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.
- He J-H: A new approach to nonlinear partial differential equations. Communications in Nonlinear Science and Numerical Simulation 1997,2(4):230-235. 10.1016/S1007-5704(97)90007-1View ArticleGoogle Scholar
- He J-H: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering 1998,167(1-2):57-68. 10.1016/S0045-7825(98)00108-XMathSciNetView ArticleMATHGoogle Scholar
- He J-H: Approximate solution of nonlinear differential equations with convolution product nonlinearities. Computer Methods in Applied Mechanics and Engineering 1998,167(1-2):69-73. 10.1016/S0045-7825(98)00109-1View ArticleMATHGoogle Scholar
- He J-H: Variational iteration method—a kind of non-linear analytical technique: some examples. International Journal of Non-Linear Mechanics 1999,34(4):699-708. 10.1016/S0020-7462(98)00048-1View ArticleMATHGoogle Scholar
- He J-H: Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering 1999,178(3-4):257-262. 10.1016/S0045-7825(99)00018-3MathSciNetView ArticleMATHGoogle Scholar
- He J-H: Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation 2003,135(1):73-79. 10.1016/S0096-3003(01)00312-5MathSciNetView ArticleMATHGoogle Scholar
- He J-H: A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics 2000,35(1):37-43. 10.1016/S0020-7462(98)00085-7MathSciNetView ArticleMATHGoogle Scholar
- He J-H: Homotopy perturbation method for solving boundary value problems. Physics Letters A 2006,350(1-2):87-88. 10.1016/j.physleta.2005.10.005MathSciNetView ArticleMATHGoogle Scholar
- He J-H: Variational iteration method for autonomous ordinary differential systems. Applied Mathematics and Computation 2000,114(2-3):115-123. 10.1016/S0096-3003(99)00104-6MathSciNetView ArticleMATHGoogle Scholar
- He J-H: Variational iteration method for delay differential equations. Communications in Nonlinear Science and Numerical Simulation 1997,2(4):235-236. 10.1016/S1007-5704(97)90008-3View ArticleGoogle Scholar
- Abdou MA, Soliman AA: New applications of variational iteration method. Physica D 2005,211(1-2):1-8. 10.1016/j.physd.2005.08.002MathSciNetView ArticleMATHGoogle Scholar
- Abdou MA, Soliman AA: Variational iteration method for solving Burger's and coupled Burger's equations. Journal of Computational and Applied Mathematics 2005,181(2):245-251. 10.1016/j.cam.2004.11.032MathSciNetView ArticleMATHGoogle Scholar
- Momani S, Abuasad S: Application of He's variational iteration method to Helmholtz equation. Chaos, Solitons & Fractals 2006,27(5):1119-1123. 10.1016/j.chaos.2005.04.113MathSciNetView ArticleMATHGoogle Scholar
- El-Shahed M: Application of He's homotopy perturbation method to Volterra's integro-differential equation. International Journal of Nonlinear Sciences and Numerical Simulation 2005,6(2):163-168. 10.1515/IJNSNS.2005.6.2.163MathSciNetView ArticleGoogle Scholar
- He J-H: Application of He's homotopy perturbation method to nonlinear wave equations. Chaos Solitons & Fractals 1998,167(1-2):69-73.Google Scholar
- Abbasbandy S: Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method. Applied Mathematics and Computation 2006,172(1):485-490. 10.1016/j.amc.2005.02.014MathSciNetView ArticleMATHGoogle Scholar
- Ganji DD, Sadighi A: Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations. International Journal of Nonlinear Sciences and Numerical Simulation 2006,7(4):411-418. 10.1515/IJNSNS.2006.7.4.411View ArticleGoogle Scholar
- He J-H: Homotopy perturbation method for bifurcation of nonlinear problems. International Journal of Nonlinear Sciences and Numerical Simulation 2005,6(2):207-208. 10.1515/IJNSNS.2005.6.2.207MathSciNetGoogle Scholar
- He J-H: The homotopy perturbation method nonlinear oscillators with discontinuities. Applied Mathematics and Computation 2004,151(1):287-292. 10.1016/S0096-3003(03)00341-2MathSciNetView ArticleMATHGoogle Scholar
- Rafei M, Ganji DD: Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method. International Journal of Nonlinear Sciences and Numerical Simulation 2006,7(3):321-328. 10.1515/IJNSNS.2006.7.3.321View ArticleMATHGoogle Scholar
- He J-H: Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos, Solitons & Fractals 2004,19(4):847-851. 10.1016/S0960-0779(03)00265-0MathSciNetView ArticleMATHGoogle Scholar
- He J-H: Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics. International Journal of Turbo and Jet Engines 1997,14(1):23-28. 10.1515/TJJ.1922.214.171.124Google Scholar
- He J-H, Wan Y-Q, Guo Q: An iteration formulation for normalized diode characteristics. International Journal of Circuit Theory and Applications 2004,32(6):629-632. 10.1002/cta.300View ArticleMATHGoogle Scholar
- He J-H, Wu G-C, Austin F: The variational iteration method which should be followed. Nonlinear Science Letters A 2010,1(1):1-30.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.