2.8) and (2.12) satisfy the following stability estimate: max1𝑘𝑁𝑢𝑘𝐿2𝐶1max1𝑘𝑁𝑓𝑘𝐿2,(2.14) where 𝐶1 does not depend on 𝜏, and 𝑓𝑘, 1𝑘𝑁.

Proof. We consider the difference scheme (2.8). We have that 𝑢𝑘(𝑥)=𝑘𝑠=1𝑅𝑘𝑠+1𝐹𝑠(𝑥)𝜏,1𝑘𝑁,(2.15) where 𝑅=𝐼+𝜏𝐴𝑥1,𝐹𝑘(𝑥)=𝑓𝑘(𝑥)𝐷𝑡1/2𝑘𝑢𝑘𝐷(𝑥),𝑡1/2𝑘𝑢𝑘(1𝑥)=𝜋𝑘𝑚=1Γ(𝑘𝑚+1/2)𝜏(𝑘𝑚)!1/2𝐷𝑡1/2𝑚𝑢𝑚(𝑥)+𝑓𝑚(.𝑥)(2.16) Using formula (2.15), we can write 𝑢𝑘(𝑥)=𝑘𝑠=1𝑅𝑘𝑠+1𝐷𝑡1/2𝑠𝑢𝑠(𝑥)+𝑓𝑠𝜏(𝑥)=𝑘𝑠=1𝑅𝑘𝑠+1𝐷𝑡1/2𝑠𝑢𝑠(𝑥)𝜏+𝑘𝑠=1𝑅𝑘𝑠+1𝑓𝑠(𝑥)𝜏,1𝑘𝑁.(2.17) First, we will prove that max1𝑘𝑁𝐷𝑡1/2𝑘𝑢𝑘𝐿2𝑀max1𝑘𝑁𝑓𝑘𝐿2.(2.18) Using formula (2.17), we get 𝑢𝑘(𝑥)𝑢𝑘1(𝑥)𝜏=𝐷𝑡1/2𝑘𝑢𝑘(𝑥)+𝑓𝑘(𝑥)𝐴𝑥𝑢𝑘(𝑥)=𝐷𝑡1/2𝑘𝑢𝑘(𝑥)+𝑓𝑘(𝑥)+𝑘𝑠=1𝐴𝑥𝑅𝑘𝑠+1𝐷𝑡1/2𝑠𝑢s(𝑥)𝜏𝑘𝑠=1𝐴𝑥𝑅𝑘𝑠+1𝑓𝑠(𝑥)𝜏.(2.19) Using formulas (2.16) and (2.19), we obtain 𝐷𝑡1/2𝑘𝑢𝑘1(𝑥)=𝜋𝑘𝑚=1Γ(𝑘𝑚+1/2)𝑢(𝑘𝑚)!𝑚(𝑥)𝑢𝑚1(𝑥)𝜏1/2=1𝜋𝑘𝑚=1Γ(𝑘𝑚+1/2)𝜏(𝑘𝑚)!1/2𝐷𝑡1/2𝑚𝑢𝑚(𝑥)+𝑓𝑚+1(𝑥)𝜋𝑘𝑘𝑠=1𝑚=𝑠Γ(𝑘𝑚+1/2)𝜏(𝑘𝑚)!3/2𝐴𝑥𝑅𝑚𝑠+1𝐷𝑡1/2𝑠𝑢𝑠1(𝑥)𝜋𝑘𝑘𝑠=1𝑚=𝑠Γ(𝑘𝑚+1/2)(𝜏𝑘𝑚)!3/2𝐴𝑥𝑅𝑚𝑠+1𝑓𝑠(𝑥).(2.20) Now, let us estimate 𝑧𝑘=𝐷𝑡1/2𝑘𝑢𝑘𝐿2,1𝑘𝑁. Applying the triangle inequality and the estimate [34] 𝐴𝑥𝑅𝑘𝐿2𝐿2𝑀,𝑅𝑘𝜏𝑘𝐿2𝐿2𝑀,1𝑘𝑁,(2.21) we get 𝑧𝑘1𝜋𝑘𝑚=1Γ(𝑘𝑚+1/2)𝜏(𝑘𝑚)!1/2𝑧𝑚+𝑓𝑚𝐿2+1𝜋𝑘𝑠=1𝑘𝑚=𝑠Γ(𝑘𝑚+1/2)𝐴(𝑘𝑚)!𝑥𝑅𝑚𝑠+1𝐿2𝐿2𝑧𝑠𝜏3/2+1𝜋𝑘𝑠=1𝑘𝑚=𝑠Γ(𝑘𝑚+1/2)𝐴(𝑘𝑚)!𝑥𝑅𝑚𝑠+1𝐿2𝐿2𝑓𝑠𝐿2𝜏3/2𝑀3𝑘1𝑠=11(𝜏𝑧𝑘𝑠)𝜏𝑠+𝑓𝑠𝐿2+𝑀4𝑧𝑠+𝑓𝑠𝐿2𝜏1/2(2.22) for any 𝑘=1,,𝑁. Then, using the difference analogy of integral inequality, we get (2.18).
Second, applying formula (2.17), estimates (2.18) and (2.21), we obtain 𝑢𝑘𝐿2=𝑘𝑠=1𝑅𝑘𝑠+1𝐿2𝐿2𝐷𝑡1/2𝑠𝑢𝑠𝐿2𝜏+𝑘𝑠=1𝑅𝑘𝑠+1𝐿2𝐿2𝑓𝑠𝐿2𝜏𝐶1max1𝑘𝑁𝑓𝑘𝐿2.(2.23) Estimate (2.14) for the solution of (2.8) is proved. The proof of estimate (2.14) for the solution of (2.12) follows the scheme of the proof of estimate (2.14) for the solution of (2.8) and rely on the estimate 𝐴𝑥𝐵𝑘𝐶2𝐿2𝐿21,𝐵𝑘𝜏𝑘𝐿2𝐿21,1𝑘𝑁.(2.24) Here, 𝜏𝐵=𝐼2𝐴𝑥𝜏𝐼+2𝐴𝑥1𝜏,𝐶=𝐼+2𝐴𝑥1.(2.25) Theorem 2.1 is proved.

