Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials

W M Abd-Elhameed

doi:10.1155/2015/672703

Abstract

The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.

Highlights

Spectral methods play prominent roles in various fields of applied science such as fluid dynamics
The main objective of this paper is to introduce a novel Galerkin operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials and employing it for solving both linear and nonlinear third-order boundary value problems (BVPs) based on the application of Galerkin and collocation methods
We extend the definition of shifted Jacobi polynomials to include the cases in which α and/or β ≤ −1

Summary

Introduction

Spectral methods play prominent roles in various fields of applied science such as fluid dynamics. In the two papers [4, 5], the authors obtained numerical algorithms for solving high even- and high odd-order boundary value problems (BVPs) by applying the Galerkin and Petrov-Galerkin methods. They constructed combinations of orthogonal polynomials satisfying the underlying boundary conditions on the given BVP, applying the Galerkin method on even-order BVPs and a Petrov-Galerkin method on oddorder BVPs for the sake of converting each equation with its boundary conditions to a system of algebraic equations. The application of Galerkin and Petrov-Galerkin methods on linear problems has a great advantage that their applications enable one to investigate carefully the resulting systems, especially their complexities and condition numbers

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Conclusion