Non-polynomial quintic spline for numerical solution of fourth-order time fractional partial differential equations

Muhammad Amin,Muhammad Kashif Iqbal,Muhammad Abbas,Dumitru Baleanu

doi:10.1186/s13662-019-2125-1

Abstract

This paper presents a novel approach to numerical solution of a class of fourth-order time fractional partial differential equations (PDEs). The finite difference formulation has been used for temporal discretization, whereas the space discretization is achieved by means of non-polynomial quintic spline method. The proposed algorithm is proved to be stable and convergent. In order to corroborate this work, some test problems have been considered, and the computational outcomes are compared with those found in the exiting literature. It is revealed that the presented scheme is more accurate as compared to current variants on the topic.

Highlights

1 Introduction In the modern era, fractional order differential equations have gained a significant amount of research work due to their wide range of applications in various branches of science and engineering such as physics, electrical networks, fluid mechanics, control theory, theory of viscoelasticity, neurology, and theory of electromagnetic acoustics [1, 2]
Zurigat et al [4] examined the approximate solution of fractional order algebraic differential equations using homotopy analysis method
This paper has been composed with the aim to develop a spline collocation method for approximate solution of fourth-order time fractional partial differential equations (PDEs)

Summary

Introduction

Fractional order differential equations have gained a significant amount of research work due to their wide range of applications in various branches of science and engineering such as physics, electrical networks, fluid mechanics, control theory, theory of viscoelasticity, neurology, and theory of electromagnetic acoustics [1, 2]. The spline approximation techniques have been applied extensively for numerical solution of ODEs and PDEs. The spline functions have a variety of significant gains over finite difference schemes. Siddiqi and Arshed [19] brought the fifth degree basis spline collocation functions into use for approximate solution of fourth-order time fractional PDEs. Rashidinia and Mohsenyzadeh [20] used non-polynomial quintic spline technique for one-dimensional heat and wave equations. This paper has been composed with the aim to develop a spline collocation method for approximate solution of fourth-order time fractional PDEs. The backward Euler scheme has been utilized for temporal discretization, whereas non-polynomial quintic spline function, comprised of a trigonometric part and a polynomial part, has been used to interpolate the unknown function in spatial direction.

Description of non-polynomial quintic spline function

Temporal discretization