r-Modified Crank-Nicholson difference scheme for fractional parabolic PDE
© Ashyralyev and Cakir; licensee Springer. 2014
Received: 2 December 2013
Accepted: 18 March 2014
Published: 31 March 2014
The second order of accuracy stable difference scheme for the numerical solution of the mixed problem for the fractional parabolic equation are presented using by r-modified Crank-Nicholson difference scheme. Stability estimate for the solution of this difference scheme is obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional parabolic partial differential equations. Numerical results for this scheme and the Crank-Nicholson scheme are compared in test examples.
At present, there is a huge number of theoretical and applied works devoted to the study of fractional differential equations. Solutions of various problems for fractional differential equations can be found, for example, in the monographs of Podlubny , Kilbas, Srivastava, and Trujillo , Diethelm , and in [4–11]. These problems were studied in various directions: qualitative properties of solutions, spectral problems, various statements of boundary value problems, and numerical investigations.
Many problems in fluid flow, dynamical and diffusion processes, control theory, mechanics, and other areas of physics can be reduced fractional partial differential equations.
In  the simple connection of fractional derivatives with fractional powers of first order differential operator was presented. This approach is important to obtain the formula for the fractional difference derivative. Presently, many mathematicians apply this approach and operator tools to investigate various problems for fractional partial differential equations which appear in applied problems (see, e.g., [13–20] and the references therein).
with boundary S, , () and (, ) are given smooth functions and , .
In the present paper, we consider an r-modified Crank-Nicholson difference scheme of the above mentioned two problems (1.1), (1.2). This r-modified scheme is of the second order of accuracy in t and in space variables difference schemes for the approximate solution of problems. The stability estimate for the solution of this difference scheme is established. We use a procedure of a modified Gauss elimination method for solving this difference scheme in the case of one-dimensional fractional parabolic partial differential equations.
2 Stability of difference scheme
for a finite system of ordinary fractional differential equations.
for the approximate solution of problem (2.2).
where C does not depend on τ, h, and , .
The proof of estimate (2.5) for the solution of (2.4) follows from (2.6), (2.9), and (2.12). Note that , are independent from τ, h, and , . Theorem 2.1 is proved. □
3 Numerical analysis
We consider two examples for numerical results.
where , is the zero matrix and is the zero matrix and .
Error analysis for Dirichlet problem
N = M = 40
N = M = 80
Error analysis for Neumann problem
N = M = 40
N = M = 80
In this study, the second order of accuracy stable difference scheme for the numerical solution of the mixed problem for the fractional parabolic equation is investigated. We have obtained a stability estimate for the solution of this difference scheme. The theoretical statements for the solution of this difference scheme for one-dimensional parabolic equations are supported by numerical examples obtained by computer.
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