Theorem 2.2. Let 𝜏 and ||=21++2𝑛 be sufficiently small positive numbers. Then, the solutions of difference scheme (2.8) satisfy the following almost coercive stability estimate: max1𝑘𝑁𝑢𝑘𝑢𝑘1𝜏𝐿2+max𝑚1𝑘𝑁𝑝=1𝑢𝑘𝑥𝑝𝑥𝑝,𝑗𝑝𝐿2𝐶21ln||||𝜏+max1𝑘𝑁𝑓𝑘𝐿2,(2.26) where 𝐶2 is independent of 𝜏, and 𝑓𝑘, 1𝑘𝑁.

Proof. We will prove the estimate max1𝑘𝑁𝑢𝑘𝑢𝑘1𝜏𝐿21𝑀minln𝜏|||𝐴,1+ln𝑥𝐿2𝐿2|||max1𝑘𝑁𝑓𝑘𝐿2.(2.27) Using formula (2.19) and estimate (2.21), we obtain max𝑚1𝑘𝑁𝑠=1𝐴𝑥𝑅𝑘𝑠+1𝑓𝑠𝜏𝐿21𝑀minln𝜏|||𝐴,1+ln𝑥𝐿2𝐿2|||max1𝑘𝑁𝑓𝑘𝐿2,max𝑚1𝑘𝑁𝑠=1𝐴𝑥𝑅𝑘𝑠+1𝐷𝑡1/2𝑠𝑢𝑠𝜏𝐿21𝑀minln𝜏|||𝐴,1+ln𝑥𝐿2𝐿2|||max1𝑘𝑁𝐷𝑡1/2𝑘𝑢𝑘𝜏𝐿2(2.28) and estimate (2.18), the triangle inequality and equation (2.8), we get (2.27). From that it follows: max1𝑘𝑁𝐴𝑥𝑢𝑘𝐿2𝑀11minln𝜏|||𝐴,1+ln𝑥𝐿2𝐿2|||max1k𝑁𝑓𝑘𝐿2.(2.29) Then, the proof of estimate (2.26) is based on estimates (2.27), (2.29), and the following theorem on coercivity inequality for the solution of the elliptic difference problem in 𝐿2.
Theorem  2.3.  For the solutions of the elliptic difference problem𝐴𝑥𝑢(𝑥)=𝑤(𝑥),𝑥Ω,𝑢(𝑥)=0,𝑥𝑆(2.30)the following coercivity inequality holds (see [14, 36])𝑚𝑝=1𝑢𝑥𝑝𝑥𝑝,𝑗𝑝𝐿2𝑤𝐶𝐿2,(2.31)where 𝐶 does not depend on and 𝑤.
Theorem 2.2 is proved.

