Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials

Umme Ruman,Md Shafiqul Islam

doi:10.11648/j.acm.20200902.11

Abstract

The main objective of this paper is to solve numerically the differential equations of fractional order with homogeneous boundary conditions by the Galerkin weighted residual method. In this method, linear combinations of some types of functions are used to find the approximate solutions which must satisfy the homogeneous boundary conditions. Such type of functions should be differentiated and integrated easily, so the piecewise polynomials, namely, Bernstein, Bernoulli and Modified Legendre polynomials are used as basis functions in this paper. The fractional derivatives are used in the conjecture of Riemann-Liouville and Caputo sense. Thus, we develop the Galerkin weighted residual formulation, in matrix form, to the linear fractional order boundary value problems, in details, which is easy to understand. The accuracy and applicability of the present method are demonstrated through few numerical examples. We observe that the approximate results converge monotonically to the exact solutions. In addition, we compare the approximate results with the exact solutions, and also with the existing solutions which are available in the literature. The absolute errors are depicted in tabular form as well as graphical representations, a reliable accuracy is achieved. The proposed method may be applied to fractional order partial differential equations also.

Highlights

In recent few decades, the wide spread of differential equations of fractional order in various forms arise in the areas of physical sciences, mathematical biology and engineering problems
Many researchers have attempted for solving several problems numerically, in a wide range, by different methods to obtain approximate solutions, such as Sinc-Galerkin method [5], generalized differential transform method [6], cubic spline solution [7], Cubic B-Spline wavelet collocation method [8], Legendre wavelet approximation [9], collocation-shooting method [10], and more, which are available in the literature
For the huge demand of numerical methods with great accuracy, we are interested to find the numerical solutions of linear fractional order two-point boundary value problems with homogeneous boundary conditions only

Summary

Introduction

The wide spread of differential equations of fractional order in various forms arise in the areas of physical sciences, mathematical biology and engineering problems. These problems have been studied for existence and uniqueness results in a limited way analytically [1, 2] or semi-analytically [3, 4], such as by variational iteration and Adomian decomposition methods. Umme Ruman and Md. Shafiqul Islam: Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials [15], and Bernoulli polynomials [16] as basis functions in the approximation. The results are compared with the existing methods, namely, Sinc-Galerkin method [5] and Cubic- Splines method [7]

Preliminaries

GWR Formulation of Fractional Differential Equations

Numerical Examples

Conclusion