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.

翻译：本文对带有弱奇异核的延迟广义Burgers-Huxley方程（一个非线性对流-扩散-反应问题）进行了分析。此外，采用协调有限元方法（CFEM）进行了数值逼近。利用Faedo-Galerkin逼近技术，详细讨论了连续问题的存在性、唯一性和正则性结果。对于数值研究，我们首先提出了空间离散的半离散协调有限元格式，并在最小正则性假设下讨论了其误差估计。接着，我们在时间上采用向后欧拉离散化，在空间上采用CFEM，以获得全离散逼近。此外，我们推导了全离散逼近解的先验误差估计。最后，我们给出了支持所推导理论结果的计算结果。