Journal of Applied Mathematics

Volume 2014 (2014), Article ID 487571, 10 pages

http://dx.doi.org/10.1155/2014/487571

Research Article

Analytical Solutions for the Elastic Circular Rod Nonlinear Wave, Boussinesq, and Dispersive Long Wave Equations

Shi Jing¹ and Yan Xin-li²

¹School of Science, Chang’an University, Xi’an 710064, China

²School of Science, Xi’an University of Architecture and Technology, Xi’an 710055, China

Received 31 December 2013; Revised 23 March 2014; Accepted 25 March 2014; Published 22 April 2014

Academic Editor: Michael Meylan

Copyright © 2014 Shi Jing and Yan Xin-li. 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.

The solving processes of the homogeneous balance method, Jacobi elliptic function expansion method, fixed point method, and modified mapping method are introduced in this paper. By using four different methods, the exact solutions of nonlinear wave equation of a finite deformation elastic circular rod, Boussinesq equations and dispersive long wave equations are studied. In the discussion, the more physical specifications of these nonlinear equations, have been identified and the results indicated that these methods (especially the fixed point method) can be used to solve other similar nonlinear wave equations.

1. Introduction

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of sciences such as biology, chemistry, communication, and especially many branches of physics like condensed matter physics, field theory, fluid dynamics, plasma physics, and optics, and so forth. In this paper, we consider nonlinear wave equation of a finite deformation elastic circular rod, Boussinesq equations, and dispersive long wave equations with same physical behavior. The elastic circular rod is one important component in the structures, in which the dynamics of these components are governed by double nonlinearity and double dispersion wave equation [1]. The Boussinesq equation, which was first introduced in 1871, arises in several physical applications. The dynamics of shallow water waves, which are seen in various places like sea beaches, lakes, and rivers, are governed by the Boussinesq equation. In recent years there has been much interest in some variants of the Boussinesq systems [2–6]. These coupled Boussinesq equations [7] arise in shallow water waves for two-layered fluid flow. This situation occurs when there is an accidental oil spill from a ship which results in a layer of oil floating above the layer of water. The -dimensional dispersive long wave equations [8, 9] were first derived by Boiti as a compatibility condition for a “weak” Lax pair. A good understanding of the solutions for these equations is very helpful to coastal and civil engineers in applying the nonlinear water model to coastal harbor design.

In the field of nonlinear science to find exact solutions for a nonlinear system is one of the most fundamental and significant studies. The evaluation of exact solutions of nonlinear wave has complicated nonlinear defects; such equations are often very difficult to be solved. Although intensive investigations have made significant progress in recent years, many methods have been proposed to construct exact solutions, such as Weierstrass elliptic function method, the homogeneous balance method, sine-cosine method, the nonlinear transformation method, the hyperbolic tangent functions finite expansion, improved mapping approach, and further extended tanh method.

The homogeneous balance method is a powerful tool to find solitary wave solutions of nonlinear partial differential equations. Fan introduced the homogeneous balance method into the search for Bäcklund transformations and obtained more solutions [10].

The mapping method is a very effective direct method to construct exact solutions of nonlinear equations. Zhang et al. make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for -dimensional dispersive long wave equations [11].

The basic idea of the fixed point methods consists in finding an iteration function, which generates successive approximations to the solution [12].

Many periodic solutions have been recently expressed in terms of various Jacobi-elliptic functions for a wide class of nonlinear evolution equations, which have been obtained by means of Jacobi elliptic expansion method [13].

The longitudinal wave equation of a finite deformation elastic circular rod, Boussinesq equations, and dispersive long wave equations are nonlinear partial differential equations of different scientific field. We find that they have the same characteristics. In this paper, the analytical solutions of the differential equations for the elastic circular rod, Boussinesq equations, and dispersive long wave equations are solved by using homogeneous balance method, Jacobi elliptic function method, fixed point method, and modified mapping method. The more physical specifications of these nonlinear equations have been identified and the results indicated.

2. Three Types of Nonlinear Wave Equations

The longitudinal wave equation of a finite deformation elastic circular rod is [1]

where is the longitudinal wave velocity for a linear elastic rod and is the shear wave velocity. is the density of the material. is the Poisson ratio. is the diameter of bar. is Yong’s modulus of material. is the elastic shear modulus of material. is the time variable.

