Bell polynomial-based semi-discretization approach for the extended Fisher-Kolmogorov equations

This work introduces an integrated numerical framework for obtaining numerical approximations to the extended Fisher-Kolmogorov (EFK) equation model. The methodology entails a two-stage process: initial linearization of the governing equation through a Taylor series approach, followed by the application of a spectral collocation method utilizing Bell polynomials to resolve the resultant linear system. Comprehensive theoretical considerations, including a detailed error estimate in the weighted $ L^2 $-norm, are established. The numerical investigation, comprising three illustrative test cases, confirms the scheme's computational efficiency and demonstrates a marked improvement in accuracy when compared to current state-of-the-art results and exact solutions.