A new generalized Jacobi Galerkin operational matrix of derivatives: two algorithms for solving fourth-order boundary value problems

Waleed M Abd-Elhameed,Hany M Ahmed,Youssri H Youssri

doi:10.1186/s13662-016-0753-2

Abstract

This paper reports a novel Galerkin operational matrix of derivatives of some generalized Jacobi polynomials. This matrix is utilized for solving fourth-order linear and nonlinear boundary value problems. Two algorithms based on applying Galerkin and collocation spectral methods are developed for obtaining new approximate solutions of linear and nonlinear fourth-order two point boundary value problems. In fact, the key idea for the two proposed algorithms is to convert the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable numerical solvers. The convergence analysis of the suggested generalized Jacobi expansion is carefully discussed. Some illustrative examples are given for the sake of indicating the high accuracy and effectiveness of the two proposed algorithms. The resulting approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by other existing techniques in the literature.

Highlights

There is a huge number of articles handling both high odd- and high even-order boundary value problems (BVPs). For example, in the sequence of papers, [ , – ], the authors have obtained numerical solutions for evenorder BVPs by applying the Galerkin method
The main idea for obtaining these solutions is to construct suitable basis functions satisfying the underlying boundary conditions on the given differential equation, applying Galerkin method to convert each equation to a system of algebraic equations
Employing the introduced operational matrix of derivatives to numerically solve linear fourth-order boundary value problems (BVPs) based on the application of Galerkin method

Summary

Introduction

There is a huge number of articles handling both high odd- and high even-order BVPs. For example, in the sequence of papers, [ , – ], the authors have obtained numerical solutions for evenorder BVPs by applying the Galerkin method. The main idea for obtaining these solutions is to construct suitable basis functions satisfying the underlying boundary conditions on the given differential equation, applying Galerkin method to convert each equation to a system of algebraic equations.

Results

Conclusion