On the solution of the nonlinear reaction–diffusion problem with time- dependent coefficients using hybrid Taylor and B-spline approximation techniques

Nonlinear reaction–diffusion problems, such as the nonlinear generalized Fitzhugh–Nagumo (NGFN) problems, have been extensively studied in the fields of biology, engineering, and physics. In this study, we propose a hybrid approximation algorithm for solving the NGFN problem, which combines a Taylor series based forward finite difference method for time discretization with cubic B-spline polynomials for spatial approximation. This approach provides accurate solutions without requiring linearization or transformation. Furthermore, the accuracy and the efficiency of the proposed technique are demonstrated through various test problems, including the computation of L2 and L∞ error norms. The results, presented in both tabular and graphical formats, demonstrate that the developed algorithm exhibits excellent concordance with the exact solutions and some existing techniques. This study represents a substantial advancement in numerical techniques for nonlinear partial differential equations, with significant applications across various scientific and engineering disciplines.