Solution of Wave Equation in Radial Form by VIM

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 [1–7]. 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 [8–13], 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: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡0𝜆(𝜉)𝐿𝑢_𝑛(𝜉)+𝑁𝑢_𝑛(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 𝑢_𝑛̃ is considered as restricted variation [15] (i.e., 𝛿𝑢_𝑛̃=0). Therefore, we first determine the Lagrange multiplier 𝜆 that will be identified optimally via integration by parts. The successive approximations 𝑢_𝑛+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=lim_𝑛→∞𝑢_𝑛.

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+1𝑢𝑟𝑟𝜕𝜕𝑢𝑟_𝑖,,0=𝑟.(3) Its correction functional can be expressed as follows: 𝑢_𝑛+1(𝑟,𝑡)=+𝑢_𝑛(𝑟,𝑡)_𝑡0(𝜉−𝑡)𝜕²𝑢_𝑛𝜕𝜉²−𝜕²𝑢_𝑛̃𝜕𝑟²−1𝑟𝜕𝜕𝑢𝑟^𝑛̃𝑑𝜉.(4) To make this correct functional stationary, 𝛿𝑢_𝑛(𝑟,0)=0, 𝛿𝑢_𝑛+1=𝛿𝑢_𝑛+𝜆₁𝛿𝑢_𝑛^^𝑡0−𝜆^1𝛿𝑢_𝑛^𝑡0+^𝑡0𝜆^1𝛿𝑢_𝑛𝑑𝜉=0.(5) Its stationary conditions can be obtained as follows: 𝛿𝑢𝛿_𝑛𝑢∶_𝑛1−𝜆^1(𝑡)=0,𝛿𝑢_𝑛∶𝜆₁(𝑡)=0,∶𝜆^1(𝜉)=0,(6) from which Lagrange multiplier can be identified as 𝜆₁=𝜉−𝑡, and the following iteration formula will be obtained: 𝑢_𝑛+1(𝑟,𝑡)=+𝑢_𝑛(𝑟,𝑡)_𝑡0(𝜉−𝑡)𝜕²𝑢_𝑛𝜕𝜉²−𝜕²𝑢_𝑛𝜕𝑟²−1𝑟𝜕𝜕𝑢𝑟^𝑛𝑑𝜉.(7) Beginning with 𝑢₀(𝑟,𝑡)=𝑟, by iteration formula (7), we have 𝑢₁(𝑟,𝑡)=𝑟+12!𝑟𝑡²𝑢₂,(𝑟,𝑡)=𝑟+12!𝑟𝑡²+1²4!𝑟³𝑢₃𝑡⁴,(𝑟,𝑡)=𝑟+12!𝑟𝑡²+1²4!𝑟³𝑡⁴+1²×3²6!×𝑟⁵⋮𝑡⁶,(8) from which the general term and so the solution will be determined as follows: 𝑢(𝑟,𝑡)=𝑟+^∞_𝑛=11²×3²×⋯×(2𝑛−3)²(2𝑛)!𝑟^2𝑛−1𝑡^2𝑛.(9)

Example 2. Let us solve two-dimensional wave equation in radial form with the boundary conditions: 𝜕²𝑢𝜕𝑡²=𝜕²𝑢𝜕𝑟²+1𝑟𝜕𝜕𝑢𝑟+𝑟1²𝜕²𝑢𝜕𝜃²𝑢_𝑟,(1,𝜃,𝑡)=−1,𝑢_𝑟(10,𝜃,𝑡)=0,𝑢_𝜃(𝑟,1,𝑡)=1.(10) Its correction functional can be expressed as follows: 𝑢_𝑛+1(𝑟,𝜃,𝑡)=+𝑢_𝑛(𝑟,𝜃,𝑡)_𝑟1𝜆₂𝜕²𝑢_𝑛𝜕𝜉²+1𝜉𝜕𝜕𝑢𝜉^𝑛+𝜉1²𝜕𝑢_𝑛̃𝜕𝜃²−𝜕²𝑢_𝑛̃𝜕𝑡²𝑑𝜉.(11) Making the above correct functional stationary, notice that 𝛿𝑢_𝑛(1,𝜃,𝑡)=0: 𝛿𝑢_𝑛+1=𝛿𝑢_𝑛+𝜆₂𝛿𝑢_𝑛^^𝑟1−𝜆^2𝛿𝑢_𝑛^𝑟1+^𝑟1𝜆^2𝛿𝑢_𝑛𝑑𝜉+𝜆²𝜉𝛿𝑢^𝑛^𝑟1−^𝑟1𝜉𝜆^2−𝜆₂𝜉²𝛿𝑢_𝑛𝑑𝜉=0.(12) Its stationary conditions can be obtained as follows: 𝛿𝑢_𝑛∶𝜆^2−𝜉𝜆^2−𝜆₂𝜉²𝛿𝑢_𝑛=0,𝛿𝑢_𝑛∶𝜆₂(𝑟)=0,∶1−𝜆^2(𝑟)+𝜆²(𝑟)𝑟=0,(13) from which the Lagrange multiplier would be identified as follows: 𝜆₂(𝜉)=𝜉(ln𝜉−ln𝑟).(14) Substituting (14) into (11) leads to the following iteration formula: 𝑢_𝑛+1(𝑟,𝜃,𝑡)=+𝑢_𝑛(𝑟,𝜃,𝑡)_𝑟1𝜉(ln𝜉−ln𝑟)𝜕²𝑢_𝑛𝜕𝜉²+1+1𝜉𝜕𝜕𝑢𝜉^𝑛𝜉²𝜕𝑢_𝑛𝜕𝜃²−𝜕²𝑢_𝑛𝜕𝑡²𝑑𝜉.(15) Starting with 𝑢₀(𝑟,𝜃,𝑡)=𝐴𝑟²+𝐵𝜃²+𝐶𝑡²,(16) we have 𝑢₁(𝑟,𝜃,𝑡)=𝐴𝑟²+𝐵𝜃²+𝐶𝑡²+^𝑟1𝜉(ln𝜉−ln𝑟)4𝐴+2𝜉²=𝐴(1+2ln𝑟)+𝐵𝜃₂𝐵−2𝐶𝑑𝜉+𝐶𝑡₂−(ln𝑟)²+𝑟²2−21−ln𝑟.(17) Imposing the boundary conditions yields to 𝐴=−1/2, 𝐵=1, 𝐶=1+ln10/99.
Thus, we have 𝑢₁(𝑟,𝜃,𝑡)=1198−100−ln10+(1+ln10)𝑟²−99(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,𝑢_𝑟(1,𝑡)=𝑡.(19) Similar to Example 2, the Lagrange multiplier can be identified as 𝜆₃(𝜉)=𝜉(ln𝜉−ln𝑟) and the following iteration formula will be obtained: 𝑢_𝑛+1(𝑟,𝑡)=+𝑢_𝑛(𝑟,𝑡)_𝑟1𝜉(ln𝜉−ln𝑟)𝜕²𝑢_𝑛𝜕𝜉²+1𝜉𝜕𝜕𝑢𝜉^𝑛−𝜕²𝑢_𝑛𝜕𝑡²𝑑𝜉.(20) Starting with 𝑢₀(𝑟,𝑡)=𝑟+^𝑟1𝑡𝑑𝜉=1+𝑡(𝑟−1),(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].