Numerical simulation method of lines together with a pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative

Mushtaq Ali,Basim Karim Albuohimad,Mohammed Almoaeet

doi:10.1108/ajms-02-2021-0052

Abstract

PurposeThis study aims to use new formula derived based on the shifted Jacobi functions have been defined and some theorems of the left- and right-sided fractional derivative for them have been presented.Design/methodology/approachIn this article, the authors apply the method of lines (MOL) together with the pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative (SFPDEs). Then, using the collocation nodes to reduce the SFPDEs to the system of ordinary differential equations, which can be solved by the ode45 MATLAB toolbox.FindingsApplying the MOL method together with the pseudospectral discretization method converts the space-dependent on fractional partial differential equations to the system of ordinary differential equations.Originality/valueThis paper contributes to gain choosing the shifted Jacobi functions basis with special parameters a, b and give the authors this opportunity to obtain the left- and right-sided fractional differentiation matrices for this basis exactly. The results of the examples are presented in this article. The authors found that the method is efficient and provides accurate results, and the authors found significant implications for success in the science, technology, engineering and mathematics domain.

Highlights

The recent development in the last few decades has shown that most of the complex system in engineering and other several phenomena can be accurately modeled using partial differential equations with fractional order
The main aim of this paper is to introduce an efficient numerical method to approximate the fractional partial differential equation (FPDE) of the form
One of the best types of methods for solving space-dependent on fractional partial differential equations (SFPDEs) numerically is by discretization of the space variable without the time variable

Summary

Introduction

The recent development in the last few decades has shown that most of the complex system in engineering and other several phenomena can be accurately modeled using partial differential equations with fractional order. Many approximation methods in the numerical analysis have been a survey to solve space-dependent on fractional partial differential equations (SFPDEs), and the target of the main subject of these methods in terms of convergence to real solutions, the stability of methods, and order of accuracy and value of error. One of the best types of methods for solving SFPDEs numerically is by discretization of the space variable without the time variable These kinds of method are referred to as method of lines (MOL). If n À 1 ≤ α < n such that n is a positive integer number and the continuous function g: I → R the left and right Riemann-Liouville fractional derivatives of order α are defined by: aDαx gðxÞ dn dxn

Γðn À αÞ

Jacobi–Gauss nodes

Th presented pseudospectral discretization method

Numerical results and comparisons

Conclusion