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

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 CrankNicholson method. And stability of this difference scheme is proved conditionally. Mathematics Subject Classification: 65N06, 65N12, 65M12

We can obtain some kind of subproblems from this problem which general type.One can see that if b(x) = 0 and c(x) = 0 then the problem transforms to time fractional diffusion problem that variable coefficient, if we choose b(x) = 0 and c(x) > 0 then the problem will be a time-fractional cable problem that variable coefficient when opposite chosen which means c(x) = 0 and b(x) > 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, 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].
A discrete approximation to the fractional derivative ∂ α u(x,t) ∂t α at (x j , t k+ 1

2
) can be obtained by the following approximation [7]: 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 also approximations for second and first derivative at (x j , t k+  ) .

Matrix Stability of the Difference Scheme
And we consructed this difference scheme from ( 6) The difference scheme above (7) can be written in matrix form, where T .Here, A and B are 3-diagonal (M − 1) × (M − 1) matrices of the form : where p = a(x j ) We denote Theorem3.1 The difference scheme ( 8) is stable.Proof.To prove the conditional stability of ( 8), let U k j and V k j be the exact and approximate solution of ( 8) with initial value U 0 j and V 0 j respectively.We denote the corresponding error by Let us prove ε k ≤ ε 0 , k = 0, 1, 2, ... by induction.In fact, if k = 0 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.
Therefore, under the condition σ 0 and c(x j ) > 0 , the stability inequality is obtained.

Convergence Of The Difference Scheme
Theorem4.1 The proposed scheme ( 8) 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 .So, corresponding difference scheme is convergent under the condition.

Numerical Example
Consider this problem, Exact solution of this problem is u(t, x) = t 2 (1 − x) sin(x).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 and space nodes.The errors in the table 1 are calculated by the formula max Figure 1: The errors for some values of M and N when t = 1.

Conclusion
In this work, O(τ 2−α + h 2 ) order approximation for the Caputo fractional derivative based on the Crank-Nicholson difference scheme was successfully applied to solve the fractional parabolic partial differential equations with variable coefficients.It is proven that the time-fractional Crank-Nicholson difference scheme is conditionally stable.Numerical results are agreement with the theoretical results.

Table 1 :
The errors for some values of M, N and α