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The analysis of a delayed generalized Burgers-Huxley equation (a non-linearadvection-diffusion-reaction problem) with weakly singular kernels is carriedout in this work. Moreover, numerical approximations are performed using theconforming finite element method (CFEM). The existence, uniqueness andregularity results for the continuous problem have been discussed in detailusing the Faedo-Galerkin approximation technique. For the numerical studies, wefirst propose a semi-discrete conforming finite element scheme for spacediscretization and discuss its error estimates under minimal regularityassumptions. We then employ a backward Euler discretization in time and CFEM inspace to obtain a fully-discrete approximation. Additionally, we derive a priorerror estimates for the fully-discrete approximated solution. Finally, wepresent computational results that support the derived theoretical results.

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