An Approximate Solution for Boundary Value Problems in Structural Engineering and Fluid Mechanics

A. Barari,D. D. Ganji,M. Omidvar,Abbas Tahmasebi Poor

doi:10.1155/2008/394103

Abstract

Variational iteration method (VIM) is applied to solve linear and nonlinear boundary value problems with particular significance in structural engineering and fluid mechanics. These problems are used as mathematical models in viscoelastic and inelastic flows, deformation of beams, and plate deflection theory. Comparison is made between the exact solutions and the results of the variational iteration method (VIM). The results reveal that this method is very effective and simple, and that it yields the exact solutions. It was shown that this method can be used effectively for solving linear and nonlinear boundary value problems.

Highlights

This paper discusses the analytical approximate solution for fourth-order equations with nonlinear boundary conditions involving third-order derivatives
The results reveal that this method is very effective and simple, and that it yields the exact solutions
For u representing an elastic beam of length L 1, which is clamped at its left side x 0, and resting on an elastic bearing at its right side x 1, and adding a load f along its length to cause deformations Figure 1, Ma and Silva 4 arrived at the following boundary value problem assuming an EI 1: u iv x f x, u x, 0 < x < 1, 1.3 the boundary conditions were taken as u 0 u/ 0 0, 1.4 u// 1 0, u/// 1 g u 1, 1.5 where f ∈ C 0, 1 × R and g ∈ C R are real functions

Summary

Introduction

This paper discusses the analytical approximate solution for fourth-order equations with nonlinear boundary conditions involving third-order derivatives. For u representing an elastic beam of length L 1, which is clamped at its left side x 0, and resting on an elastic bearing at its right side x 1, and adding a load f along its length to cause deformations Figure 1 , Ma and Silva 4 arrived at the following boundary value problem assuming an EI 1:. With the rapid development of nonlinear science, many different methods were proposed to solve differential equations, including boundary value problems BVPS. These two methods are the homotopy perturbation method HPM 5–7 and the variational iteration method VIM 8–17. It is aimed to apply the variational iteration method proposed by He 14 to different forms of 1.1 subject to boundary conditions of physical significance

Basic idea of He’s variational iteration method

The applications of VIM method

Conclusion