Control of error in the homotopy analysis of nonlinear Klein–Gordon initial value problems
Introduction
Consider the nonlinear Klein–Gordon equationHere, is an arbitrary analytic function which will serve as the nonlinear term, while are the analytic initial data. We seek a solution 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 . 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 , 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 (), the modified Liouville [30], [31] equation (), the sinh-Gordon [32], [33], [34], [35], [36], [37], [38] equation () and the tanh-Gordon [39] equation (). 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 , which is not differentiable at 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 operatorswhere N is the original nonlinear operator governing (1), L is an auxiliary linear operator of our choice, is an unknown function which is the solution
Quasilinear Klein–Gordon equation
The quasilinear Klein–Gordon equation corresponds to , where is a real-valued parameter. We then find that the first several terms of are given byThe order zero approximation remains (13), while the higher order approximations are governed byetcetera. We find from (15) that is
Modified Liouville equation
The modified Liouville equation corresponds to , where is a real-valued parameter.
We shall consider initial data of the form . We then haveand from (15) we getTaking and , we haveNow,
sinh-Gordon
Let us consider the sinh-Gordon equation () with general initial conditions.
We shall consider initial data of the form . We then haveand from (15) we getTaking and , we have
tanh-Gordon
Finally, we will consider the so-called tanh-Gordon equation () 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, . 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 into the time dependent approximate solution desired. In this sense, the method deforms an initial order zero approximation
References (49)
- et al.
On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach
Communications in Nonlinear Science and Numerical Simulation
(2009) The application of homotopy analysis method to nonlinear equations arising in heat transfer
Physics Letters A
(2006)Homotopy analysis method for heat radiation equations
International Communications in Heat and Mass Transfer
(2007)An explicit, totally analytic approximation of Blasius’ viscous flow problems
International Journal of Non-Linear Mechanics
(1999)- et al.
Implicit differential equation arising in the steady flow of a sisko fluid
Applied Mathematics and Computation
(2009) - et al.
On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder
Physics Letters A
(2007) - et al.
Solving solitary waves with discontinuity by means of the homotopy analysis method
Chaos, Solitons & Fractals
(2005) - et al.
Analytical solutions to a generalized Drinfel’d–Sokolov equation related to DSSH and KdV6
Applied Mathematics and Computation
(2010) - et al.
Solving the one-loop soliton solution of the Vakhnenko equation by means of the homotopy analysis method
Chaos, Solitons & Fractals
(2005) - et al.
Series solutions of nano-boundary-layer flows by means of the homotopy analysis method
Journal of Mathematical Analysis and Applications
(2008)
Nano boundary layers over stretching surfaces
Communications in Nonlinear Science and Numerical Simulation
Homotopy analysis method for singular IVPs of Emden–Fowler type
Communications in Nonlinear Science and Numerical Simulation
Analytic and numerical solutions to the Lane–Emden equation
Physics Letters A
A new analytic algorithm of Lane–Emden type equations
Applied Mathematics and Computation
Solving the Klein–Gordon equation by means of the homotopy analysis method
Applied Mathematics and Computation
Some solutions of the linear and nonlinear Klein–Gordon equations using the optimal homotopy asymptotic method
Applied Mathematics and Computation
Homotopy analysis method for the sine-Gordon equation with initial conditions
Applied Mathematics and Computation
Numerical solution of a nonlinear Klein–Gordon equation
Journal of Computational Physics
Sinh-Gordon equation, Painlevé property and Bäcklund transformation
Physica A
The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations
Applied Mathematics and Computation
Analytical method for the construction of solutions to the Föppl–von Kármán equations governing deflections of a thin flat plate
International Journal of Non-Linear Mechanics
Gaussian waves in the Fitzhugh–Nagumo equation demonstrate one role of the auxiliary function in the homotopy analysis method
Communications in Nonlinear Science and Numerical Simulation
The comparison between homotopy analysis method and optimal homotopy asymptotic method for nonlinear age-structured population models
Communications in Nonlinear Science and Numerical Simulation
Mathematical properties of h-curve in the frame work of the homotopy analysis method
Communications in Nonlinear Science and Numerical Simulation
Cited by (24)
An improved optimal homotopy analysis algorithm for nonlinear differential equations
2020, Journal of Mathematical Analysis and ApplicationsCitation Excerpt :Since Liao's book [15], the method has been widely implemented for solving several nonlinear problems in physical science and engineering [1–4,8,12,13,20,23–26,28,29,35,37,39,41–43,45]. Recently, optimal approaches to improve the convergence and the efficiency of the HAM for nonlinear differential equations have been introduced [5–7,9–11,18,21,22,36,38,40,44]. These approaches include the optimal selection of the control parameter, initial approximation, auxiliary linear operator and auxiliary function in the implementation of HAM to nonlinear differential problems.
On the optimal selection of the linear operator and the initial approximation in the application of the homotopy analysis method to nonlinear fractional differential equations
2019, Applied Numerical MathematicsCitation Excerpt :These reliable modifications extend the application of the HAM to solve efficiently nonlinear differential equations with fractional derivatives. Recently, optimal approaches to improve the convergence and the efficiency of the HAM for nonlinear differential equations have been introduced [25,7,22,38,47,6,41,14,43,12,13]. These approaches include the optimal selection of the control parameter, auxiliary linear operator and auxiliary function in the implementation of HAM to nonlinear differential problems.
Solution of the one-phase inverse Stefan problem by using the homotopy analysis method
2015, Applied Mathematical ModellingCitation Excerpt :The homotopy analysis method has been also used for investigating the heat conduction problems [14–20], whereas in papers [21,22] the method has been applied for solving the inverse heat conduction problem. Theoretical results concerning, among others, convergence of the method are included, for example, in works [3,4,23–30]. Only in few simple cases it is possible to find an exact solution of the inverse Stefan problem.
Numerical and analytical solutions for Falkner-Skan flow of MHD Maxwell fluid
2014, Applied Mathematics and ComputationCitation Excerpt :Numerical solution is computed by Runge–Kutta method [32]. Optimal HAM is also utilized in the studies [33–40]. The effects of important parameters of interest on the flow quantities are also discussed.
Approximate Solution of Nonlinear Volterra-Fredholm Fuzzy Integral Equations
2022, AIP Conference ProceedingsSolving Volterra-Fredholm fuzzy integro-differential equations by using homotopy analysis method
2022, AIP Conference Proceedings