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The basic aim of this paper is to introduce and describe an efficient numerical scheme based on spectral approach coupled with Chebyshev wavelets for the approximate solutions of Klein-Gordon and Sine-Gordon equations. The main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. To show the accuracy and the efficiency of the method, several benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that the proposed method is superior to other existing ones and is highly accurate

Many physical phenomena encountered in science and engineering are governed by ordinary as well as partial differential equations. Some disciplines that use partial differential equations to describe the phenomena of interest are fluid mechanics, solid mechanics, quantum mechanic, propagation of acoustic and electromagnetic waves and problems in heat and mass transfer. Many linear and nonlinear phenomena appear in several areas of scientific fields like physics, chemistry and biology can be modeled by different type of partial differential equation such as evolution equation, reaction diffu- sion equation, Schrodinger type wave equations, Vander Poll’s equation, Telegraph equation, Lyapunov equation etc. A broad class of analytical methods and numerical methods available in the literature are used to handle these problems. In this present work we are dealing with two partial differential equation named as Klein-Gordon and Sine-Gordon equations. The Klein-Gordon equation is as follows:

where

Wavelet analysis had made a lot of successes in different fields of science and engineering due to its beautiful properties such as orthogonality, multi-resolution analysis and computational efficiency. Wavelet permits the accurate representation of a variety of functions and operators. Wavelet analysis and wavelet transform are recently developed mathematical tool for solving the linear and non-linear ordinary differential equations, partial differential equations and integral equation. Wavelets also applied in numerous disciplines such as image compression, data compression and deionising data. Most commonly wavelets are Haar, Legendre, Chebyshev are used to find the numerical solution of partial differential equations. In addition wavelet approach can make a connection with some fast and reliable numerical methods. The spectral method has the advantage of exponential convergence property when orthogonal basis functions are involved. As a result, it plays a vital role in solving partial differential equation. It is important to choose the basis function for possible coupling with spectral method. The wavelet basis can combine the advantages of both infinitely differentiable and small compact support which is far better than the spectral and finite element basis.

In recent year, spectral method [

The rest of the paper is as follows: In Section 2, Chebyshev wavelet and its properties are discussed. Operational matrix of derivative required for our subsequent development is presented in Section 3. Section 4 is devoted to present the Chebyshev wavelets spectral collocation method for solving Klein-Gordon and Sine-Gordon equations then approximate the unknown function. Section 5 deals with the illustrative examples and their solutions by the proposed approach compared with exact as well as with existing literature. Finally, concluding remarks are made in Section 6.

In the past decades, wavelets [

If we choose

These family of functions are a wavelet basis for

Chebyshev wavelets

where

with respect to the weight function

Moreover, the set of Chebyshev wavelet are an orthogonal set with respect to the weight function

Any function

where the wavelet coefficients of the series representation in (5) become

If the infinite series in (5) is truncated then Equation (5) can be written as

where C and

In this section, we first derive the operational matrix D of derivative which plays a great role in order to reducing the given problem into solving the system of algebraic equation. For this, we concern with some Theorem and Corollary as follows.

Theorem 1 [

where D is

in which O is an

Corollary 1. By using Equation (10), the operational matrix for nth derivative can be derived as

where

In different type of numerical methods, spectral methods are one of the most popular methods of discretization for the numerical solution of partial differential equations and integral equations. The main advantage of this method lies in their accuracy for a given number of unknowns. For smooth problems in simple geometries, they offer exponential rates of convergence or spectral accuracy. In the recent literature, Galerkin, collocation, and Tau methods are the three most widely used spectral versions, in which collocation methods have become increasingly popular for solving differential equations, also they are very useful in providing highly accurate solutions to nonlinear differential equations. Now, we focus on the solution nature of this method as follows:

Let us consider the equation in the form:

with the initial conditions

or boundary conditions

In order to transform the arbitrary domain

By employing q-weight scheme [

where

Now Equation (16) becomes

In the light of Equation (7),the term

Submitting Equation (18) into Equation (17), we have

in which

Also, by using the boundary conditions given in Equation (15), one can get

Collocating Equation (19) in

Equation (20) and (21) can be written as matrix form

where A and B are

Again using the first and second initial conditions given in Equation (14), we have

and

Equation (24) can be written as

Equation (22) using Equation (23) gives a linear system of equations with

where

In this section, we use Chebyshev wavelets spectral collocation method described in section 4 to solve nonlinear type of Klein-Gordon and Sine-Gordon equations. The proposed method provides a reliable technique which is computer oriented if compared with traditional techniques. To give the clear overview of this method we consider three examples of Klein-Gordon equation and Sine-Gordon equation. All the results are calculated by using the symbolic calculus software MATLAB 2013a and Mathematica.

Example 1 [

and the Dirichlet boundary condition

The analytical solution is given by

The obtained

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Example 2 [

and the Dirichlet boundary condition

The analytical solution is given by

The

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Example 3 [

where

and the Dirchlet boundary conditions

The exact solution is given by

The numerical solution of Sine-Gordon equation has presented in

In this article, we have proposed an efficient and accurate method based on Chebyshev wavelets to solve both Klein-Gordon and Sine-Gordon equations arising in different field of sciences, engineering and technology. The main advantage of this method is that it transforms the problem into algebraic equation so that the computation is effective and simple. To appraise the performance and efficiency of the method, three benchmark problems are included and discussed. The numerical results are compared with a few existing methods reported recently in the literature. The numerical experi- ments confirm that the spectral method coupled with Chebyshev wavelets is superior to other existing ones.

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We thank the Editor and the referee for their comments.

Iqbal, J. and Abass, R. (2016) Numerical Solution of Klein/Sine- Gordon Equations by Spectral Method Coupled with Chebyshev Wavelets. Applied Mathematics, 7, 2097-2109. http://dx.doi.org/10.4236/am.2016.717167