Cubic B-spline based numerical schemes for delayed time-fractional advection-diffusion equations involving mild singularities

Abstract

Abstract This article presents two efficient layer-adaptive numerical schemes for a class of time-fractional advection-diffusion equations with a large time delay. The fractional derivative of order $ \alpha $ with $ \alpha\in(0,1) $ is taken in the Caputo sense. The solution to this type of problem generally has a layer due to the mild singularity near the time $ t=0. $ Consequently, the polynomial interpolation discretizing scheme degrades the convergence rate in the case of uniform meshes. In the presence of a singularity, the tempered fractional operator is discretized by employing the L1 technique on a layer-resolving mesh. In contrast, the cubic B-spline collocation method is used in the spatial direction. The convergence analysis and estimation of error are presented for the proposed scheme under reasonable regularity assumptions on the coefficients. The scheme achieves its optimal convergence rate $ (2-\alpha) $ for suitable choice of grading parameter $ (\gamma \geq (2-\alpha)/\alpha). $ Furthermore, we modified the proposed scheme by discretizing the fractional operator with the help of the L1-2 technique. The modified scheme gets a quadratic order convergence for $ \gamma \geq 2/\alpha. $ In addition, we extend the proposed schemes to solve the corresponding semilinear problem. Numerical examples demonstrate the efficiency and applicability of the proposed techniques.