Make the traveling wave transformation

Substituting (2) into (1) and integrating it with respect to twice, we can obtain

where is the second integral constant. The first integral constant is zero and

The variant Boussinesq equations were discussed in [7]. They are coupling wave equations which can be expressed as follows:

where is the velocity and is the total depth. Make the traveling wave transformation

Substituting (6) into (5) and integrating with respect to , we have

where , are integral constants.

Substituting the second formula of (7) into the first one and assuming , , , , we can obtain (3). So the solutions of (7) are equivalent to that of (3) and the second formula of (7).

-dimension dispersive long wave equations were discussed in [8, 9]. They can be written as follows:

Suppose that the traveling wave solutions for (8) can be given in the following forms:

Substituting (9) into (8), we have

where is the second integral constant.

Obviously, the first formula of (10) and (3) have the same characteristics. On the other hand, the solutions of (10) and (7) are same with each other except for the coefficients. Therefore, in following parts, the solutions of (5) are presented.

3. Homogeneous Balance Method

In [14, 15] the homogeneous balance method is used to solve nonlinear wave equations. For example, the following equation was solved in [15]:

The exact solution of the following corresponding equation was given by using the same method:

To solve (5), the homogeneous balance method was improved in this paper.

Now, suppose that

where , , and are the undetermined functions and , are undetermined constants.

Substituting (13) into (5), the equations can be rewritten as follows:

where , , . In (14), let

Integrating the second formula of (15), we have

Furthermore, we can obtain

Substituting (17) into the first formula of (15), we have

Integrating (18), we have

We know the solution of (19) is

Obviously, we have

Substituting (17) and (21) into (14), the equations can be rewritten as follows:

Letting all coefficients of in (22) be zero, we have

From first and fifth second formula of (23), it can be obtained that

where is undetermined function and is arbitrary function.

From second formula of (23), we have

where is the integral constant.

Therefore, from (13), (20), (24), and (25), we know the solutions of (5) are

4. Jacobi Elliptic Function Method

In [16], the NLS equation and Zakharov equation were studied by using the Jacobi elliptic function method. In this paper, (1) and (5) are discussed by using this method.

4.1. Jacobi Elliptic Sine Function Method

Let

We know

where and are the Jacobi elliptic cosine function and the third type Jacobi elliptic function, respectively, and is module.

Substituting (27) into (3), we have

Let the coefficients of all derivatives of be zero and we have

Therefore the solution of (3) is

where , , and are denoted by (30), and is arbitrary constant.

When , ,

Thus, the solutions of (5) are (32) and the following formula:

Assuming , , , , and , it can be obtained that , , and . The solutions of (5) are and . The solitary wave and behavior of the solutions and are shown in Figures 1 and 2, respectively, for and . The waveform is similar to the result in [17].

Figure 1: The solitary wave and behavior of the solutions for and .

Figure 2: The solitary wave and behavior of the solutions for and .

4.2. Jacobi Elliptic Cosine Function Method

Let

Thus

Substituting (35) into (3), we can obtain

Let the coefficients of all derivatives of be zero and we have

Therefore, the solution of (3) is

where , and are denoted by (37), and is arbitrary constant.

The solutions of (7) are (38) and the following formula:

Taking , , (38) can be rewritten as follows:

Assuming , , , and , it can be obtained that , , and . The solution of (3) is . The solitary wave and behavior of the solutions are shown in Figure 3, respectively, for and .

Figure 3: The solitary wave and behavior of the solutions for and .

4.3. Third Kind of Jacobi Elliptic Function Method

Let

Thus

Substituting them into (3), we can obtain

Letting the coefficients of all derivatives of be zero, we have

Therefore the solution of (3) is

where , , and are denoted by (44), and is arbitrary constant.

Thus the solutions of (7) are (45) and the following formula:

Letting , then (45) equals (38).

4.4. Jacobi Elliptic Function Method

Let

Then

Substituting (47) and (48) into (3), we can obtain

Let the coefficients in above formula be zero, and we have

According to (47), the solution of (3) is

where , , and are denoted by (50), and is arbitrary constant.

Therefore, the solutions of (7) are (51) and the following formula:

When ,

Assuming , , , , and , we can obtain (3). Choosing , , then , . Substituting , , into (50), it can be obtained that , , and . The solutions of (8) are and . The solitary wave and behavior of the solutions and are shown in Figures 4 and 5, respectively, for , , , and .

Figure 4: The solitary wave and behavior of the solutions for , , and .

