An integrated analytical–numerical framework for studying nonlinear PDEs: The GBF case study

In this study, we investigate the complex dynamics of the [Formula: see text]-dimensional generalized Burgers–Fisher (GBF) model, a nonlinear partial differential equation that encapsulates the interplay between wave propagation, diffusion, and reaction processes. Our work employs a combination of the modified Khater (MKhat) method, the unified (UF) method, and He’s variational iteration (HVI) scheme to derive and validate analytical and numerical solutions. We present a comprehensive analysis of solitary wave, shock wave, and diffusion-driven phenomena within the GBF framework. The novelty of our study lies in the integration of these methods to provide deeper insights into the model’s physical implications, specifically highlighting the interactions between nonlinear advection, diffusion, and reaction mechanisms. This approach not only enhances the accuracy and applicability of the derived solutions, but also contributes to the advancement of nonlinear wave theory and related interdisciplinary fields.