On the Homotopy Analysis Method and Optimal Value of the Convergence Control Parameter: Solution of Euler-Lagrange Equation

Jafar Saberi-Nadjafi,Hassan Saberi Nik,Reza Buzhabadi

doi:10.4236/am.2012.38129

Jafar Saberi-Nadjafi, Hassan Saberi Nik + Show 1 more

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https://doi.org/10.4236/am.2012.38129

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Journal: Applied mathematics	Publication Date: Jan 1, 2012
Citations: 2	License type: CC BY 4.0

Affiliation: Ferdowsi University of Mashhad

Abstract

This paper presents, an efficient approach for solving Euler-Lagrange Equation which arises from calculus of variations. Homotopy analysis method to find an approximate solution of variational problems is proposed. An optimal value of the convergence control parameter is given through the square residual error. By minimizing the the square residual error, the optimal convergence-control parameters can be obtained. It is showed that the homotopy analysis method was valid and feasible to the study of variational problems.

Highlights

There has been a considerable renewal of interest in the classical problems of the calculus of variations both from the point of view of mathematics and of applications in physics, engineering, and applied mathematics
This paper presents, an efficient approach for solving Euler-Lagrange Equation which arises from calculus of variations
We will adopt the homotopy analysis method (HAM), for solving the Euler-Lagrange equation, which arises from problems in calculus of variations

Summary

Introduction

There has been a considerable renewal of interest in the classical problems of the calculus of variations both from the point of view of mathematics and of applications in physics, engineering, and applied mathematics. Chen and Hsiao [4] introduced the Walsh series method to variational problems. One of the semi-exact methods for solving nonlinear Equation which does not need small/large parameters is HAM, first proposed by Liao in 1992 [9,10,11,12,13]. Since Liao [10] for the homotopy analysis method was published in 2003, more and more researchers have been successfully applying this method to various nonlinear problems in science and engineering, such as the viscous flows of non-Newtonian fluids [14], the KdV-type equations [15],. We will adopt the homotopy analysis method (HAM), for solving the Euler-Lagrange equation, which arises from problems in calculus of variations

Basic Idea of HAM

Statement of the Problem

Numerical Results

Conclusion

HAM Exact