Cubic B-spline quasi-interpolation and an application to numerical solution of generalized Burgers-Huxley equation

Lan-Yin Sun,Chun-Gang Zhu

doi:10.1177/1687814020971061

Abstract

Nonlinear partial differential equations are widely studied in Applied Mathematics and Physics. The generalized Burgers-Huxley equations play important roles in different nonlinear physics mechanisms. In this paper, we develop a kind of cubic B-spline quasi-interpolation which is used to solve Burgers-Huxley equations. Firstly, the cubic B-spline quasi-interpolation is presented. Next we get the numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and modified Euler scheme to approximate the time derivative of the dependent variable. Moreover, the efficiency of the proposed method is illustrated by the agreement between the numerical solution and the analytical solution which indicate the numerical scheme is quite acceptable.

Highlights

Nonlinear phenomena play a crucial role in various nonlinear fields of science which has undergone many studies.[1,2,3,4,5] It is known that various phenomena in scientific fields can be described by nonlinear partial differential equations
The Burgers-Huxley equations arise from the mathematical modeling of many nonlinear scientific phenomena
We compare the numerical solution with the analytical solution at x =À 12, x =À 5, x = 0:6, x = 4:6 for t = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, respectively. These results show that the numerical solution obtained by our proposed method is in good agreement with the analytical solution

Summary

Introduction

Nonlinear phenomena play a crucial role in various nonlinear fields of science which has undergone many studies.[1,2,3,4,5] It is known that various phenomena in scientific fields can be described by nonlinear partial differential equations. Keywords Nonlinear physics mechanisms, Burgers-Huxley equation, numerical solution, cubic B-spline quasi-interpolation Consider the following generalized Burgers-Huxley equation (1) Hodgkin and Huxley[12] developed an efficient numerical scheme for Burgers’ equation.

Results

Conclusion