A numerical solution of the coupled sine-Gordon equation using the modified decomposition method

https://doi.org/10.1016/j.amc.2005.08.018Get rights and content

Abstract

The modified decomposition method has been implemented for solving a coupled sine-Gordon equation. We consider a system of coupled sine-Gordon equation, which models one-dimensional nonlinear wave processes in two-component media. By using an initial value system, the numerical solutions of coupled sine-Gordon equation have been represented graphically.

Introduction

In this paper, we shall consider a system of coupled sine-Gordon equations in the formutt-uxx=-δ2sin(u-w),wtt-c2wxx=sin(u-w),which was introduced recently by Khusnutdinova and Pelinovsky [1]. The coupled sine-Gordon equations generalize the Frenkel–Kontorova dislocation model [2], [3]. The system (1.1) with c = 1 was also proposed to describe the open states in DNA [4].

In Ref. [1], the authors construct the linear and nonlinear wave process involving the exchange of energy between the two physical components of the system. They construct the linear solutions by considering ∣u  w  1, the exact nonlinear solutions for the case c = 1 and the weakly nonlinear solutions, for general case, by means of asymptotic methods. We used to get the explicit series solutions of the coupled sine-Gordon equations without use of any linearization or transformation method by the Adomian’s decomposition method (in short ADM) [5], [6]. The method presented here is also very easy to use for obtain numerical solutions of the Eq. (1.1) without using any discretization techniques.

In this paper, we use the modified decomposition method (in short MDM) to obtain the numerical solutions of the coupled sine-Gordon Eq. (1.1). Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the Adomian decomposition method [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. A reliable modification of Adomian decomposition method has been done by Wazwaz [27]. The decomposition method provides an effective procedure for analytical solution of a wide and general class of dynamical systems representing real physical problems [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Recently, the implementations of Adomian decomposition method for the solutions of sine-Gordon equation, generalized Hirota–Satsuma coupled KdV equation, generalized regularized long-wave (RLW) and Korteweg–de Vries (KdV) equations have been well established by the notable researchers [13], [14], [15], [16], [17], [18]. This method efficiently works for initial-value or boundary-value problems and for linear or nonlinear, ordinary or partial differential equations and even for stochastic systems. Moreover, we have the advantage of a single global method for solving ordinary or partial differential equations as well as many types of other equations. Recently, the solution of fractional differential equation has been obtained through Adomian decomposition method by the researchers [19], [20], [21], [22], [23], [24], [25], [26]. The method has features in common with many other methods, but it is distinctly different on close examinations, and one should not be mislead by apparent simplicity into superficial conclusions [5], [6].

Section snippets

Analysis of the method

We consider the coupled sine-Gordon equations (1.1) in the operator formLttu=Lxxu-δ2N(u,w),Lttw=c2Lxxw+N(u,w),where Ltt2t2, Lxx2x2 symbolize the linear differential operators and the notation N symbolize the nonlinear operator.

Applying the two-fold integration inverse operator Ltt-10t0t()dtdt to the system (2.1) and using the specified initial conditions yieldsu(x,t)=u(x,0)+tut(x,0)+Ltt-1Lxxu-δ2Ltt-1N(u,w),w(x,t)=w(x,0)+twt(x,0)+c2Ltt-1Lxxw+Ltt-1N(u,w).The Adomain decomposition method

Implementation of the method

We consider the coupled sine-Gordon equationsutt-uxx=-δ2sin(u-w),wtt-c2wxx=sin(u-w)with initial conditions [1]u(x,0)=Acoskx,ut(x,0)=0,w(x,0)=0,wt(x,0)=0.We adopt modified decomposition method for solving Eq. (3.1). In the light of this method we can writeu0=0,u1=Acoskx+Ltt-1Lxx(u0)-δ2Ltt-1(A0)=Acoskx,u2=Ltt-1Lxx(u1)-δ2Ltt-1(A1)=-12Ak2t2coskx-12At2δ2coskx,u3=Ltt-1Lxx(u2)-δ2Ltt-1(A2)=124Ak2t4(k2+δ2)coskx+124At4δ2(1+k2+δ2)coskx,u4=Ltt-1Lxx(u3)-δ2Ltt-1(A3)=-1720Ak2t6(k4+δ2+2k2δ2+δ4)coskx-1720At2δ2

Conclusion

In this paper, the modified decomposition method was used for finding the numerical solutions for the coupled sine-Gordon equations with initial conditions. The approximate solutions to the equations have been calculated by using the MDM without any need to a transformation techniques and linearization of the equations. Additionally, it does not need any discretization method to get numerical solutions. This method thus eliminates the difficulties and massive computation work. The decomposition

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