Finite Difference Method of Fractional Parabolic Partial Differential Equations with Variable Coefficients

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1 International Journal of Contemporary Mathematical Sciences Vol. 9, 014, no. 16, HIKARI Ltd, Finite Difference Method of Fractional Parabolic Partial Differential Equations with Variable Coefficients Ibrahim Karatay and Nurdane Kale Department of Mathematics, Fatih University Hadimoy Campus, Hadimoy road 34500, Buyucemece, Istanbul, Turey Copyright c 014 Ibrahim Karatay and Nurdane Kale. 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 wor is properly cited. Abstract In this study, we consider the fractional parabolic partial differential equations that variable coefficients with the Caputo fractional derivative. We constructed a difference scheme based on Cran- Nicholson method. And stability of this difference scheme is proved conditionally. Mathematics Subject Classification: 65N06, 65N1, 65M1 Keywords: Difference scheme, Stability, Caputo fractional derivative 1 Introduction In this study, we propose a Cran-Nicholson type of difference scheme and we see that under some conditions the difference scheme is stable for the following time fractional partial differential problem that variable coefficients; α ux,t = a x ux,t t α x c x ux, t fx, t, 0 < x < 1, 0 < t < 1, ux, 0 = rx, 0 x 1, u0, t = 0, u1, t = 0, 0 t 1. b x ux,t x 1

2 768 Ibrahim Karatay and Nurdane Kale α ux, t 1 = t α Γ1 α where Γ. is the Gamma function. t 0 u t x, τ dτ, 0 < α < 1, t τ α We can obtain some ind of subproblems from this problem which general type. One can see that if bx = 0 and cx = 0 then the problem transforms to time fractional diffusion problem that variable coefficient, if we choose bx = 0 and cx > 0 then the problem will be a time-fractional cable problem that variable coefficient when opposite chosen which means cx = 0 and bx > 0 the problem is called fractional advection dispersion problem that variable coefficient. Discretization of the Problem For some positive integers M and N, the grid sizes in space and time for the finite difference algorithm are defined by h = 1/M and τ = 1/N, respectively. The grid points in the space interval [0, 1] are the numbers x j = jh, j = 0, 1,,..., M, and the grid points in the time interval [0, 1] are labeled t = τ, = 0, 1,,..., N. The values of the functions u and f at the grid points are denoted u j = ux j, t and fj = fx j, t, respectively. Let ux, t, u t x, t and u tt x, t are continuous on [0, 1]. A discrete approximation to the fractional derivative α ux,t at x t α j, t 1 can be obtained by the following approximation[7]: [ α ux j, t 1 = w t α 1 u Oτ α. ] w m1 w m u m w u 0 u1 j u j 1 α Where = 1 1 and w Γ α τ α j = j 1/ 1 α j 1/ 1 α In addition for = 0 there is no these terms w 1 u and w u 0. On the other hand, we have also approximations for second and first derivative at x j, t 1. [ ux j, t 1 = 1 u 1 j1 ] u1 j u 1 j u j1 u j u j Oh. 4 x h h 3 ux j, t 1 x = 1 [ u 1 j1 ] u1 j u j1 u j h h 5

3 Matrix stability 769 Using these approximations 3, 4 and 5 into 1, we obtain the following difference scheme for 1 which is accurate of order Oτ α h ; [ w 1 u j [ ax j ] w m1 w m u m j w u 0 j u1 j u j 1 α ] u 1 j1 u1 j u 1 j u j1 u j u j bx h h j cx j u1 j u j = fx j, t τ, 0 N 1, 1 j M 1, u 0 j = rx j, 0 j M, u 0 = 0, u M = 0, 0 N. u 1 j1 u1 j u j1 u j 4h 4h ax j u j1 ax j cx j 1 α h ax j bx j = fx j, t τ ax j u 1 u 1 w 1 u j j j1 u j ax j bx j u 1 j, 0 N 1, 1 j M 1, ax j cx j 1 α h u j w m1 w m u m j w u 0 j u 0 j = rx j, 0 j M, u 0 = 0, u M = 0, 0 N. 6 3 Matrix Stability of the Difference Scheme And we consructed this difference scheme from 6 ax j bx j ax j bx j u 1 j u j ax j cx j 1 α h 1 α ax j w 1 u j w m1 w m u m j w u 0 j cx j h u 1 j u j ax j ax j u 1 j1 u j1 = fx j, t τ, 0 N 1, 1 j M 1, u 0 j = rx j, 0 j M, u 0 = 0, u M = 0, 0 N. 7

