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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 767380, 16 pages
http://dx.doi.org/10.1155/2013/767380
Research Article

New Exact Solutions for New Model Nonlinear Partial Differential Equation

1Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt
2Department of Mathematics, Faculty of Science and Education, Aden University, Aden, Yemen

Received 10 February 2013; Accepted 20 April 2013

Academic Editor: Anjan Biswas

Copyright © 2013 A. Maher et al. This 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.

Abstract

In this paper we propose a new form of Padé-II equation, namely, a combined Padé-II and modified Padé-II equation. The mapping method is a promising method to solve nonlinear evaluation equations. Therefore, we apply it, to solve the combined Padé-II and modified Padé-II equation. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions, and elliptic functions.

1. Introduction

In recent years, directly searching for exact solutions of nonlinear partial differential equations (PDEs) has become more and more attractive field in different branches of physics and applied mathematics. These equations appear in condensed matter, solid state physics, fluid mechanics, chemical kinetics, plasma physics, nonlinear optics, propagation of fluxions in Josephson junctions, theory of turbulence, ocean dynamics, biophysics star formation, and many others.

In order to get exact solutions directly, many powerful methods have been introduced such as the -expansion method [1], inverse scattering method [2, 3], Hirota’s bilinear method [4, 5], the tanh method [6, 7], the sine-cosine method [8, 9], Bäcklund transformation method [10, 11], the homogeneous balance [12, 13], Darboux transformation [14], and the Jacobi elliptic function expansion method [15].

Recently, Peng [16] introduced a new approach, namely, the mapping method for a reliable treatment of the nonlinear wave equations. The useful mapping method is then widely used by many authors [17, 18].

2. Description of the Method

Consider the general nonlinear partial differential equations (PDEs); say, in two variables,

Let ,  ; then (1) reduces to a nonlinear ordinary differential equation (ODE) Assume the solution of (2) takes the form where the coefficients ,  and   are constants to be determined, and satisfies a nonlinear ordinary differential equation where the coefficients ,  and are constants to be determined and satisfies (4); the parameter will be found by balancing the highest-order nonlinear terms with the highest-order partial derivative term in the given equation. Substituting (3) into (2), using (4) repeatedly and setting the coefficients of the each order of to zero, we obtain a set of nonlinear algebraic equations for ,  and  . With the aid of the computer program Maple, we can solve the set of nonlinear algebraic equations and obtain all the constants , and . The ODE (4) has the following solutions:(1) ,  ,(2) ,  ,(3) ,  ,(4) ,  ,(5) ,  ,  ,(6) ,  ,  ,(7) ,  ,(8) ,  ,(9) ,  ,(10) ,  ,  ,(11) ,  ,  ,(12) ,  ,(13) ,  ,(14) ,  ,(15) ,  ,  ,(16) ,  ,(17) ,  ,  ,(18) ,  ,  ,(19) ,  ,  ,(20) ,  ,(21) ,  ,(22) ,  ,  ,(23) ,  ,  ,(24) ,  ,  ,(25) ,  ,  ,(26) ,  ,  ,(27) ,  ,(28) ,  ,(29) ,  ,(30) ,  .

3. Application

In this section, we present our proposed equation, namely, a combined Padé-II and modified Padé-II equation, as the form where ,  ,  and   are real numbers [19].

Now, we apply the mapping method to solve our equation. Consequently we get the original solutions for our new equation, as the follows:

Substituting ,   in (5) and integrating once yield Balancing the order of the nonlinear term with the highest derivative gives that gives . Thus, the solution of (6) has the form where

Substituting (7) in (6) and using (8), collecting the coefficients of each power of , setting each coefficient to zero, and solving the resulting system, we obtain the following sets of solutions:(1) ,  ,(2) , (3) ,  ,  ,  ,(4) ,  ,  ,  ,  ,(5) ,  ,  ,  ,  ,(6) ,  ,  ,  ,  ,(7) ,  ,  , ,  ,(8),  , ,  ,  ,(9) ,  , , ,  ,(10) ,  , ,  ,  .

Using (7), the solution of (8) when ,  and  , and the sets of solutions (1)–(10), we get for for

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get for

For

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get, .

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get, .

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain , and when , we obtain for for

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where and are defined in the sets of solutions (3)–(10).

Note that, when we obtainconstant solutions, when we obtain,   and .

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain, , when , we obtain   and  .

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain constant solution, and when , we obtain   and  .

Using (7), the solution of (8) when  ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain   and  , when , we obtain   and  .

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain   and  , and   when , we obtain   and  .

Using (7), the solution of (8) when ,  and , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain   and  , and  when , we obtain constant solutions.

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain   and  , and when , we obtain constant solutions.

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain   and  , when , we obtain .

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain   and  , when , we obtain .

Using (7), the solution of (8), when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain   and  , and   when , we obtain also   and .

Using (7), the solution of (8), when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain   and  , and when , we obtain also   and  .

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain constant solutions, and  when , we obtain , and for

For

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets of solutions (3)–(10).

Note that, when , we obtain for For

When , we obtain   and  .

Using (7), the solution of (8) when ,  and  , and the sets of solutions (3)–(10), we get , where , and are defined in the sets o