Higher order finite difference method for the reaction and anomalous-diffusion equation. (English) Zbl 1429.65188
Summary: In this paper, our aim is to study the high order finite difference method for the reaction and anomalous-diffusion equation. According to the equivalence of the Riemann-Liouville and Grünwald-Letnikov derivatives under the suitable smooth condition, a second-order difference approximation for the Riemann-Liouville fractional derivative is derived. A fourth-order compact difference approximation for second-order derivative in spatial is used. We analyze the solvability, conditional stability and convergence of the proposed scheme by using the Fourier method. Then we obtain that the convergence order is \(O(\tau^2 + h^4)\), where \(\tau\) is the temporal step length and \(h\) is the spatial step length. Finally, numerical experiments are presented to show that the numerical results are in good agreement with the theoretical analysis.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
reaction and anomalous-diffusion equation; finite difference scheme; Riemann-Liouville derivative; Fourier method; numerical stabilityReferences:
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