Numerical investigation of the time-fractional Fisher equation in physics-based diffusion-reaction systems

In this study, we investigate the numerical solutions of the time-fractional Fisher equation by employing the collocation finite element method (FEM). The fractional derivative is considered in the Caputo sense, which provides a suitable framework for modeling memory-dependent diffusion-reaction processes. The temporal discretization is carried out using the $L1$ algorithm, while the spatial discretization is carried out using the unified hyperbolic polynomial B-spline basis within a collocation finite element framework. Two test problems are presented to demonstrate the efficiency of the proposed method, and the obtained numerical results are compared with available exact and reference solutions. The results confirm that the collocation finite element approach is a reliable and effective technique for solving fractional partial differential equations (FPDEs) arising in nonlinear diffusion-reaction models.