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ISRN Computational Mathematics
Volume 2012 (2012), Article ID 138718, 3 pages
http://dx.doi.org/10.5402/2012/138718
Research Article

Solution of Wave Equation in Radial Form by VIM

Department of Applied Mathematics, School of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht 41938, Iran

Received 23 December 2011; Accepted 1 February 2012

Academic Editors: G. Bella and E. Weber

Copyright © 2012 Hossein Aminikhah. 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.

Abstract

An analytic approximation to the solution of wave equation is studied. Wave equation is in radial form with indicated initial and boundary conditions, by variational iteration method it has been used to derive this approximation and some examples are presented to show the simplicity and efficiency of the method.

1. Introduction

Wave equation has attracted much attention and solving these kind of equations has been one of the interesting tasks for mathematicians. Variational iteration method is known as a powerful device for solving functional equations [17]. Numerical methods which are commonly used as finite-difference methods and characteristics method need large size of computational works and usually the effect of round-off error causes the loss of accuracy in the results. Analytical methods commonly used for solving wave equation are very restricted and can be used in special cases, so they cannot be used to solve equations resulted by mathematical modeling of numerous realistic scenarios. In this article, the variational iteration method has been applied to solve more general forms of wave equation.

2. He’s Variational Iteration Method

The variational iteration method [813], which is a modified of general Lagrange multiplier method [14], has been shown to solve effectively, easily, and accurately large class of nonlinear problems with approximations which converge rapidly to accurate solutions. To illustrate the method, consider the following nonlinear equation:𝐿𝑢(𝑡)+𝑁𝑢(𝑡)=𝑔(𝑡),(1) where 𝐿 is a linear operator, 𝑁 is a nonlinear operator, and 𝑔(𝑡) is a known analytic function. According to the variational iteration method, we can construct the following correction functional: 𝑢𝑛+1(𝑡)=𝑢𝑛(𝑡)+𝑡0𝜆(𝜉)𝐿𝑢𝑛(𝜉)+𝑁𝑢𝑛(𝜉)𝑔(𝜉)𝑑𝜉,(2) where 𝜆 is general Lagrange multiplier which can be identified via variational theory, 𝑢0(𝑡) is an initial approximation with possible unknowns, and ̃𝑢𝑛 is considered as restricted variation [15] (i.e., 𝛿̃𝑢𝑛=0). Therefore, we first determine the Lagrange multiplier 𝜆 that will be identified optimally via integration by parts. The successive approximations 𝑢𝑛+1(𝑡) of the solution 𝑢(𝑡) will be readily obtained upon using the obtained Lagrange multiplier and by using any selective function 𝑢0. Consequently, the exact solution may be obtained by 𝑢=lim𝑛𝑢𝑛.

3. Numerical Results

To illustrate the method and to show ability of the method, some examples are presented.

Example 1. Let us have one-dimensional wave equation in radial form with initial condition: 𝜕2𝑢𝜕𝑡2=𝜕2𝑢𝜕𝑟2+1𝑟𝜕𝑢,𝑢𝑟𝜕𝑟𝑖,0=𝑟.(3) Its correction functional can be expressed as follows: 𝑢𝑛+1(𝑟,𝑡)=𝑢𝑛+(𝑟,𝑡)𝑡0𝜕(𝜉𝑡)2𝑢𝑛𝜕𝜉2𝜕2̃𝑢𝑛𝜕𝑟21𝑟𝜕̃𝑢𝑛𝜕𝑟𝑑𝜉.(4) To make this correct functional stationary, 𝛿𝑢𝑛(𝑟,0)=0, 𝛿𝑢𝑛+1=𝛿𝑢𝑛+𝜆1𝛿𝑢𝑛𝑡0𝜆1𝛿𝑢𝑛𝑡0+𝑡0𝜆1𝛿𝑢𝑛𝑑𝜉=0.(5) Its stationary conditions can be obtained as follows: 𝛿𝑢𝑛1𝜆1(𝑡)=0,𝛿𝑢𝑛𝜆1(𝑡)=0,𝛿𝑢𝑛𝜆1(𝜉)=0,(6) from which Lagrange multiplier can be identified as 𝜆1=𝜉𝑡, and the following iteration formula will be obtained: 𝑢𝑛+1(𝑟,𝑡)=𝑢𝑛+(𝑟,𝑡)𝑡0𝜕(𝜉𝑡)2𝑢𝑛𝜕𝜉2𝜕2𝑢𝑛𝜕𝑟21𝑟𝜕𝑢𝑛𝜕𝑟𝑑𝜉.(7) Beginning with 𝑢0(𝑟,𝑡)=𝑟, by iteration formula (7), we have 𝑢11(𝑟,𝑡)=𝑟+𝑡2!𝑟2,𝑢21(𝑟,𝑡)=𝑟+𝑡2!𝑟2+124!𝑟3𝑡4,𝑢31(𝑟,𝑡)=𝑟+𝑡2!𝑟2+124!𝑟3𝑡4+12×326!×𝑟5𝑡6,(8) from which the general term and so the solution will be determined as follows: 𝑢(𝑟,𝑡)=𝑟+𝑛=112×32××(2𝑛3)2(2𝑛)!𝑟2𝑛1𝑡2𝑛.(9)

