Open Access

Comparison between the Variational Iteration Method and the Homotopy Perturbation Method for the Sturm-Liouville Differential Equation

Boundary Value Problems20102010:317369

DOI: 10.1155/2010/317369

Received: 28 October 2009

Accepted: 10 April 2010

Published: 17 May 2010

Abstract

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.

1. Introduction

The variational iteration method (VIM) [14] and homotopy perturbation method (HPM) [58], proposed by He, are powerful analytical methods for various kinds of linear and nonlinear problems. For example, the variational iteration method has been applied to autonomous ordinary differential equation [9] and delay differential equation [10]. Abdou and Soliman applied this method to Schrodinger-KDV, generalized KDV, and Shallow water equations [11], Burger's equations, and coupled Burger's equations [12]. Furthermore, Momani and Abuasad [13] used VIM for Helmoltz partial equation. Also homotopy perturbation method was successfully applied to Voltra's integrodifferential equation [14], boundary value problem [8], nonlinear wave equations [15], and so forth; see [1620]. In this paper, we exert these methods for linear Sturm-Liouville eigenvalue and boundary value problems (BVPs). A linear Sturm-Liouville operator has the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ1_HTML.gif
(1.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ2_HTML.gif
(1.2)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq1_HTML.gif is known analytic function representing the nonhomogeneous term. Associated with the differential equation (1.1) are the following separated homogeneous boundary conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq3_HTML.gif are arbitrary constants. For simplicity, we will assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq5_HTML.gif are continuous. The values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq6_HTML.gif for which BVP has a nontrivial solution are called eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq7_HTML.gif , 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

To illustrate the basic concept of He's variational iteration method [14], we consider the following nonlinear differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ4_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq8_HTML.gif is a linear operator, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq9_HTML.gif is a nonlinear operator, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq10_HTML.gif is a nonhomogeneous term. He has modified the general Lagrange multiplier method into an iteration method which is called correction functional as follows [14, 9]:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ5_HTML.gif
(2.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq11_HTML.gif is a general Lagrange multiplier, which can be identified optimally via the variational theory [21]. The subscript https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq12_HTML.gif denotes the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq13_HTML.gif th approximation, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq14_HTML.gif is considered as a restricted variation [14], that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq15_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq16_HTML.gif can be exactly identified.

3. Homotopy Perturbation Method

In this section, we will present a review of the homotopy perturbation method. To clarify the basic idea of the HPM [58], we consider the following nonlinear differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ6_HTML.gif
(3.1)
with boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ7_HTML.gif
(3.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq17_HTML.gif is a general differential operator, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq18_HTML.gif is a boundary operator, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq19_HTML.gif is a known analytic function, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq20_HTML.gif is the boundary of the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq21_HTML.gif . The operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq22_HTML.gif can, generally speaking, be divided into parts https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq24_HTML.gif while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq25_HTML.gif is nonlinear. Equation (3.1), therefore, can be rewritten as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ8_HTML.gif
(3.3)
By the homotopy technique, we construct a homotopy as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ9_HTML.gif
(3.4)
which satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ10_HTML.gif
(3.5)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ11_HTML.gif
(3.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq26_HTML.gif is an embedding parameter, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq27_HTML.gif is an initial approximation of (3.1) which satisfies the boundary conditions. Obviously, from (3.5), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ12_HTML.gif
(3.7)
The changing process of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq28_HTML.gif from zero to unity is just that of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq29_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq30_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq31_HTML.gif . In topology, this is called deformation and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq32_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq33_HTML.gif are called homotopic. According to HPM, we can assume that the solution of (3.5) can be written as a power series in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq34_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ13_HTML.gif
(3.8)
Setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq35_HTML.gif results in the approximate solution  (3.2):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ14_HTML.gif
(3.9)

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

To solve (1.1), by means of homotopy perturbation method, we choose linear operator
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ15_HTML.gif
(4.1)
with the property https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq36_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq37_HTML.gif is constant of integration and suggests that we define a nonlinear operator as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq38_HTML.gif . Also https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq39_HTML.gif is known analytic function representing the nonhomogeneous term. Therefore, (1.1) can be rewritten as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ16_HTML.gif
(4.2)
By the homotopy perturbation technique proposed by He [58], we can construct a homotopy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ17_HTML.gif
(4.3)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ18_HTML.gif
(4.4)
One may now try to obtain a solution of (4.2) in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ19_HTML.gif
(4.5)
where the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq40_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq41_HTML.gif are functions yet to be determined. Substituting (4.5) into (4.4) yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ20_HTML.gif
(4.6)
Collecting terms of the same powers of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq42_HTML.gif yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ21_HTML.gif
(4.7)

The initial approximation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq43_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq44_HTML.gif 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.

Example 5.1.