Figure 5: The solitary wave and behavior of the solutions for , , and .

5. Fixed Point Method

In general, fixed point theory can be divided into two types. The first type is only used to discuss the existence of solution. The second type is used not only to discuss the existence and uniqueness of solution but also to search the fixed point. We are more interested in the second type.

Lemma 1 (see [18–20]). Suppose is a real Banach space which has normal cone. Consider , . The operator is mixed monotone and satisfies

(1)

(2)for all , constant , satisfies

Then there exists a unique which satisfies

where is called the fixed point of .

In for all , letting , , we have

In this paper, is taken as a continuous function space which is defined in the closed interval . Introducing equivalent norm, it is easy to know that is real Banach space which has normal cone. Let

Integrating (3) with respect to , we can obtain

where is integral constant.

Integrating the above formula with respect to and using the formula in [12], we have

where is integral constant.

If , (i.e., ), and (i.e., ), suppose

Theorem 2. In (61), supposing , , , , , and , there exists unique fixed point for the operator which is defined by (61). Then (57) is true.

Proof. Obviously, is mixed monotone. Consider

Thus satisfies the term (1) of Lemma 1.

For all , ,

Then, we have

where , and when is large enough, .

Therefore, operator which is defined by (61) satisfies the term (2) of Lemma 1. The proof is completed.

Obviously, the fixed point of the operator defined by (61) is equivalent to the solution of (3). We denote

By using the classical iteration algorithms, such as Mann iteration method, Ishikawa iteration method, and Noor iteration method, the solutions can be obtained. Then, the traveling wave solutions of (5) are (65) and the following formula:

6. Modified Mapping Method

The modified mapping method was introduced and the nonlinear evolution equation was solved in [21, 22]. In [22], several hundreds of Jacobi elliptic function expansion solutions were obtained. We also use this method in this paper.

Let

where . , , and are constants.

Thus

Substituting (68) into (3), we have

Notice that we get the same algebraic equations when the coefficients of and in (69) are zero. Therefore there are only five algebra equations. Solve them and we have

In order to make , be real number, and should be different signs with .

(i) Choosing , , and , we have . Then the solitary wave solutions of (3) are

Assuming , , , and , it can be obtained that , , , , and . The solutions of (8) are and . The solitary wave and behavior of them are shown in Figures 6 and 7, respectively, for , , , and .

Figure 6: The solitary wave and behavior of the solutions for , , and .

Figure 7: The solitary wave and behavior of the solutions for , , and .

(ii) Choosing , , and , thus

where is the module of Jacobi elliptic function.

Therefore the expansion solutions of (3) are

Letting , then (73) equals (71).

(iii) Choosing , , and , we have

Therefore the solutions of (3) are

where , , of are denoted by (70).

Taking , (75) can be rewritten as follows:

Assuming , , , and , it can be obtained that , , , , and . The solutions of (8) are and . The solitary wave and behavior of them are shown in Figures 8 and 9, respectively, for , , , and .

Figure 8: The solitary wave and behavior of the solutions   for , , and .

Figure 9: The solitary wave and behavior of the solutions   for , , and .

7. Summary and Conclusions

In summary, by using different methods, nonlinear wave equation of a finite deformation elastic circular rod, Boussinesq equations, and dispersive long wave equations are solved in this paper, and several analytical solutions are obtained. Kink soliton solutions are obtained when we use the homogenous balance method. The Jacobi elliptic function method is utilized and periodic solutions are obtained. When , the solutions reduce to solitary solution. Their wave forms are the bell type, kink type, exotic type, and peakon type. By using the modified mapping method, peakon solutions of (3), (5), (8) are obtained. On the other hand, after proving the existence and uniqueness of fixed point of operator , we can use the classical iteration algorithms to get the solutions. The method has wide application and can be used to solve other nonlinear wave equations. Fixed point method may be significant and important for the explanation of some special physical problems whose analytical solution cannot be obtained. The results indicate the following. First, the natural phenomena and the physical properties of the equations are abundant, which should be further discovered. Second, there are many kinds of methods for solving these equations. Third, the arbitrary constant can be determined by the initial and boundary conditions and we can obtain the exact solutions for certain engineering or scientific problems. It is shown that the methods, proposed in this paper for three types of nonlinear wave equations, are feasible for determining exact solutions of other nonlinear equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Special Fund for Basic Scientific Research of Central Colleges (Grant no. CHD2009JC140).

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