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APPLICATION OF LAPLACE DECOMPOSITION METHOD FOR SOLVING SINE- GORDON EQUATION.

Byline: Z. Zafar A. M. Sultan A. Pervaiz M.O.Ahmed and S. Hussain

Abstract

In this study we considered the initial value problem for the sine-Gordon equation by using Laplace Decomposition Method (LDM). The advantage of this method was its ability and flexibility to provide the analytical or approximate solutions to linear and nonlinear equations without linearization or discretization which makes it reliable for solving sine-Gordon equation.

Key words: Approximate solutions Laplace Decomposition method Adomian decomposition method Sine Gordon Equation.

INTRODUCTION

The sine-Gordon equation which arises in the study of differential geometry of surfaces with Gaussian curvature has wide applications in the propagation of fluxon in Josephson junctions (Perring and Skyrme 1962 Whitham 1999 Sirendaoreji and Jiong 2002 Fu et al. 2004) between two superconductors the motion of a rigid pendulum attached to a stretched wire solid state physics nonlinear optics stability of fluid motions dislocations in crystals and other scientific fields (Whitham 1999). Since its wide applications and important mathematical properties many methods have been presented to study the different solutions and physical phenomena related to this equation (Bratsos and Twizell 1998 Kaya 2003 Wang 2006 Wei 2000 Wazwaz 2006 Peng 2003).

Reviewing these improvements by means of linearization discretization or transformations this equation is transformed to more simple equation and then different types of solutions are followed. Unlike the classical schemes we will consider the sine-Gordon equation by a new approach in this study.

METHODOLOGY

In this section we established an Algorithm using Laplace Decomposition method on the partial differential equations which were nonlinear. We considered the general form of inhomogeneous nonlinear partial differential equations with initial conditions as given below Equations the remaining linear operator represents a general nonlinear differential operator and was source term. The first step we will take Laplace transform on equation (1). Equation

By applying the Laplace transform differentiation property we have

Equations

The 2nd step in LDM is that we signify the solution as an infinite series as Equation

The nonlinear operator is written as

Equations

where equation represented the expression from source term and with the initial conditions. First we applied Laplace transform on the right hand side of Eq. (16) then by taking the inverse Laplace transform we obtained the values of Equation correspondingly.

APPLICATION 1

Let us consider the homogeneous partial differential equation

Equations

The initial conditions were Equation

The nonlinear term was handled with the help of Adomian Polynomials as below Equations

This was our required result in Series form which was the same as obtained in Partial Differential Equations and Solitary Waves Theory (Wazwaz 2009).

APPLICATION 2

Let us consider the homogeneous partial differential equation

Equations

The nonlinear term was handling with the aid of Adomian Polynomials as below

Equation

obtained upon Taylor expansion for the Trigonometric functions involved. This was our required result in Series form as obtained in Partial Differential Equations and Solitary Waves Theory (Wazwaz 2009).

RESULTS AND DISCUSSION

In this work we have applied Laplace Decomposition Method to Sine Gordon equation. A series of solutions of the Sine Gordon equation were developed by using this method and an efficient result has been obtained. The Laplace Decomposition method was a powerful tool to search for solutions of various linear and nonlinear initial value problems. In the real world problems we dealt with ambiguous conditions for example the uncertainties and vagueness in the values of the initial conditions. This method overcame these difficulties. A comparison of this method with Adomian Decomposition revealed healthy similar results. We observed that the working of the method was simple and straight forward as it has been checked by two applications. The method gave more realistic series solutions that converged very rapidly in physical problems.

REFERENCES

Bratsos A.G. and E.H. Twizell A family of parametric finite-difference methods for the solution of the sine-Gordon equation Appl. Math. Comput. 93:117-137 (1998).

Fu Z. S. Liu and S. Liu Exact solutions to double and triple sinh-Gordon equations Z Naturforsch 59: 933-937 (2004).

Kaya D. A numerical solution of the sine-Gordon equation using the modified decomposition method Appl. Math. Comput. 143:309-317(2003).

Peng Y.Z. Exact solutions for some nonlinear partial differential equations Phys. Lett. A 314 401-408 (2003).

Perring J.K.and T.H. Skyrme A model unified field equation Nucl. Phys.31:550-555 (1962).

Sirendaoreji and S. Jiong A direct method for solving sinh-Gordon type equation Phys. Lett. A 298:133-139 (2002).

Wang Q. An application of the modified Adomian decomposition method for (N+1)-dimensional Sine-Gordon field Appl. Math. Comput. 181:147-152 (2006).

WazwazA.M. Exact solutions for the generalized sine-Gordon and the generalized sinh-Gordonequations Chaos Solitons and Fractals 28 127-135 (2006).

Wazwaz A. M. Partial Differential Equations and Solitary Waves Theory Higher Education Press Beijing (2009).

Wei G.W. Discrete singular convolution for the sine-Gordon equation Phys. D 137:247-259 (2000).

Whitham G.B. Linear and nonlinear waves Wiley-Interscience New York (1999).
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Publication:Pakistan Journal of Science
Article Type:Report
Date:Sep 30, 2014
Words:817
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