Theorem 2.4. Let 𝜏 and ||=21++2𝑚 be sufficiently small positive numbers. Then, the solutions of difference scheme (2.12) satisfy the following almost coercive stability estimate: max1𝑘𝑁𝑢𝑘𝑢𝑘1𝜏𝐿2+max1𝑘𝑁12𝑚𝑝=1𝑢𝑘+𝑢𝑘1𝑥𝑝𝑥𝑝,𝑗𝑝𝐿2𝐶31ln||||𝜏+max1𝑘𝑁𝑓𝑘𝐿2,(2.32) where 𝐶3 does not depend on 𝜏, and 𝑓𝑘, 1𝑘𝑁.

The proof of Theorem 2.4 follows the proof of Theorem 2.2 and on the estimate (2.24) and the self-adjointness and positive definiteness of operator 𝐴𝑥 in 𝐿2 and Theorem  2.3.

Remark 2.5. The stability estimates of Theorems 2.1, 2.2, and 2.4 are satisfied in the case of operator 𝐴𝑢=𝑛𝑘=1𝑎𝑘𝜕(𝑥)2𝑢𝜕𝑥2𝑘+𝑛𝑘=1𝑏𝑘(𝑥)𝜕𝑢𝜕𝑥𝑘+𝑐(𝑥)𝑢(2.33) with Dirichlet condition 𝑢=0 in 𝑆. In this case, 𝐴 is not self-adjoint operator in 𝐻. Nevertheless, 𝐴𝑢=𝐴0𝑢+𝐵𝑢 and 𝐴0 is a self-adjoint positive definite operator in 𝐻 and 𝐵𝐴01 is bounded in 𝐻. The proof of this statement is based on the abstract results of [14] and difference analogy of integral inequality.
The method of proofs of Theorems 2.1, 2.2, and 2.4 enables us to obtain the estimate of convergence of difference schemes of the first and second order of accuracy for approximate solutions of the initial-boundary value problem 𝜕𝑢(𝑡,𝑥)𝜕𝑡𝑛𝑝=1𝑎𝑝(𝑥)𝑢𝑥𝑝𝑥𝑝+𝑛𝑝=1𝑏𝑝(𝑥)𝑢𝑥𝑝+𝐷𝛼𝑡𝑢(𝑡,𝑥)=𝑓𝑡,𝑥;𝑢(𝑡,𝑥),𝑢𝑥1(𝑡,𝑥),,𝑢𝑥𝑛,𝑥(𝑡,𝑥)𝑥=1,,𝑥𝑛Ω,0<𝑡<𝑇,𝑢(0,𝑥)=0,𝑥Ω,𝑢(𝑡,𝑥)=0,𝑥𝑆(2.34) for semilinear fractional parabolic partial differential equations.

Note that, one has not been able to obtain a sharp estimate for the constant figuring in the stability estimates of Theorems 2.1, 2.2, and 2.4. Therefore, our interest in the present paper is studying the difference schemes (2.8) and (2.12) by numerical experiments. Applying these difference schemes, the numerical methods are proposed in the following section for solving the one-dimensional fractional parabolic partial differential equation. The method is illustrated by numerical experiments.

