Solving time fractional Burgers\u2019 and Fisher\u2019s equations using cubic B-spline approximation method

Abdul Majeed,Mohsin Kamran,Muhammad Kashif Iqbal,Dumitru Baleanu

doi:10.1186/s13662-020-02619-8

Abstract

This article presents a numerical algorithm for solving time fractional Burgers’ and Fisher’s equations using cubic B-spline finite element method. The L1 formula with Caputo derivative is used to discretized the time fractional derivative, whereas the Crank–Nicolson scheme based on cubic B-spline functions is used to interpolate the solution curve along the spatial grid. The numerical scheme has been implemented on three test problems. The obtained results indicate that the proposed method is a good option for solving nonlinear fractional Burgers’ and Fisher’s equations. The error norms L_{2} and L_{infty } have been calculated to validate the efficiency and accuracy of the presented algorithm.

Highlights

1 Introduction In recent years, most of the practical problems arising in different fields of science like biology, chemistry, physics, engineering, and mathematics lead to nonlinear fractional partial differential equations [1,2,3]
Still it is noted that a large number of nonlinear fractional differential equations that are encountered in these areas are very difficult to solve numerically
The finite element method based on third degree basis spline functions has not been widely used for solving time fractional problems and this will be the focus of our paper

Summary

Introduction

Most of the practical problems arising in different fields of science like biology, chemistry, physics, engineering, and mathematics lead to nonlinear fractional partial differential equations [1,2,3]. To solve these nonlinear mathematical models, several techniques have been developed. The finite element method based on third degree basis spline functions has not been widely used for solving time fractional problems and this will be the focus of our paper. We consider the following generalized time fraction partial differential equation:

Discussion

Conclusion