A robust higher-order finite difference technique for a time-fractional singularly perturbed problem. (English) Zbl 1540.65306
Summary: A higher-order finite difference method is developed to solve the variable coefficients convection-diffusion singularly perturbed problems (SPPs) involving fractional-order time derivative with the order \(\alpha \in (0,1)\). The solution to this problem class possesses a typical weak singularity at the initial time \(t=0\) and an exponential boundary layer at the right lateral surface as the perturbation parameter \(\varepsilon\to 0\). Alikhanov’s \(L2-1_\sigma\) approximation is applied in the temporal direction on a suitable graded mesh, and the spatial variable is discretized on a piecewise uniform Shishkin mesh using a combination of midpoint upwind and central finite difference operators. Stability estimates and the convergence analysis of the fully discrete scheme are provided. It is shown that the fully discrete scheme is uniformly convergent with a rate of \(O(M^{-p}+N^{-2} (\log N)^2)\), where \(p= \min \{2,r\alpha\}\), \(r\) is the graded mesh parameter and \(M\), \(N\) are the number of mesh points in the time and space direction, respectively. Two numerical examples are taken in counter to confirm the sharpness of the theoretical estimates.
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:
Caputo fractional derivative; singularly perturbed problem; non-uniform mesh; finite difference scheme; stability; uniform convergenceReferences:
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