3. Numerical Results

For the numerical result, the mixed problem 𝜕𝑢(𝑡,𝑥)𝜕𝑡+𝐷𝑡1/2𝜕𝑢(𝑡,𝑥)𝜕𝑥(1+𝑥)𝜕𝑢(𝑡,𝑥)𝜕𝑥=𝑓(𝑡,𝑥),𝑓(𝑡,𝑥)=3+16𝑡5𝜋+𝜋2𝑡𝑡(1+𝑥)2sin𝜋𝑥𝜋𝑡3cos𝜋𝑥,0<𝑡<1,0<𝑥<1,𝑢(𝑡,0)=𝑢(𝑡,1)=0,0𝑡1,𝑢(0,𝑥)=0,0𝑥1(3.1) for the one-dimensional fractional parabolic partial differential equation is considered. The exact solution of problem (3.1) is 𝑢(𝑡,𝑥)=𝑡3sin𝜋𝑥.(3.2) First, applying difference scheme (2.8), we obtain 𝑢𝑘𝑛𝑢𝑛𝑘1𝜏+1𝜋𝑘𝑟=1Γ(𝑘𝑟+1/2)𝑢(𝑘𝑟)!𝑟𝑛𝑢𝑛𝑟1𝜏1/211+𝑥𝑛+1𝑢𝑘𝑛+1𝑢𝑘𝑛1+𝑥𝑛𝑢𝑘𝑛𝑢𝑘𝑛1=𝜑𝑘𝑛,𝜑𝑘𝑛𝑡=𝑓𝑘,𝑥𝑛,𝑡𝑘=𝑘𝜏,𝑥𝑛𝑢=𝑛,1𝑘𝑁,1𝑛𝑀1,𝑘0=𝑢𝑘𝑀𝑢=0,0𝑘𝑁,0𝑛=0,0𝑛𝑀.(3.3) We can rewrite it in the system of equations with matrix coefficients 𝐴𝑈𝑛+1+𝐵𝑈𝑛+𝐶𝑈𝑛1=𝐷𝜑𝑛𝑈,1𝑛𝑀1,0=̃0,𝑈𝑀=̃0.(3.4) Here and in the future ̃0 is the (𝑁+1)×1 zero matrix and 𝐴=𝑎𝑛𝐷,𝐶=𝑐𝑛𝐷, 𝐷=0000001000001000001000001(𝑁+1)×(𝑁+1),𝑏𝐵=11𝑏000021𝑏22𝑏00031𝑏32𝑏33𝑏00𝑁,1𝑏𝑁,2𝑏𝑁,3𝑏𝑁,𝑁0𝑏𝑁+1,1𝑏𝑁+1,2𝑏𝑁+1,3𝑏𝑁+1,𝑁𝑏𝑁+1,𝑁+1(𝑁+1)×(𝑁+1),𝜑𝑛=𝜑0𝑛𝜑1𝑛𝜑2𝑛𝜑𝑛𝑁1𝜑𝑁𝑛(𝑁+1)×1,𝑈𝑞=𝑢0𝑞𝑢1𝑞𝑢2𝑞𝑢𝑞𝑁1𝑢𝑁𝑞(𝑁+1)×1𝑎,𝑞=𝑛±1,𝑛,𝑛=1+𝑥𝑛+12,𝑐𝑛=1+𝑥𝑛2,𝑏11=1,𝑏211=𝜏1𝜏,𝑏22=1𝜏+1𝜏+2+𝑥𝑛+1+𝑥𝑛2,𝑏31=Γ(1+1/2)𝜋𝜏,𝑏32=Γ(1+1/2)Γ(1/2)1𝜋𝜏𝜏,𝑏33=1𝜏+1𝜏+2+𝑥𝑛+1+𝑥𝑛2,𝑏𝑖𝑗=Γ(𝑖2+1/2),𝜋𝜏(𝑖2)!𝑗=1,Γ(𝑖𝑗+1/2)𝜋𝜏(𝑖𝑗)!Γ(𝑖𝑗1+1/2),𝜋𝜏(𝑖𝑗1)!2𝑗𝑖2,Γ(1+1/2)Γ(1/2)1𝜋𝜏𝜏,1𝑗=𝑖1,𝜏+1𝜏+2+𝑥𝑛+1+𝑥𝑛2,𝑗=𝑖,0,𝑖<𝑗𝑁+1(3.5) for 𝑖=4,5,,𝑁+1 and 𝜑𝑘𝑛=3+16𝑘𝜏5𝜋+𝜋2(𝑘𝜏)(1+𝑛)(𝑘𝜏)2sin(𝜋𝑛)𝜋(𝑘𝜏)3cos(𝜋𝑛).(3.6)