Consider the Sturm-Liouville equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ22_HTML.gif
(5.1)
with initial approximation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ23_HTML.gif
(5.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq46_HTML.gif are constants. To solve (5.1) using the VIM, we have correction functional
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ24_HTML.gif
(5.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq47_HTML.gif is Lagrange multiplier. Making the above correction functional stationary, we can obtain the following stationary conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ25_HTML.gif
(5.4)
The Lagrange multiplier can, therefore, be identified as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ26_HTML.gif
(5.5)
Substituting (5.5) for correction functional (5.3), we have the following iteration formula:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ27_HTML.gif
(5.6)
Using the iteration formula (5.6) and initial approximation (5.2), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ28_HTML.gif
(5.7)
In the same way, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ29_HTML.gif
(5.8)
which means that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ30_HTML.gif
(5.9)

is the exact solution of (5.1).

In order to solve (5.1) using the HPM according to (4.4), we can readily construct a homotopy which satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ31_HTML.gif
(5.10)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ32_HTML.gif
(5.11)
We consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq48_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ33_HTML.gif
(5.12)
Substituting (5.12) into (5.11), collecting terms of the same power, and using initial approximation, we have the following set of linear equations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ34_HTML.gif
(5.13)
Solving the above equations, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ35_HTML.gif
(5.14)
Continuing in this manner, we can obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ36_HTML.gif
(5.15)

which is exactly the same as that obtained by VIM.

Example 5.2.

As another example, we consider Sturm-Liouville problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ37_HTML.gif
(5.16)
with initial conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ38_HTML.gif
(5.17)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq50_HTML.gif are constants. To solve (5.16) by means of variational method, we construct a correction functional
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ39_HTML.gif
(5.18)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq51_HTML.gif is the Lagrange multiplier and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq52_HTML.gif denotes restricted variation that is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq53_HTML.gif . Then, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ40_HTML.gif
(5.19)
Calculus of variations and integration by parts give the stationary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ41_HTML.gif
(5.20)
for which the Lagrange multiplier https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq54_HTML.gif should satisfy. The Lagrange multiplier can, therefore, be identified as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ42_HTML.gif
(5.21)
Substituting (5.21) into correction functional (5.18) results in the following iteration formula:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ43_HTML.gif
(5.22)
According to initial conditions (5.17), it is natural to choose initial approximation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq55_HTML.gif Using the above variational formula (5.22), we can obtain the following result:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ44_HTML.gif
(5.23)
In order to solve system (5.16)-(5.17) using HPM, after applying HPM and rearranging based on powers of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq56_HTML.gif -terms, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ45_HTML.gif
(5.24)
Solving the above equations, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ46_HTML.gif
(5.25)

Example 5.3.

Finally, we consider eigenvalue Sturm-Liouville problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ47_HTML.gif
(5.26)
along with the Dirichlet boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ48_HTML.gif
(5.27)
To solve (5.26) by means of variational method, we construct a correction functional for (5.26) that reads as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ49_HTML.gif
(5.28)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq57_HTML.gif is Lagrange multiplier. Following the discussion presented in the previous example, we obtain the following iteration formula:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ50_HTML.gif
(5.29)
Let us begin with an initial approximation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq58_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq60_HTML.gif are constants to be determined. Substituting the proposed initial iterate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq61_HTML.gif in (5.29) gives
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ51_HTML.gif
(5.30)
In the same way, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ52_HTML.gif
(5.31)
So, we can derive that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ53_HTML.gif
(5.32)

is the exact solution of (5.26).

In order to solve (5.26) using HPM, similar to previous examples, after applying HPM and rearranging based on powers of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq62_HTML.gif -terms, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ54_HTML.gif
(5.33)
Now, we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq63_HTML.gif . Solving the above sets of equations yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ55_HTML.gif
(5.34)
Hence, from (4.4) we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ56_HTML.gif
(5.35)
which is exactly the same as that obtained by VIM. Now, we use the boundary condition (5.27) to obtain eigenvalue and eigenfunctions of (5.26). Imposing the boundary conditions in (5.35) yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ57_HTML.gif
(5.36)
So, there are two infinite sequences of eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq64_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ58_HTML.gif
(5.37)
Thus, corresponding linearly nontrivial solutions are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ59_HTML.gif
(5.38)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq66_HTML.gif are of class https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq67_HTML.gif , that is, are continuous real-valued functions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq68_HTML.gif , using the definition of inner product on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq69_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ60_HTML.gif
(5.39)
and the norm induced by inner product
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ61_HTML.gif
(5.40)
we get the normalization constants as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ62_HTML.gif
(5.41)
Consequently, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_Equ63_HTML.gif
(5.42)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq70_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq71_HTML.gif are normalized eigenfunctions, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F317369/MediaObjects/13661_2009_Article_915_IEq73_HTML.gif .

6. Conclusion

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 [24], 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.

Authors’ Affiliations

(1)
Department of Mathematics, University of Mazandaran
(2)
Department of Mathematics, Islamic Azad University Neka Branch

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Copyright

© A. Neamaty and R. Darzi. 2010

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.