An elegant operational matrix based on harmonic numbers: Effective solutions for linear and nonlinear fourth-order two point boundary value problems

W M Abd-Elhameed

doi:10.15388/na.2016.4.2

Abstract

This paper analyzes the solution of fourth-order linear and nonlinear two point boundary value problems. The suggested method is quite innovative and it is completely different from all previous methods used for solving such kind of boundary value problems. The method is based on employing an elegant operational matrix of derivatives expressed in terms of the well-known harmonic numbers. Two algorithms are presented and implemented for obtaining new approximate solutions of linear and nonlinear fourth-order boundary value problems. The two algorithms rely on employing the new introduced operational matrix for reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. For this purpose, the two spectral methods namely, Petrov-Galerkin and collocation methods are applied. Some illustrative examples are considered aiming to ascertain the wide applicability, validity, and efficiency of the two proposed algorithms. The obtained numerical results are satisfactory and the approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by some other existing techniques in literature.

Highlights

Spectral methods on bounded domains typically employ grids consisting of zeros of Chebyshev polynomials, or zeros of Legendre polynomials, or some other points related to various orthogonal polynomials
We aim to introduce a novel operational matrix of derivatives in terms of the well-known harmonic numbers, and employ the introduced operational matrix to numerically solve both of linear and nonlinear fourth-order boundary value problems
A novel operational matrix of derivatives is introduced. This operational matrix is given in terms of the well-known harmonic numbers

Summary

Introduction

Spectral methods on bounded domains typically employ grids consisting of zeros of Chebyshev polynomials, or zeros of Legendre polynomials, or some other points related to various orthogonal polynomials (see [34]). The aim of spectral methods is to approximate functions (solutions of differential equations) by means of truncated series of orthogonal polynomials. The choice of the suitable used spectral method suggested for solving the given equation c Vilnius University, 2016 depends certainly on the type of the differential equation and on the type of the boundary conditions governed by it. In Galerkin method, the test functions are the same as the trial functions and they are chosen such that each member of them satisfies the boundary conditions governed by the given differential equation. The main difference between Galerkin and Petrov–Galerkin methods is that the test functions in Petrov– Galerkin methods are not identical with the trial functions unlike Galerkin methods (see, for example, [9])

Objectives

Methods

Conclusion