So, we have the second-order difference equation with respect to 𝑛 matrix coefficients. This type system was developed by Samarskii and Nikolaev [37]. To solve this difference equation we have applied a procedure for difference equation with respect to 𝑘 matrix coefficients. Hence, we seek a solution of the matrix equation in the following form: 𝑈𝑗=𝛼𝑗+1𝑈𝑗+1+𝛽𝑗+1,𝑈𝑀=0,𝑗=𝑀1,,2,1,(3.7) where 𝛼𝑗(𝑗=1,2,,𝑀) are (𝑁+1)×(𝑁+1) square matrices and 𝛽𝑗(𝑗=1,2,,𝑀) are (𝑁+1)×1 column matrices defined by 𝛼𝑗+1=𝐵+𝐶𝛼𝑗1𝛽𝐴,(3.8)𝑗+1=𝐵+𝐶𝛼𝑗1𝐷𝜑𝑗𝐶𝛽𝑗,𝑗=1,2,,𝑀1,(3.9) where 𝑗=1,2,,𝑀1, 𝛼1 is the (𝑁+1)×(𝑁+1) zero matrix and 𝛽1 is the (𝑁+1)×1 zero matrix.

Second, applying difference scheme (2.12), we obtain 𝑢𝑘𝑛𝑢𝑛𝑘1𝜏+𝐷𝑡1/2𝑘𝜏/2𝑢𝑘𝑛121+𝑥𝑛𝑢𝑘𝑛+12𝑢𝑘𝑛+𝑢𝑘𝑛12+𝑢𝑘𝑛+1𝑢𝑘𝑛1+21+𝑥𝑛𝑢𝑘1𝑛+12𝑢𝑛𝑘1+𝑢𝑘1𝑛12+𝑢𝑘1𝑛+1𝑢𝑘1𝑛12=𝜑𝑘𝑛,𝜑𝑘𝑛𝑡=𝑓𝑘𝜏2,𝑥𝑛,𝑡𝑘=𝑘𝜏,𝑥𝑛𝑢=𝑛,1𝑘𝑁,1𝑛𝑀1,𝑘0=𝑢𝑘𝑀𝑢=0,0𝑘𝑁,0𝑛=0,0𝑛𝑀,(3.10) where 𝐷𝑡1/2𝑘𝜏/2𝑢𝑘𝑛=223𝜋𝜏𝑢0𝑛+223𝜋𝜏𝑢1𝑛+2𝜏3𝜋𝑢0,𝑥𝑛,𝑘=1,6𝜋𝜏45𝑢0𝑛+25𝑢1𝑛+25𝑢2𝑛6𝜏5𝜋𝑢0,𝑥𝑛𝑑,𝑘=2,𝑘1𝑚=2(𝑘𝑚)𝑏1(𝑘𝑚)+𝑏2𝑢(𝑘𝑚)𝑛𝑚2+(2𝑚2𝑘1)𝑏1(𝑘𝑚)2𝑏2(𝑢𝑘𝑚)𝑛𝑚1+(𝑘𝑚+1)𝑏1(𝑘𝑚)+𝑏2𝑢(𝑘𝑚)𝑚𝑛+𝑐𝑢𝑛𝑘24𝑢𝑛𝑘1+5𝑢𝑘𝑛,3𝑘𝑁(3.11) for any 𝑛, 1𝑛𝑀1. We get the system of equations in the matrix form 𝐴𝑈𝑛+1+𝐵𝑈𝑛+𝐶𝑈𝑛1=𝐷𝜑𝑛𝑈,1𝑛𝑀1,0=̃0,𝑈𝑀=̃0,(3.12) where 𝐴=𝑎𝑛𝐹,𝐶=𝑐𝑛𝐹, 𝐹=0000011000011000001000011(𝑁+1)×(𝑁+1),𝑏𝐵=11𝑏000021𝑏22𝑏00031𝑏32𝑏33𝑏00𝑁,1𝑏𝑁,2𝑏𝑁,3𝑏𝑁,𝑁0𝑏𝑁+1,1𝑏𝑁+1,2𝑏𝑁+1,3𝑏𝑁+1,𝑁𝑏𝑁+1,𝑁+1(𝑁+1)×(𝑁+1),𝐷=0000001000001000001000001(𝑁+1)×(𝑁+1),𝜑𝑛=𝜑0𝑛𝜑1𝑛𝜑2𝑛𝜑𝑛𝑁1𝜑𝑁𝑛(𝑁+1)×1,𝑈𝑞=𝑈0𝑞𝑈1𝑞𝑈2𝑞𝑈𝑞𝑁1𝑈𝑁𝑞(𝑁+1)×1𝑎,𝑞=𝑛±1,𝑛,𝑛1=21+𝑥𝑛2+12,𝑐𝑛1=21+𝑥𝑛21,𝑏211=1,𝑏212=231𝜋𝜏𝜏+1+𝑥𝑛2,𝑏22=223+1𝜋𝜏𝜏+1+𝑥𝑛2,𝑏31=465𝜋𝜏,𝑏32=2651𝜋𝜏𝜏+1+𝑥𝑛2,𝑏33=265+1𝜋𝜏𝜏+1+𝑥𝑛2,𝑏41=𝑑1𝑏1(1)+𝑏2(1),𝑏42=𝑑3𝑏1(1)2𝑏2𝑏(1)𝑐,43=𝑑2𝑏1(1)+𝑏21(1)4𝑐𝜏+1+𝑥𝑛2,𝑏441=5𝑐+𝜏+1+𝑥𝑛2,𝑏51=𝑑2𝑏1(2)+𝑏2(2),𝑏52=𝑑5𝑏1(2)2𝑏2(2)+1𝑏1(1)+𝑏2,𝑏(1)53=𝑑3𝑏1(2)+𝑏2(2)3𝑏1(1)2𝑏2𝑏(1)𝑐,54=𝑑2𝑏1(1)+𝑏21(1)4𝑐𝜏+1+𝑥𝑛2,𝑏551=5𝑐+𝜏+1+𝑥𝑛2,𝑏𝑖𝑗=𝑑(𝑖3)𝑏1(𝑖3)+𝑏2,𝑑(𝑖3)𝑗=1,(52𝑖)𝑏1(𝑖3)2𝑏2(𝑖3)+(𝑖4)𝑏1(𝑖4)+𝑏2,𝑑(𝑖4)𝑗=2,(𝑖𝑗+1)𝑏1(𝑖𝑗)+𝑏2(𝑖𝑗)+(2𝑗2𝑖+1)𝑏1(𝑖𝑗1)2𝑏2(𝑖𝑗1)+(𝑖𝑗2)𝑏1(𝑖𝑗2)+𝑏2,𝑑(𝑖𝑗2)3𝑗𝑖3,3𝑏1(2)+𝑏2(2)3𝑏1(1)2𝑏2𝑑(1)𝑐,𝑗=𝑖2,2𝑏1(1)+𝑏21(1)4𝑐𝜏+1+𝑥𝑛2,1𝑗=𝑖1,5𝑐+𝜏+1+𝑥𝑛2,𝑗=𝑖,0,𝑖<𝑗𝑁+1(3.13) for 𝑖=6,7,,𝑁+1 and 𝜑𝑘𝑛=3+16𝑘𝜏5𝜋+𝜋2(𝑘𝜏)(1+𝑛)(𝑘𝜏)2sin(𝜋𝑛)𝜋(𝑘𝜏)3cos(𝜋𝑛).