High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations. II. (English) Zbl 1325.65121
Summary: In this paper, we first establish a high-order numerical algorithm for \(\alpha\)-th \((0 < \alpha < 1)\) order Caputo derivative of a given function \(f(t)\), where the convergence rate is \((4 - \alpha)\)-th order. Then by using this new formula, an improved difference scheme with high order accuracy in time to solve Caputo-type fractional advection-diffusion equation with Dirichlet boundary conditions is constructed. Finally, numerical examples are carried out to confirm the efficiency of the constructed algorithm.
For Part I, see [C. Li, R. Wu and H. Ding, “High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations”, Commun. Appl. Ind. Math. 6, No. 2, Article ID 536, electronic only (2014; doi:10.1685/journal.caim.536)].
For Part I, see [C. Li, R. Wu and H. Ding, “High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations”, Commun. Appl. Ind. Math. 6, No. 2, Article ID 536, electronic only (2014; doi:10.1685/journal.caim.536)].
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 |
| 26A33 | Fractional derivatives and integrals |
Software:
ma2dfcReferences:
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