Finite element approximation for a delayed generalized Burgers-Huxley equation with weakly singular kernels: Part I Well-posedness, Regularity and Conforming approximation

TL;DR

This paper investigates the well-posedness, regularity, and finite element approximation of a delayed generalized Burgers-Huxley equation with weakly singular kernels, providing theoretical analysis and computational validation.

Contribution

It introduces a conforming finite element method for a complex delayed nonlinear PDE with singular kernels, including error estimates and computational results.

Findings

Existence and uniqueness of solutions established

Error estimates for semi-discrete and fully-discrete schemes derived

Computational results confirm theoretical error bounds

Abstract

The analysis of a delayed generalized Burgers-Huxley equation (a non-linear advection-diffusion-reaction problem) with weakly singular kernels is carried out in this work. Moreover, numerical approximations are performed using the conforming finite element method (CFEM). The existence, uniqueness and regularity results for the continuous problem have been discussed in detail using the Faedo-Galerkin approximation technique. For the numerical studies, we first propose a semi-discrete conforming finite element scheme for space discretization and discuss its error estimates under minimal regularity assumptions. We then employ a backward Euler discretization in time and CFEM in space to obtain a fully-discrete approximation. Additionally, we derive a prior error estimates for the fully-discrete approximated solution. Finally, we present computational results that support the derived theoretical results.