An Adaptive Scheme Based on the Cubic B-Spline Collocation Technique for Time-Fractional Singular Parabolic Problems With Semilinearity Available to Purchase

This article presents an efficient layer-adaptive numerical scheme for time-fractional semilinear advection-reaction-diffusion equations with variable coefficients. In general, the solution to such type of problem exhibits mild singularity near $t=0$⁠. The semilinear problem is linearized by applying Newton's linearization technique. The fractional component is discretized employing the L2-$1σ$ formula, and the semidiscrete scheme is constructed as a set of boundary value problems (BVPs). To solve the resulting semidiscrete problems, the cubic B-spline collocation method is used. The presence of singularities creates a layer at the origin, and as a result, proposed scheme fails to achieve its optimal convergence rate on a uniform mesh. A graded mesh is used in the temporal direction with an user-chosen grading parameter to accelerate the convergence rate. On a suitable norm, convergence analysis and error-bound estimation are performed. The computational evaluation and comparison with the existing results demonstrate the robustness and effectiveness of the proposed scheme.