A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher’s equation. (English) Zbl 1304.35713
Summary: In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher’s equation, which is obtained from the standard one-dimensional Fisher’s equation by replacing the first-order time derivative with a fractional derivative (of order \(\alpha\), with \(0<\alpha <1\)). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order \(O(h^{k+1}+\tau^{2-\alpha})\), where \(h\), \(\tau\) and \(k\) are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.
MSC:
35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
35R11 | Fractional partial differential equations |
26A33 | Fractional derivatives and integrals |
Keywords:
time-fractional Fisher’s equation; local discontinuous Galerkin finite element method; Caputo fractional derivative; fractional differential equationReferences:
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