The use of He's variational iteration method for solving variational problems

Abstract

In this paper, the variational iteration method is implemented to give approximate solution of the Euler–Lagrange, Euler–Poisson and Euler–Ostrogradsky equations as ordinary (or partial) differential equations which arise from the variational problems. In this method, general Lagrange multipliers are introduced to construct correction functional for the variational problems. The multipliers in the functionals can be identified optimally via the variational theory. The initial approximations can be freely chosen with possible unknown constant, which can be determined by imposing the boundary or initial conditions. Illustrative examples are included to demonstrate the validity and applicability of the variational iteration method.

Keywords:

• variational iteration method
• variational problems
• Euler–Lagrange equation
• Euler–Poisson equation
• Euler–Ostrogradsky equation

2000 AMS Subject Classifications: :

• 49M05
• 65L05
• 65L10
• 65M99
• 34K60

Acknowledgements

The authors are very grateful to four reviewers for carefully reading the paper.