Control of error in the homotopy analysis of nonlinear Klein–Gordon initial value problems

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Abstract

In the present paper, we discuss the application of homotopy analysis to general nonlinear Klein–Gordon type equations. We first outline the method for general forms of the nonlinearity, as well as for general functional forms of the initial conditions. In particular, we discuss a method of controlling the residual error in approximate solutions which may be found via homotopy analysis, through adequate selection of the convergence control parameter. With the general problem outlined, we apply the method to various equations, including the quasilinear cubic Klein–Gordon equation, the modified Liouville equation, the sinh-Gordon equation, and the tanh-Gordon equation. For each of these equations and related initial data, we obtain residual error minimizing solutions which demonstrate the qualitative behavior of the true solutions in each case.

Introduction

Consider the nonlinear Klein–Gordon equationutt-uxx=F(u),u(x,0)=f(x),ut(x,0)=g(x).Here, FC(R) is an arbitrary analytic function which will serve as the nonlinear term, while f,gC(R) are the analytic initial data. We seek a solution u:R×[0,)R to the initial value problem (1), (2).

In the present paper, we shall apply the homotopy analysis method to obtain approximate analytical solutions to the initial value problem (1), (2). First conceived by Liao [1], the method of homotopy analysis has been applied to a number of various problems in recent years [2], [3], [4], [5]. Recent successful applications include a number of non-trivial and traditionally hard to solve nonlinear differential equations, for instance nonlinear equations arising in heat transfer [6], [7], fluid mechanics [8], [9], [10], [11], [12], [13], solitons and integrable models [14], [15], [16], [17], nanofluids [18], [19] and the Lane–Emden equation which appears in stellar astrophysics [20], [21], [22], [23], to name a few areas. In regards to the present initial value problem, some authors have considered the homotopy analysis of some special cases for F(u). Sun [24] considered the quasilinear cubic Klein–Gordon equation, and used homotopy analysis to obtain a solution form in terms of a trigonometric basis. However, such solutions were for the restriction to the traveling wave case, which reduces (1) from a partial differential equation to an ordinary differential equation. More recently, Iqbal et al. [25] considered the same equation from the standpoint of the optimal homotopy analysis method, and similar results were obtained. None of these studies considered arbitrary initial data. Approximate homotopy analysis solutions for the sin-Gordon equation were recently discussed by Yücel [26], however no discussion of error was provided (the author simply applied the so-called h-curve in order to deduce possible regions of convergence of the solutions).

First, we provide a general outline of the homotopy analysis method as it applied to problems of the form (1), for general forms of F(u), and arbitrary initial data. The primary benefit to the method is the arbitrary convergence control parameter, which may be picked so that the obtained approximate solution is an accurate approximation to the true solution. Since exact solutions to (1), (2) are rare (particularly for arbitrary initial data), we shall discuss the error of our approximations in terms of residual error. Indeed, we shall select the convergence control parameter so that the residual errors over a region of the problem domain are minimized.

Once the general framework is laid out, we will consider several applications to specific problems, particularly the quasilinear Klein–Gordon [27], [28], [29] equation (F(u)=u3-αu), the modified Liouville [30], [31] equation (F(u)=eβu), the sinh-Gordon [32], [33], [34], [35], [36], [37], [38] equation (F(u)=sinh(u)) and the tanh-Gordon [39] equation (F(u)=tanh(u)). For these qualitatively different equations, we shall discuss the method of obtaining solutions subject to general initial data. Then, in order to obtain more concrete examples, we shall consider specific closed-form initial data. At times, in order to demonstrate the voracity of the method, we will weaken the assumption of analytic initial data; indeed, solutions valid almost everywhere can be obtained for initial data which is not differentiable over a discrete subset of the real line. One such example will be f(x)=e-x, which is not differentiable at x=0 yet still permits solutions to (1), (2).

