Numerical solutions of strongly non-linear generalized Burgers–Fisher equation via meshfree spectral technique

Abstract

In this article, a meshfree spectral interpolation technique combined with Crank–Nicolson difference scheme is proposed to solve a class of strongly non-linear Burgers–Fisher type equation numerically. The proposed technique utilizes meshless shape functions for approximation of unknown spatial function and its derivatives. These shape functions are obtained by combining radial basis functions and point interpolation method in the spectral framework. The Crank–Nicolson finite difference scheme is employed for time integration. Stability of the proposed method is analysed theoretically and supported by numerical evidences for RBFs shape parameter $\left(c\right)$, which is an equally important task. Measure of fitness quality is assessed via $L_{\mathrm{\infty }}$, $L_2$ and $L_{\mathrm{r}\mathrm{m}\mathrm{s}}$ error norms. Efficiency and accuracy of the proposed technique is further examined via variation of time-step size $\delta t$ and number of nodal points N. Comparison made with existing techniques in the literature confirms excellent performance of the proposed scheme.

Keywords:

• Reaction-diffusion
• Burgers–Fisher equation
• Meshless spectral interpolation technique
• radial basis functions
• Crank–Nicolson scheme
• stability analysis

2010 AMS Subject Classifications:

• 65M06
• 60M70
• 35K20
• 35Q92

Acknowledgments

The authors would like to express their sincere thanks to the editor and referees of this manuscript whose suggestions greatly improved the presentation of this manuscript.

Disclosure statement

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