The use of Adomian decomposition method for solving problems in calculus of variations

Mehdi Dehghan,Mehdi Tatari

doi:10.1155/mpe/2006/65379

Abstract

In this paper, a numerical method is presented for finding the solution of some variational problems. The main objective is to find the solution of an ordinary differential equation which arises from the variational problem. This work is done using Adomian decomposition method which is a powerful tool for solving large amount of problems. In this approach, the solution is found in the form of a convergent power series with easily computed components. To show the efficiency of the method, numerical results are presented.

Highlights

In the large number of problems arising in analysis, mechanics, geometry, and so forth, it is necessary to determine the maximal and minimal of a certain functional
We find the solution of variational problem by applying Adomian decomposition method on the Euler-Lagrange equations
Adomian decomposition method is used for finding the solution of the ordinary differential equations which arise from problems of calculus of variations

Summary

Introduction

In the large number of problems arising in analysis, mechanics, geometry, and so forth, it is necessary to determine the maximal and minimal of a certain functional. Because of the important role of this subject in science and engineering, considerable attention has been received on this kind of problems. The problem of brachistochrone is proposed in 1696 by Johann Bernoulli which is required to find the line connecting two certain points A and B that do not lie on a vectorial line and possessing the property that a moving particle slides down this line from A to B in the shortest time. In the problem of geodesics we want to determine the line of minimum length connecting two given points on a certain surface This problem was solved in 1698 by Jacob Bernoulli and a general method for solving such problems was given in the works of Euler and Lagrange.

Application of Adomian’s method in calculus of variations

Test examples

Conclusion