Non-polynomial B-spline and shifted Jacobi spectral collocation techniques to solve time-fractional nonlinear coupled Burgers\u2019 equations numerically

Adel R. Hadhoud,Abdulqawi A. M. Rageh,H. M. Srivastava

doi:10.1186/s13662-021-03604-5

Abstract

This paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval [0, L]. The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

Highlights

Many phenomena in engineering and applied sciences can be represented successfully using fractional calculus models such as anomalous diffusion, materials, and mechanics, signal processing, biological systems, finance, etc
There is a tremendous interest in fractional differential equations, as the theory of fractional derivatives itself and its applications have been intensively developed
In this paper we develop two accurate numerical methods to approximate the numerical solutions of the coupled TFBEs

Summary

Introduction

Many phenomena in engineering and applied sciences can be represented successfully using fractional calculus models such as anomalous diffusion, materials, and mechanics, signal processing, biological systems, finance, etc. (see, for instance, [1,2,3,4,5,6,7]). In [25] Liu and Hou explicitly applied the generalized two-dimensional differential transform method to solve the coupled space- and time-fractional Burgers equations (STFBEs). Heydari and Avazzadeh [26] proposed an effective numerical method based on Hahn polynomials to solve the nonsingular variableorder time-fractional coupled Burgers’ equations. Hussein [33] proposed a continuous and discrete-time weak Galerkin finite element approach for solving two-dimensional time-fractional coupled Burgers’ equations. In this paper we develop two accurate numerical methods to approximate the numerical solutions of the coupled TFBEs. The first method is the non-polynomial B-spline method [8, 40,41,42] based on the L1-approximation and finite difference approximations for spatial derivatives. Using Lemma 1, the Liouville–Caputo fractional derivative can be approximated as follows: σ1 n–1. Equations (23) and (24) design two methods for choices of parameters β and γ as follows: If 2γ

Convergence analysis

Stability analysis of the method

Conclusion

Methods