J. Comput. Complex. Appl. 2 (3) (2016) 46 - 48
New correction functional and Lagrange multipliers for two point value problems Guo-Cheng Wu* Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, Sichuan, China
Abstract For the two point value problem of differential equation, this paper considers a novel correctional functional and identify the Lagrange multipliers in non–differentiable functions. New variational iteration formula has a simpler form but more efficiently lead to approximate solutions than the existing results. Finally, some new possible applications are discussed. Keywords: Variational iteration method; New Lagrange multipliers; Correction functionals
1
Introduction
In the past ten years, with the benefit of the advancement of the computation science, some excellent analytical methods have been well developed for various initial or boundary value problems. All of these methods such as the Adomian decomposition method [1] and the variational iteration method (VIM) [2] should involve in or more or less use the integral equations. However, to the two point value problem, it’s not easy to determine one or all the equivalent integral equations like the Cauchy problem. In view of this point, all the methods generally use the trial function with unknown parameters [3]. According to the shooting method’s idea, one will re-consider the initial problem. In this paper, we consider the following simple two point value problem d2 u + u = 0, u(0) = u(1) = 0 dt2
(1)
and give a new general way to identify the Lagrange multipliers. This idea can be applied to obtain new variational iteration formulae of other two point value or boundary value problems.
2
New correction functional
According to the normal VIM [1], as we summarized in [3], to use the method to obtain variational iteration solutions, one generally follows to the following steps • Step I
Establish the correctional functionals;
• Step II Identify the Lagrange multipliers via the variation theories; • Step III Determine the initial iteration. ∗
Corresponding author. Email:
[email protected] (G.C. Wu) c Copyright⃝2015 SI JI PUBLISHING COMPANY LIMITED.
