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A Fourier method for the fractional diffusion equation describing sub-diffusion. (English) Zbl 1165.65053
The paper focuses on the initial-boundary value problem of the fractional diffusion equation describing sub-diffusion. A brief review of relevant literature is included, thus setting the paper in context. A paper by T. A. M. Langlands and B. I. Henry [J. Comput. Phys. 205, No. 2, 719–736 (2005; Zbl 1072.65123)] in which they proposed an implicit numerical scheme (${L}_{1}$ approximation) and discussed its accuracy and stability, is referred to. The motivation behind this paper is to build on this work, by deriving the global accuracy of the presented implicit scheme and establishing unconditional stability for all $\lambda$ in the range $0<\lambda \le 1$. A Fourier method is used. In Section 2 an implicit difference approximation scheme (IDAS) is presented and its unconditional stability and ${L}_{2}$-convergence are investigated in Sections 3 and 4. In Section 5 the implicit difference scheme is written in matrix form and is proved to be uniquely solvable. The paper concludes with two numerical examples, each describing sub-diffusion (one wit a non-homogeneous term and the second with a homogeneous term), to confirm their theoretical results. The examples demonstrate that the IDAS is unconditionally stable and convergent ant that it can be applied to simulate fractional dynamical systems. The authors state that the Fourier method technique used to analyse stability and convergence can be extended to other fractional partial differential equations.
##### MSC:
 65M12 Stability and convergence of numerical methods (IVP of PDE) 65M70 Spectral, collocation and related methods (IVP of PDE) 35B35 Stability of solutions of PDE