Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines

Abstract

In this work, we propose a numerical scheme to obtain approximate solutions of generalized Burgers–Fisher and Burgers–Huxley equations. The scheme is based on collocation of modified cubic B-spline functions and is applicable for a class of similar diffusion–convection–reaction equations. We use modified cubic B-spline functions for space variable and for its derivatives to obtain a system of first-order ordinary differential equations in time. We solve this system by using SSP-RK54 scheme. The stability of the method has been discussed and it is shown that the method is unconditionally stable. The approximate solutions have been computed without using any transformation or linearization. The proposed scheme needs less storage space and execution time. The test problems considered by the different researchers have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in the literature. The scheme is simple as well as computationally efficient. The scheme provides approximate solution not only at the grid points but also at any point in the solution range.

Keywords:

• Burgers–Fisher equation
• Burgers–Huxley equation
• Fitzhugh–Nagumo equation
• modified cubic B-splines
• SSP-RK54 scheme
• collocation method

2010 AMS Subject Classifications::

• 65D07
• 65N30
• 65N35
• 74S05

Acknowledgements

The authors are very thankful to the reviewers for their through reviews, valuable comments and suggestions to improve the quality of this paper. The author Amit Tripathi thankfully acknowledges the financial assistance provided by MHRD India.