4 770 Ibrahim Karatay and Nurdane Kale The difference scheme above 7 can be written in matrix form, AU 1 = BU 0 ϕ 0, = 0 AU 1 = BU w 1 U w m w m1 U m w U 0 ϕ, 1 N 1 Uj 0 = rx j, 0 j M, U0 = 0, UM = 0, 0 N. where ϕ = [ϕ 1, ϕ, ϕ 3,..., ϕ M ]T, ϕ 0 j = rx j, ϕ j = fx j, t 1/, 1 N, 1 j M, and U = [ U1, U, U3,..., UM] T.Here, A and B are 3-diagonal M 1 M 1 matrices of the form : p ax j 1 α A = B = ax j bx j p 1 α ax j where p = ax j cx j h p ax j 1 α ax j bx j 1 α p ax j ax j ax j bx j p 1 α ax j bx j ax j ax j 1 α p { M } We denote A = A = max a jm, where A = [a jm ] M M. 1 j M Lemma3.1 If ax j cx j > 0, ax j 1 α h bx j > h ax j > 0, bx j > 0 and cx j > 0,then A B 1. Proof. Therefore, A B A 1 B { a jj M = min 1 j M ax j cx j 1 α h ax j cx j 1 α h cx j 1 α cx j 1 α 1. { ax j { ax j m j, bx j a jm } B ax j bx j ax j } } Lemma3. If ax j cx j w 1 α h 1 > 0, ax j > h, ax bx j j > 0, bx j > 0 and cx j > 0 then A B w 1 I A w 1 1

5 Matrix stability 771 Proof. A B w 1 I w1 A A B w1 I w1 A 1 cx j w 1 α 1 cx j 1 α w 1 cx j 1 α Theorem3.1 The difference scheme 8 is stable. Proof. To prove the conditional stability of 8, let Uj and Vj be the exact and approximate solution of 8 with initial value Uj 0 and Vj 0 respectively. We denote the corresponding error by ε j = Uj Vj and ε = [ε 1, ε,..., ε M ]t where 0 j M, 0 N. Then ε satisfies if = 0 if > 0 Aε 1 = Bε w 1 ε Aε 1 = Bε 0 w m w m1 ε m w ε 0. Let us prove ε ε 0, = 0, 1,,... by induction. In fact, if = 0 ε 1 = A Bε 0 from that ε 1 = A Bε 0 A B ε 0. Since A B 1, from the Lemma 1, we have ε 1 ε 0. If = 1, then we have From the last equation, we obtain ε = A B w 1 Iε 1 w 1 A ε 0. ε = A B w 1 Iε 1 w 1 A ε 0 A B w 1 I ε 1 w1 A ε A B w 1 I ε w1 A ε { A B w 1 I w1 A } ε. If the condition above is satisfied, then ε ε 0 is obtained. Now, assume

