An efficient Mickens' type NSFD scheme for the generalized Burgers Huxley equation

Abstract

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 $\theta \in \mathbb{N}.$ 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 $\alpha ,\text{β},\gamma ,\theta$ 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.

Keywords:

• Non-standard finite difference scheme
• stability
• positivity
• consistency
• exact finite difference scheme
• generalized Burgers Huxley
• non-linear

2010 Mathematics Subject Classifications:

• 35K61
• 65M06

Disclosure statement

No potential conflict of interest was reported by the author(s).