A transformed \(L 1\) method for solving the multi-term time-fractional diffusion problem. (English) Zbl 1540.65435
Summary: In this paper, we present a novel scheme for solving a time-fractional initial-boundary value problem, where the equation contains a sum of Caputo derivatives with orders between 0 and 1. In order to overcome the difficulty of initial layer, we introduce a change of variable in the temporal direction and investigate the regularity of the solutions of the resulting system. A modified \(L 1\) approximation is used to approximate the Caputo derivatives and a standard Galerkin-Spectral method is applied to approximate the spatial derivatives. Unconditional stability and convergence of the fully-discrete scheme are proved by applying a novel discrete fractional Grönwall inequality. Finally, numerical examples are given to confirm our theoretical results.
MSC:
| 65M70 | Spectral, collocation and related 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:
multi-term time-fractional equation; modified \(L 1\) scheme; Chebyshev-Galerkin spectral method; error estimatesReferences:
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