Example 2. Let us solve two-dimensional wave equation in radial form with the boundary conditions: 𝜕2𝑢𝜕𝑡2=𝜕2𝑢𝜕𝑟2+1𝑟𝜕𝑢+1𝜕𝑟𝑟2𝜕2𝑢𝜕𝜃2,𝑢𝑟(1,𝜃,𝑡)=1,𝑢𝑟(10,𝜃,𝑡)=0,𝑢𝜃(𝑟,1,𝑡)=1.(10) Its correction functional can be expressed as follows: 𝑢𝑛+1(𝑟,𝜃,𝑡)=𝑢𝑛+(𝑟,𝜃,𝑡)𝑟1𝜆2𝜕2𝑢𝑛𝜕𝜉2+1𝜉𝜕𝑢𝑛+1𝜕𝜉𝜉2𝜕̃𝑢𝑛𝜕𝜃2𝜕2̃𝑢𝑛𝜕𝑡2𝑑𝜉.(11) Making the above correct functional stationary, notice that 𝛿𝑢𝑛(1,𝜃,𝑡)=0: 𝛿𝑢𝑛+1=𝛿𝑢𝑛+𝜆2𝛿𝑢𝑛𝑟1𝜆2𝛿𝑢𝑛𝑟1+𝑟1𝜆2𝛿𝑢𝑛𝜆𝑑𝜉+2𝜉𝛿𝑢𝑛𝑟1𝑟1𝜉𝜆2𝜆2𝜉2𝛿𝑢𝑛𝑑𝜉=0.(12) Its stationary conditions can be obtained as follows: 𝛿𝑢𝑛𝜆2𝜉𝜆2𝜆2𝜉2=0,𝛿𝑢𝑛𝜆2(𝑟)=0,𝛿𝑢𝑛1𝜆2(𝜆𝑟)+2(𝑟)𝑟=0,(13) from which the Lagrange multiplier would be identified as follows: 𝜆2(𝜉)=𝜉(ln𝜉ln𝑟).(14) Substituting (14) into (11) leads to the following iteration formula: 𝑢𝑛+1(𝑟,𝜃,𝑡)=𝑢𝑛+(𝑟,𝜃,𝑡)𝑟1𝜉𝜕(ln𝜉ln𝑟)2𝑢𝑛𝜕𝜉2+1𝜉𝜕𝑢𝑛+1𝜕𝜉𝜉2𝜕𝑢𝑛𝜕𝜃2𝜕2𝑢𝑛𝜕𝑡2𝑑𝜉.(15) Starting with 𝑢0(𝑟,𝜃,𝑡)=𝐴𝑟2+𝐵𝜃2+𝐶𝑡2,(16) we have 𝑢1(𝑟,𝜃,𝑡)=𝐴𝑟2+𝐵𝜃2+𝐶𝑡2+𝑟12𝜉(ln𝜉ln𝑟)4𝐴+𝜉2𝜃𝐵2𝐶𝑑𝜉=𝐴(1+2ln𝑟)+𝐵2(ln𝑟)2𝑡+𝐶2+𝑟2212.ln𝑟(17) Imposing the boundary conditions yields to 𝐴=1/2, 𝐵=1, 𝐶=1+ln10/99.
Thus, we have 𝑢11(𝑟,𝜃,𝑡)=198100ln10+(1+ln10)𝑟299(ln𝑟)2(200+2ln10)ln𝑟+99𝜃2+2(1+ln10)𝑡2,(18) which is an exact solution.

Example 3. Consider Example 1 with boundary conditions following: 𝑢(1,𝑡)=1,𝑢𝑟(1,𝑡)=𝑡.(19) Similar to Example 2, the Lagrange multiplier can be identified as 𝜆3(𝜉)=𝜉(ln𝜉ln𝑟) and the following iteration formula will be obtained: 𝑢𝑛+1(𝑟,𝑡)=𝑢𝑛+(𝑟,𝑡)𝑟1𝜉𝜕(ln𝜉ln𝑟)2𝑢𝑛𝜕𝜉2+1𝜉𝜕𝑢𝑛𝜕𝜕𝜉2𝑢𝑛𝜕𝑡2𝑑𝜉.(20) Starting with 𝑢0(𝑟,𝑡)=𝑟+𝑟1𝑡𝑑𝜉=1+𝑡(𝑟1),(21) by iteration formula (20), we have 𝑢1(𝑟,𝑡)=1+𝑡ln𝑟,(22) which is an exact solution.

4. Conclusion

In this work, we present an analytical approximation to the solution of wave equation in radial form in different cases. We have achieved this goal by applying variational iteration method. The small size of computations in comparison with the computational size required in numerical methods and the rapid convergence shows that the variational iteration method is reliable and introduces a significant improvement in solving the wave equation over existing methods. The main advantage of the VIM over A.D.M. is that this method provides the solution without a need for calculating Adomian’s polynomials [16].

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