Comparison between the Variational Iteration Method and the Homotopy Perturbation Method for the Sturm-Liouville Differential Equation
© A. Neamaty and R. Darzi. 2010
Received: 28 October 2009
Accepted: 10 April 2010
Published: 17 May 2010
We applied the variational iteration method and the homotopy perturbation method to solve Sturm-Liouville eigenvalue and boundary value problems. The main advantage of these methods is the flexibility to give approximate and exact solutions to both linear and nonlinear problems without linearization or discretization. The results show that both methods are simple and effective.
where and are arbitrary constants. For simplicity, we will assume that and are continuous. The values of for which BVP has a nontrivial solution are called eigenvalues of , and a nontrivial solution corresponding to an eigenvalue is called an eigenfunction.
The paper is organized as follows: in Sections 2 and 3, an analysis of the variational iteration and homotopy perturbation methods will be given. In Section 4, we apply HPM to solve Sturm-Liouville problems. We present 3 examples to show the efficiency and simplicity of the proposed methods in Section 5. Finally, we give our conclusions in Section 6.
2. He's Variational Iteration Method
where is a general Lagrange multiplier, which can be identified optimally via the variational theory . The subscript denotes the th approximation, and is considered as a restricted variation [1–4], that is, . Employing the restricted variation in (2.2) makes it easy to compute the Lagrange multiplier; see [22, 23]. It is shown that this method is very effective and easy and can solve a large class of nonlinear problems. For linear problems, its exact solution can be obtained only one iteration because can be exactly identified.
3. Homotopy Perturbation Method
The coupling of the perturbation method and the homotopy method is called the homotopy perturbation method which has eliminated limitations of the traditional perturbation method. On the other hand, the proposed technique can take full advantage of the traditional perturbations techniques.
4. Applying HPM to Solve Sturm-Liouville Problem
The initial approximation or can be freely chosen.
5. The Applications
To incorporate our discussion above, three special cases of the Sturm-Liouville equation (1.1) will be studied.
is the exact solution of (5.1).
which is exactly the same as that obtained by VIM.
is the exact solution of (5.26).
where and are normalized eigenfunctions, that is, and .
In this work, we proposed variational method and compared with homotopy perturbation method to solve ordinary Sturm-Liouville differential equation. The variational iteration algorithm used in this paper is the variational iteration algorithm-I; there are also variational iteration algorithm-II and variational iteration algorithm-III , which can also be used for the present paper. It may be concluded that the two methods are powerful and efficient techniques to find exact as well as approximate solutions for wide classes of ordinary differential equations.
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