Abstract
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.
Highlights
There is a huge number of theoretical and applied works devoted to the study of fractional 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. Many mathematicians apply this approach and operator tools to investigate various problems for fractional partial differential equations which appear in applied problems
The role played by stability inequalities in the study of boundary-value problems for parabolic partial differential equations is well known
Summary
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 [ – ] These problems were studied in various directions: qualitative properties of solutions, spectral problems, various statements of boundary value problems, and numerical investigations. In previous paper [ ] authors investigated stability estimates for Crank-Nicholson schemes for the Dirichlet problem for the fractional parabolic equation. In [ ] the authors investigated stability estimates for Crank-Nicholson schemes for the Neumann problem for the fractional parabolic equation m p=. The role played by stability inequalities (well posedness) in the study of boundary-value problems for parabolic partial differential equations is well known (see, e.g., [ – ]). 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
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