6 77 Ibrahim Karatay and Nurdane Kale ε s ε 0 for all s, we will prove it is also true for s = 1. ε 1 = A Bε w 1 ε w m w m1 ε m w ε 0 A B w 1 I ε w m w m1 A ε m w A ε A B w 1 I ε 0 w A ε w m w m1 A ε A B w 1 I ε {w w... w 1 w w } A ε A B w 1 I w1 A ε ε. Therefore, under the condition ax j cx j w 1 α h 1 > 0, ax j > h, ax bx j j > 0, bx j > 0 and cx j > 0, the stability inequality is obtained. 4 Convergence Of The Difference Scheme Theorem4.1 The proposed scheme 8 is convergent and the following estimate holds: e Zατ α h, with the condition ax j cx j w 1 α h 1 > 0, ax j > h, ax bx j j > 0, bx j > 0 and cx j > 0. Here Zα does not depend on τ and h. Proof. Let R = [cτ α h,..., cτ α h ] T M set w 0 = 1. Since e j = u j Uj, notice that e 0 = 0. Firstly, we prove that e 1 wn for all n by induction. We have the following error equation when = 0. e 1 = A Be 0 A R = A R e 1 = A R A w 0. Similarly, for = we have the error equation e = A B w 1 Ie 1 w 1 A e 0 A R If we tae the norm of this equality, we obtain e = A B w 1 I e 1 A R [ A B w 1 I A w 1 ].w 1 w 1.

7 Matrix stability 773 Assume the inequality e s ws is true for all s. Now, we will prove that it is also true for s = 1. We have the following error equation e 1 = A Be w 1 e w m w m1 e m w e 0 R e 1 A B w 1 I e A w m w m1 e m A R A B w 1 I w A w m w m1 w A B w 1 I w A w m w m1 w m A w w A w w A B w 1 I A w m w m1 A w w A B w 1 I A w 1 w w Since, lim α = α 1 1 α 1 α there exists constant C > 0 such that e 1 w = α α w = Γ α C τ α h 1 α Zα τ α h, α τ α α Γ α 1 1 α 1 1 α where Zα = C Γ α. So, corresponding difference scheme is convergent under the 1 α condition.

8 774 Ibrahim Karatay and Nurdane Kale 5 Numerical Example Consider this problem, α ut,x t α t α = x ut,x x sinx ut,x x x ut, x 1 x sinx [ Γ3 α t 1 x sinxx cosx x x cosx sin x ], 0 < x < 1, 0 < t < 1, u0, x = 0, 0 x 1, ut, 0 = 0, ut, 1 = 0, 0 t 1. Exact solution of this problem is ut, x = t 1 x sinx. The errors for some M and N are given in figure 1. The errors when solving this problem are listed in the table1 for various values of time andspace nodes. The errors in the table 1 are calculated by the formula max ut, x n Un. 0 n M 0 N Figure 1: The errors for some values of M and N when t = 1.

9 Matrix stability 775 Table 1: The errors for some values of M, N and α N M α = 0.3 α = 0.5 α = 0.8 Errorα, τ Err. rate Errorα, τ Err. rate Errorα, τ Err. rate Conclusion In this wor, Oτ α h order approximation for the Caputo fractional derivative based on the Cran-Nicholson difference scheme was successfully applied to solve the fractional parabolic partial differential equations with variable coefficients. It is proven that the time-fractional Cran-Nicholson difference scheme is conditionally stable. Numerical results are agreement with the theoretical results. References [1] A. Ashyralyev and Z. Cair, On the numerical solution of fractional parabolic partial differential equations with the Dirichlet condition, Discrete Dynamics in Nature and Society Volume 01 01, Article ID , 15 pages, [] D. A. Benson, S. Wheatcraft, M. M. Meerschaert, Application of a fractional advection dispersion equation, Water Resour. Res., , [3] Z. Cair, Stability of Difference Schemes for Fractional Parabolic PDE with the Dirichlet-Neumann Conditions, Volume 01 01, Article ID , 17 pages. [4] C. Chen, F. Liu, I. Turner, V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, Journal of Computational Physics, vol , [5] Z. Deng, V.P. Singh, L. Bengtsson, Numerical solution of fractional advection-dispersion equation, J. Hydraulic Eng [6] I. Karatay, S. R. Bayramoglu, A. Sahin, Implicit difference approximation for the time fractional heat equation with the nonlocal con-

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