(3.14) So, we have again the second-order difference equation with respect to 𝑛 matrix coefficients. Therefore, applying the same procedure of modified Gauss elimination method (3.7) and (3.8) difference equation (3.12).

Finally, we give the results of the numerical analysis. The numerical solutions are recorded for different values of 𝑁 and 𝑀 and 𝑢𝑘𝑛 represents the numerical solutions of these difference schemes at (𝑡𝑘,𝑥𝑛). The error is computed by the following formula: 𝐸𝑁𝑀=max1𝑘𝑁,1𝑛𝑀1||𝑢𝑡𝑘,𝑥𝑛𝑢𝑘𝑛||.(3.15) Table 1 is constructed for 𝑁=𝑀=20,40, and 80, respectively.

tab1
Table 1: Error analysis.

Thus, by using the Crank-Nicholson difference scheme, the accuracy of solution increases faster than the first order of accuracy difference scheme.

4. Conclusion

In this study, the first and second order of accuracy stable difference schemes for the numerical solution of the mixed problem for the fractional parabolic equation are investigated. We have obtained stability and almost coercive stability estimates for the solution of these difference schemes. The theoretical statements for the solution of these difference schemes for one-dimensional parabolic equations are supported by numerical example in computer. We showed that the second order of accuracy difference scheme is more accurate comparing with the first order of accuracy difference scheme.

Acknowledgment

The authors are grateful to Professor Pavel E. Sobolevskii for his comments and suggestions to improve the quality of the paper.

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