He's Variational Iteration Method

He's Variational Iteration Method

DOI: 10.4018/978-1-5225-2713-8.ch003
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In this chapter, a variational iteration method (VIM) has been applied to nonlinear heat transfer equation. The concept of the variational iteration method is introduced briefly for applying this method for problem solving. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. The results reveal that the VIM is very effective and convenient in predicting the solution of such problems.
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Basic Idea of He's Variational Iteration Method

To illustrate the basic concepts of variational iteration method, we consider the following deferential equation:

(1) where 978-1-5225-2713-8.ch003.m02 is a linear operator, 978-1-5225-2713-8.ch003.m03 a nonlinear operator, and 978-1-5225-2713-8.ch003.m04 a heterogeneous term. According to VIM, we can construct a correction functional as follows:
(2) where 978-1-5225-2713-8.ch003.m06 is a general Lagrangian multiplier (He, 1998a, 1999b), which can be identified optimally via the variational theory (He, 1999b), the subscript 978-1-5225-2713-8.ch003.m07 indicates the nth order approximation, 978-1-5225-2713-8.ch003.m08 which is considered as a restricted variation, i.e. 978-1-5225-2713-8.ch003.m09.



Most of engineering problems, especially some heat transfer equations are nonlinear, and in most cases it is difficult to solve them, especially analytically. Perturbation method is one of the well-known methods to solve nonlinear problems, it is based on the existence of small/large parameters, the so-called perturbation quantity (Cole, 1968; Nayfeh, 2000). Many nonlinear problems do not contain such kind of perturbation quantity, and we can use non perturbation methods, such as the artificial small parameter method (Lyapunov, 1992), the 978-1-5225-2713-8.ch003.m10 -expansion method (Karmishin, Zhukov & Kolosov, 1990), the Adomian’s decomposition method (Adomian, 1994), the homotopy perturbation method (HPM) (He, 2005, 2007; Ganji & Rajabi, 2006; He, 2006, 2000), and the variational iteration method (VIM) (He, 2000, 1998a; Ganji & Sadighi, 2006; Ganji, Jannatabadi & Mohseni, 2006).

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