Numerical Solutions of a Class of Second Order Boundary Value Problems on Using Bernoulli Polynomials

Md Shafiqul Islam,Afroza Shirin

doi:10.4236/am.2011.29147

Abstract

The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions. We use the Bernoulli polynomials as linear combination to the approximate solutions of 2nd order boundary value problems. Here the Bernoulli polynomials over the interval [0, 1] are chosen as trial functions so that care has been taken to satisfy the corresponding homogeneous form of the Dirichlet boundary conditions in the Galerkin weighted residual method. In addition to that the given differential equation over arbitrary finite domain [a, b] and the boundary conditions are converted into its equivalent form over the interval [0, 1]. All the formulas are verified by considering numerical examples. The approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. A reliable good accuracy is obtained in all cases.

Highlights

There are many linear and nonlinear problems in science and engineering, namely second order differential equations with various types of boundary conditions, are solved either analytically or numerically
The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions
The Bernoulli polynomials over the interval [0, 1] are chosen as trial functions so that care has been taken to satisfy the corresponding homogeneous form of the Dirichlet boundary conditions in the Galerkin weighted residual method

Summary

Introduction

There are many linear and nonlinear problems in science and engineering, namely second order differential equations with various types of boundary conditions, are solved either analytically or numerically. In the literature of numerical analysis solving a two point second order boundary value problem (BVP) of differential equations, many authors have attempted to obtain higher accuracy rapidly by using a numerous methods. The purpose of this paper is to present the Galerkin weighted residual method to solve both linear and nonlinear second order BVP with all types of boundary conditions as well. We derive the individual formulas for each BVP consisting of Dirichlet, Neumann and Robin boundary conditions, respectively Numerical examples, for both linear and nonlinear boundary value problems, are considered to verify the effectiveness of the derived formulas, and are compared with the exact solutions.

Formulation of Second Order BVP

Numerical Examples

Conclusions