Stability and Convergence of a Finite Difference Scheme for Fractional Partial Differential Equations by Matrix Method

Here, we searched stability and convergence of a difference scheme which is constructed for solving a fractional partial differential equation with Caputo fractional derivative. Stability is proved by matrix method. Numerical experiments are presented. Mathematics Subject Classification: 65N06, 65N12, 65M12


Introduction
There are two fundamental types of fractional heat equations, time fractional heat equations and space fractional heat equations.Some difference schemes for the space-fractional heat equations are presented in [6] [15] [16].The Crank-Nicholson method was applied directly to obtain a numerical solution for time fractional advection dispersion equations with Riemann Liouville derivative in [8].
In this work, we use a numerical approximation based on the Crank-Nicholson method for fractional derivatives.Then, the matrix stability of the method is proved conditionally.Here, we consider the following time fractional heat equation; Here, the term ∂ α u(t,x) ∂t α denotes α-order Caputo derivative, with the formula: where Γ(.) is the Gamma function.

Finite Difference Approximation to Derivatives
In this section, we introduce the basic ideas for the numerical solution of the time fractional heat equation (1) by Crank-Nicholson difference scheme.
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, 2, ..., M , and the grid points in the time interval [0, 1] are labeled t k = kτ , k = 0, 1, 2, ..., N .The values of the functions u and f at the grid points are denoted u k j = u(x j , t k ) and f k j = f (x j , t k ), respectively.Let u(x, t), u t (x, t) and u tt (x, t) are continuous on [0, 1].
In addition for k = 0 there is no these terms w 1 u k and w k u 0 .On the other hand, we have 3 The Difference Scheme Using the equations ( 3) and ( 4), we obtain the following difference scheme [7] which is accurate of order O(τ 2−α + h 2 ); (5)

Matrix Stability Of The Difference Scheme
The difference scheme above ( 5) can be written in matrix form, where Here, A and B are 3-diagonal matrices of the form : .
Theorem 4.1 The difference scheme (6) is stable.Proof To prove the conditional stability of (6), let U k j and V k j be the exact and approximate solution of (6) with initial value U 0 j and V 0 j respectively.We denote the corresponding error by From the last equation, we obtain If the condition above is satisfied, then ε 2 ≤ ε 0 is obtained.Now, assume ε s ≤ ε 0 for all s ≤ k, we will prove it is also true for s = k + 1.

Convergence Of The Difference Scheme
Theorem 5.1.The proposed scheme is convergent and the following estimate holds: Here Z(α) does not depend on τ and h.
notice that e 0 = 0. Firstly, we prove that e k+1 ≤ w −1 n R for all n by induction.We have the following error equation when k = 0.
Similarly, for k = 2 we have the error equation If we take the norm of this equality, we obtain Assume the inequality e s ≤ w −1 s−1 R is true for all s ≤ k.Now, we will prove that it is also true for s = k + 1.We have the following error equation where Z(α) = C Γ(2−α) 1−α .
Exact solution of this problem is u(x, t) = (1 − x)x(t 3 + 1).The solution by the proposed scheme is given in Figure 1.The errors when solving this problem are listed in the Table 1 for various values of time and space nodes.

Figure 1 :
Figure 1: The errors for some values of M and N when t = 1.

Table 1 :
Error table