Error analysis of a fast high order spline approximation method for time-fractional nonlinear reaction-diffusion problems exhibiting singular behavior

This research presents an accelerated non-polynomial spline numerical scheme to solve time-fractional nonlinear reaction diffusion problems with an initial time layer. Our approach utilizes a fast L1 temporal discretization, where the singular kernel is approximated using a sum-of-exponentials method, resulting in a highly efficient computational algorithm. We implement a non-uniform graded mesh that concentrates more grid points near the singularity at t=0. The spatial derivatives are discretized using a higher order approximation scheme. We perform a thorough theoretical analysis of the fully discrete numerical method using the discrete energy approach. The analysis reveals that the method achieves a convergence rate of O(N−min{rγ,2−γ},h4.5), where N is the total number of time grid points, h is the spatial step size, and r is the grading parameter for the temporal mesh. The proposed scheme significantly reduces memory from O(N) to O(log2N) and computational cost from O(N2) to O(Nlog2N). It also achieves higher spatial accuracy and optimal temporal convergence, outperforming existing methods. For instance, with N = 210, it numerically yields a 79% reduction in CPU time and 89% in memory compared to the classical L1. Numerical results confirm the theoretical convergence and highlight the method's superiority for nonlinear time-fractional problems. Additional numerical experiments on the time-fractional Fitzhugh-Nagumo and generalized Fisher equations further demonstrate its accuracy and efficiency.