Numerical treatment of the generalized time-fractional Huxley-Burgers’ equation and its stability examination

Adel R. Hadhoud,Faisal E. Abd Alaal,Taha Radwan,Ayman A. Abdelaziz

doi:10.1515/dema-2021-0040

Abstract

Abstract In this paper, we show how to approximate the solution to the generalized time-fractional Huxley-Burgers’ equation by a numerical method based on the cubic B-spline collocation method and the mean value theorem for integrals. We use the mean value theorem for integrals to replace the time-fractional derivative with a suitable approximation. The approximate solution is constructed by the cubic B-spline. The stability of the proposed method is discussed by applying the von Neumann technique. The proposed method is shown to be conditionally stable. Several numerical examples are introduced to show the efficiency and accuracy of the method.

Highlights

Fractional calculus is a generalization of classical calculus that is concerned with operations of differentiation and integration to fractional order
On there have been several fundamental works on the fractional derivative and fractional differential equations, and many books were performed in this field, in which we can refer to the books of Ross and Miller [1], Samko et al [2], and Podlubny [3]
Several numerical methods have been developed to obtain an approximate solution of fractional differential equations

Summary

Introduction

Fractional calculus is a generalization of classical calculus that is concerned with operations of differentiation and integration to fractional order. Several numerical methods have been developed to obtain an approximate solution of fractional differential equations. The generalized time-fractional Huxley-Burgers’ equation whose solutions will be approximated is of the form [20]:. We can obtain approximate solutions of the generalized time-fractional Huxley-Burgers’ equation that have applications in various fields of science and engineering. We have proposed a collocation method for solving the time-fractional Burgers-Huxley equation using the mean value theorem for integrals and cubic B-spline basis functions. In this method, we use cubic B-splines for spatial variables and their derivatives which produce a system of fractional ordinary differential equations.

Derivation of the method

Initial vector

Stability analysis

Application and discussion

Conclusion