Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation

A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimensional sub-diffusion equation in the case where the Grünwald-Letnikov definition is used for the time-fractional derivative. The stability and convergence of the proposed Crank-Nicolson scheme are also analyzed. Finally, numerical examples are presented to test that the numerical scheme is accurate and feasible.


Introduction
Fractional calculus is essentially arbitrary order differentiation and integration.Comprehensive studies on fractional calculus and its applications can be found in [1][2][3][4].Certain phenomena and processes can best be described by the fractional diffusion equation having fractional order derivatives in time or space or space-time [5].Most papers on the numerical solution of the time fractional sub-diffusion equation have utilized the Caputo definition for the time fractional derivative [6,7,8,9].There have not been many studies that utilize the Grünwald-Letnikov definition.The limited studies that have used the Grünwald-Letnikov (or related to Grünwald-Letnikov) definition include [10,11,12,13,14].This paper discusses the use of a Crank-Nicolson scheme for solving the two-dimensional time fractional sub-diffusion equation is constructed by applying the Grünwald-Letnikov definition instead of the Caputo definition for the timefractional derivative.If the initial condition is zero then the Grünwald-Letnikov definition and Caputo definition are equivalent [15,16].It should be noted however that the Grünwald-Letnikov definition has the advantage of being less complex and more easily applied.This paper considers the following two dimensional time fractional sub-diffusion equation subject to u(x, y, 0) = f 1 (x, y), (1.2) u(0, y,t) = f 2 (y,t), u(x, 0,t) = f 3 (x,t), u(1, y,t) = f 4 (y,t), u(x, 1,t) = f 5 (x,t), 0 ≤ x, y ≤ L, 0 ≤ t ≤ T, (1.3) where f 1 , f 2 , f 3 , f 4 and f 5 are known functions, and u is the unknown dependent variable.
The time fractional derivative of order α(0 < α ≤ 1) of u can be defined by and According to [9], ( 4) is known as the Caputo formula and ( 5) is known as the Riemann-Liouville formula.The Grünwald-Letnikov time fractional derivative formula is defined by [15] ∂ α u(x, y,t) ( The Caputo fractional derivative and Grünwald-Letnikov fractional derivative are equivalent if u(x, y, 0) = 0.The Grünwald-Letnikov formula can also be written as [13] ∂ α u(x, y,t) where t/τ is an integer, ω k−1 and k = 0, 1, 2, ..t/τ.The right shifted Grünwald-Letnikov formula can be defined as Lemma 1.1.The relation between Caputo and Reimann-Liouville fractional derivative is [16]: These two fractional derivatives are equivalent if and only if u(x, y, 0) = 0.The proof is given by the Lemma 6.4.2 in [18].

Stability analysis of Crank-Nicolson method
We follow the approach in [14] for the analysis of stability.Suppose that U k i, j , is the approximate solution of (11) and the error is defined as Due to linearity, the error satisfies equation ( 11) and we have The error and initial conditions are given by By defining the following grid functions for k = 0, 1, 2, . . ., N − 1 International Scientific Publications and Consulting Services then Ψ k (x, y) can be expanded as a Fourier series: From the definition of l 2 norm and Parseval equality: Supposing that where σ 1 = 2πl 1 /L, σ 2 = 2πl 2 /L and substituting (3.20) in (3.14), where, µ = 4 Proof.The proof utilizes mathematical induction; take k = 0, in (3.21) and as 0 < α < 1, µ ≥ 0, then and as 0 < α < 1 and µ ≥ 0, from (3.21) and Lemma 1.2, we obtain This complete the proof of Proposition 3.1 by induction method.
By using Proposition 3.1 and (3.19), it can be seen that the solution of (2.11) satisfies which means that the Crank-Nicolson difference scheme in (2.11) is unconditionally stable.
International Scientific Publications and Consulting Services

Convergence analysis of Crank-Nicolson method
We follow the approach in [14] for analyzing the convergence.Let u(x i , y j ,t k ) be the exact solution represented by Taylor series.Then the truncation error of Crank-Nicolson method is From (1.1), we have = O(τ + (∆x) 2 + (∆y) 2 ).(4.24) Since i, j and k are finite, a positive constant C 1 exists, for all i, j and k, such that ) The error is defined as From (4.23), we have To obtain the error equation, subtract (4.27) from (2.11) to obtain + τ α T k+ 1 2 i, j , (4.28) International Scientific Publications and Consulting Services with error boundary conditions ϕ k 0 = ϕ k M = 0, k = 0, 1, 2, . . ., N − 1, and the initial condition ϕ 0 i, j = 0, i = 1, 2, . . ., M x , j = 1, 2, . . ., M y .Next, the following grid functions are defined for k = 0, 1, 2, . . ., N − 1 and Here, the ϕ k (x, y) and T k (x, y) can be expanded in Fourier series such as −1π(l 1 x/L+l 2 y/L) dxdy, (4.29) From the definition of l 2 norm and the Parseval equality: and Based on the above, suppose that where µ is as mentioned in section 3.
Table 1: The errors E ∞ between the exact solution and the numerical solution of (TFSDE) at T = 1.0 Table 1 shows that, for various values of α, the errors decrease as we reduce the time and space step size τ and (∆x, ∆y).This indicates the method is convergent.Figures 1 and 2 shows the numerical solution of the equation (5.40) and compares it with exact solution at T = 1.0.Clearly the numerical solution is in good agreement with the exact solution.These results seem to confirm the theoretical analysis.
(5.47)The above numerical results show that the exact solution and the numerical solution are in good agreement.The result displayed and discussed in this section seems to confirm the results of our theoretical analysis.

Conclusions
The Crank-Nicolson difference method for two-dimensional sub-diffusion equation of fractional order has been described the Grünwald-Letnikov formula was used for time fractional derivative.The scheme was found to be convergent with order (τ + (∆x) 2 + (∆y) 2 ).Further it is unconditionally stable.The results of an application to certain examples indicated that the scheme is feasible and accurate.

Table 2 :
The errors E ∞ between the exact solution and the numerical solution of (TFSDE) at T = 1.0