Numerical simulation for time-fractional nonlinear reaction-diffusion system on a uniform and nonuniform time stepping. (English) Zbl 1473.65256
Summary: In this article, two nonstandard high-order schemes on a uniform and nonuniform time stepping combined with the multi-parameter exponential fitting technique (MPEF) have been developed to solve the time-fractional nonlinear reaction-diffusion system. The first method based on the MPEF combined with the 3-weighted shifted-Grünwald-Letnikov approximation with uniform time stepping, this scheme leads to a numerical solution that suffers from the singularity near \(t = 0\). In order to frustrate this singularity, a nonstandard higher-order L1-approximation for a nonuniform time-stepping scheme is developed. The developed scheme’s convergence and unconditionally stability analysis have been verified. Numerical results effectively validate the theoretical aspects.
MSC:
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 35K57 | Reaction-diffusion equations |
| 35R11 | Fractional partial differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
Caputo fractional derivative; convergence analysis; exponential fitting; nonuniform time stepping; stabilityReferences:
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