A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions.(English)Zbl 1242.65160

Summary: One-dimensional fractional anomalous sub-diffusion equations on an unbounded domain are considered. Beginning with the derivation of the exact artificial boundary conditions, the original problem on an unbounded domain is converted into mainly solving an initial-boundary value problem on a finite computational domain. The main contribution of our work, as compared with the previous work, lies in the reduction of fractional differential equations on an unbounded domain by using artificial boundary conditions and construction of the corresponding finite difference scheme with the help of the method of order reduction. The difficulty is the treatment of the Neumann condition on the artificial boundary, which involves the time-fractional derivative operator. The stability and convergence of the scheme are proven using the discrete energy method. Two numerical examples clarify the effectiveness and accuracy of the proposed method.

MSC:

 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35K05 Heat equation 35R11 Fractional partial differential equations

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