An accurate numerical method and its analysis for time-fractional Fisher’s equation

Abstract

This article aims to develop an optimal superconvergent numerical method for approximating the solution of the nonlinear time-fractional generalized Fisher’s (TFGF) equation. The time-fractional derivative in the model problem is considered in the sense of Caputo and is approximated using the \(L2-1_{\sigma }\) scheme. Spatial discretization is performed using an optimal superconvergent quintic B-spline (OSQB) technique. To derive the method, a high-order perturbation of the semi-discretized equation of the original problem is generated using spline alternate relations. Convergence and stability of the method are analyzed, demonstrating that the method converges with \(O(\Delta t^{2}+\Delta x^6)\), where \(\Delta x\) and \(\Delta t\) are step sizes in space and time, respectively. Three numerical examples are provided to demonstrate the robustness of the proposed method. Our method is compared with an existing method in the literature and the elapsed computational time for the present scheme is provided.