We consider the generalized Burgers–Huxley (GBH) equation subject to certain initial and boundary conditions (BCs). Using a solitary wave solution, we derive an exact finite difference (EFD) scheme for the GBH equation. Furthermore, we propose a non-standard finite difference (NSFD) scheme which operates for all The qualitative properties, i.e. positivity and boundedness, are satisfied by the proposed NSFD scheme. Moreover, the stability and consistency of the NSFD scheme are also discussed. Our scheme is stable under certain conditions with the first-order accuracy in both time and space. We compute solutions of the GBH equation for various values of at a different time using the NSFD scheme and calculate their respective maximum errors. The maximum error of NSFD solutions is compared with the maximum error of several other methods to depict the supremacy of the proposed method. We also compute CPU time for all the computations which reveal that our scheme gives an accurate result within few seconds which saves our time. Our scheme gives precise results with only a few spatial divisions.

中文翻译：

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广义 Burgers Huxley 方程的有效 Mickens 型 NSFD 方案

我们考虑受某些初始和边界条件 (BC) 约束的广义 Burgers-Huxley (GBH) 方程。使用孤立波解，我们推导出 GBH 方程的精确有限差分 (EFD) 方案。此外，我们提出了一种非标准有限差分（NSFD）方案，该方案适用于所有定性属性，即正性和有界性，都满足所提出的 NSFD 方案。此外，还讨论了NSFD方案的稳定性和一致性。我们的方案在一定条件下是稳定的，在时间和空间上都具有一阶精度。我们使用 NSFD 方案计算不同时间的不同值的 GBH 方程的解，并计算它们各自的最大误差。将NSFD解的最大误差与其他几种方法的最大误差进行比较，以说明所提出方法的优越性。我们还计算了所有计算的 CPU 时间，这表明我们的方案在几秒钟内给出了准确的结果，从而节省了我们的时间。我们的方案只用几个空间划分就给出了精确的结果。