Research Article
A Homotopy-Analysis Approach for Nonlinear Wave-Like
Equations with Variable Coefficients
Afgan Aslanov
Mathematics and Computing Department, Beykent University, 34396 Istanbul, Turkey
Correspondence should be addressed to Afgan Aslanov; afganaslanov@beykent.edu.tr
Received 20 February 2015; Accepted 23 April 2015
Academic Editor: R. Naz
Copyright © 2015 Afgan Aslanov. Tis is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We are interested in the approximate analytical solutions of the wave-like nonlinear equations with variable coefcients. We use
a wave operator, which provides a convenient way of controlling all initial and boundary conditions. Te proposed choice of the
auxiliary operator helps to fnd the approximate series solution without any discretization, linearization, or restrictive assumptions.
Several examples are given to verify the reliability and efciency of the method.
1. Introduction
We consider the equation
−(,)
=(,
,
,
)+(,),
>0,>0
(1)
with initial conditions
(,0)=(),
(,0)=()
(2)
and boundary condition
(0,)=ℎ(), (3)
where , , , , , and ℎ are known functions.
Note that the proposed method can be applied for
equations like
(,)
−
=(,
,
,
)+(,), (4)
with the same type of initial-boundary conditions.
Problems like (1)–(3) model many problems in classical
and quantum mechanics, solitons, and matter physics [1, 2]. If
is a function of only and (,)= const we obtain Klein-
Gordon or sine-Gordon type equations. Tese models can
describe some nonlinear phenomena; for example, wave-like
equation can describe earthquake stresses [3], coupling cur-
rents in a fat multistrand two-layer superconducting cable
[4], and nonhomogeneous elastic waves in soils [5]. Typical
examples of the wave-like equations with variable coef-
cients are Euler-Tricomi equation [6] or Chaplygin equa-
tion [7] given by
−
=0,
−()
=0,
(5)
which are useful in the study of transonic fow, where =
(,) is the stream function of a plane-parallel steady-state
gas fow, () is positive at subsonic and negative at supersonic
speed, and is the angle of inclination of the velocity vector.
Some Chaplygin type of equation of the special form
+
2
1−
2
/
2
+
=0, (6)
where =() is the speed of sound, has also applications in
the study of transonic fow [8].
Note that we use the term wave-like equation to describe
the partial diferential equations with the terms
and
;
that is, the term “wave-like” may not correspond to the real
physical waves, in general.
Recently, there has been a growing interest for obtaining
the explicit solutions to wave-like and heat-like models by
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2015, Article ID 628310, 7 pages
http://dx.doi.org/10.1155/2015/628310