Section snippets

Homotopy analysis for the general nonlinear wave equation

The homotopy analysis method has proven a useful and versatile tool for the study of nonlinear differential equations. In the present section, we shall present an overview of the method and it’s application to the general nonlinear wave equation discussed in (1), (2).

We define a homotopy of operatorsH[ψ]=(1-q)L[ψ(x,t;q)-u0(x,t)]-qhN[ψ(x,t;q)],where N is the original nonlinear operator governing (1), L is an auxiliary linear operator of our choice, ψ is an unknown function which is the solution

Quasilinear Klein–Gordon equation

The quasilinear Klein–Gordon equation corresponds to F(u)=u3-αu, where α is a real-valued parameter. We then find that the first several terms of F(ψ(x,t;h)) are given byF(ψ(x,t;h))=u03-αu0+3u02u1-αu1q+3(u02u2+u0u12)-αu2q2+.The order zero approximation remains (13), while the higher order approximations are governed byL[u1]=h(u0)tt-(u0)xx-u03+αu0,L[u2]=(u1)tt+2(u1)t+u1+h(u1)tt-(u1)xx-3u02u1+αu1,L[u3]=(u2)tt+2(u2)t+u2+h(u2)tt-(u2)xx-3(u02u2+u0u12)+αu2,etcetera. We find from (15) that u1 isu1(x,t

Modified Liouville equation

The modified Liouville equation corresponds to F(u)=eβu, where β is a real-valued parameter.F(ψ(x,t;h))=e-u0-e-u0u1q+e-u0u12-2u2q2+.L[u1]=h(u0)tt-(u0)xx-e-u0.

We shall consider initial data of the form f(x)=1,g(x)=-1. We then haveu0(x,t)=e-tand from (15) we getu1(x,t;h)=h2t2e-t-h0t(t-s)e-(t-s)e-sds.Taking J1(t,h)=L[u1]+hM[u1] and K1(t,h)=-e-u02u12, we haveu2(x,t;h)=0t(t-s)e-(t-s)J1(s,h)-hK1(s,h)ds=h2t2e-t+h22t2e-t-2h3t33-t-h23t3e-t+h6t4e-t+h224t4e-t+h2e-t20ts2(t-s)ee-sdsNow, N[u]=M[u]-eu=utt-

sinh-Gordon

Let us consider the sinh-Gordon equation (F(u)=sinh(u)) with general initial conditions.F(ψ(x,t;h))=sinh(u0)+cosh(u0)u1q+2u2cosh(u0)+u12sinh(u0)q2+.L[u1]=h(u0)tt-(u0)xx-sinh(u0).

We shall consider initial data of the form f(x)=1,g(x)=-1. We then haveu0(x,t)=e-tand from (15) we getu1(x,t;h)=h2t2e-t-h0t(t-s)e-(t-s)sinh(e-s)ds.Taking J1(t,h)=L[u1]+hM[u1] and K1(t,h)=cosh(u0)u1, we haveu2(x,t;h)=0t(t-s)e-(t-s)J1(s,h)-hK1(s,h)ds=7h224t4e-t-h23t3e-t+h22t2e-t+h2t2e-t-he-t0t(t-r)(h+1)sinh(e-r)er+cosh

tanh-Gordon

Finally, we will consider the so-called tanh-Gordon equation (F(u)=tanh(u)) with general initial conditions (2). While not near as prevalent in the literature, according to [39] the problem of optical dynamic holography in a medium with a non-local response can be reduced to a nonlinear equation of the sine-Gordon family [40], [41] in the transmission geometry. However, [39] note that in the reflection geometry it is a similar non-local tanh-Gordon equation. With such an application in mind,

Conclusions

We have applied the homotopy analysis method to a general nonlinear Klein–Gordon type equation, utt-uxx=F(u). We first outline the method for general forms of the nonlinearity, as well as for general functional forms of the initial conditions. We treat the auxiliary linear operator as a time evolution operator, which evolves the initial data u(x,0)=f(x),ut(x,0)=g(x) into the time dependent approximate solution desired. In this sense, the method deforms an initial order zero approximation

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