Comparative Analysis of the Singularly Perturbed Generalized Burgers-Huxley Problem via Approximate Lie Symmetry and Exponentially Fitted Finite Element Method

Abstract

Singularly perturbed generalized Burgers-Huxley equation (SPGBHE) models nonlinear wave phenomena in fluid dynamics, combustion, and biological systems. This study addresses the solutions of SPGBHE through approximate Lie symmetry analysis (LSA) and finite element method (FEM), with rigorous comparison validating solutions obtained through both methodologies. The method of approximate symmetry, which involves expanding the infinitesimal generator in a perturbation series is implemented to solve the governing equation, yielding approximate infinitesimal generators essential for constructing the optimal system of Lie sub-algebra. A set of group invariant solutions is determined for each reduction derived from corresponding sub-algebras mentioned in the system. On the second part, exponentially fitted finite element method (EF-FEM) along with the explicit Euler scheme is implemented using a piecewise uniform Shishkin mesh to examine the equation from a numerical perspective. Stability and uniform convergence of outlined approach is discussed to provide credibility to the numerical scheme. Moreover, solutions derived through each technique are authenticated by firm comparison following a detailed error analysis, including error table and comparison plots. Visual representations of solution profiles for specific outcomes of LSA are also presented to see the impact of the singular perturbation parameter \((\epsilon )\) alongside other parameters.