Application of Kudryashov and functional variable  methods to the strain wave equation in microstructured solids

Z. Ayati; K. Hosseini; M. Mirzazadeh

doi:10.1515/nleng-2016-0020

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Application of Kudryashov and functional variable methods to the strain wave equation in microstructured solids

Z. Ayati , K. Hosseini and M. Mirzazadeh

Published/Copyright: January 27, 2017

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From the journal Nonlinear Engineering Volume 6 Issue 1

Abstract

The aim of this paper is to obtain the exact solutions of the strain wave equation applied for illustrating wave propagation in microstructured solids. The effective Kudryashov and functional variable methods along with the symbolic computation system have been used to accomplish the purpose.

Keywords: Strain wave equation; Kudryashov method; Functional variable method; Exact solutions

1 Introduction

The search for the exact solutions of nonlinear partial differential quations (PDEs) has been one of the most important concerns of mathematicians throughout the world for a long time. Since for understanding the nonlinear phenomena, which are usually described by partial differential equations, the study of the exact solutions is essential. In recent years, several effective methods, including extended tanh method [1, 2], first integral method [3, 4], He’s semi-inverse method [5, 6], sine–cosine method [7, 8], exp-function method [9, 10], (G′ /G)-expansion method [11, 12], Kudryashov method [13, 14] and functional variable method [15, 16] have been recommended to find the exact solutions of a wide variety of the nonlinear PDEs. Some recent applications of Kudryashov and functional variable methods are listed, here. Lee and Sakthivel applied Kudryashov method to extract the exact travelling wave solutions of two nonlinear physical models, including the sine-Gordon and Dodd–Bullough–Mikhailov equations [17]. Mirzazadeh et al. adopted Kudryashov method to generate new exact solutions of the nonlinear Schrödinger’s equation in Kerr law medium with fourth order dispersion and the Schrödinger–Hirota equation with power law nonlinearity [18]. Cevikel et al. utilized functional variable method to construct the exact periodic and solitary wave solutions of the higher-order nonlinear Schrödinger equation [19], and Nazarzadeh et al. used functional variable method to find the exact solutions of other well-known nonlinear PDEs, such as the Klein–Gordon and Camassa–Holm Kadomtsev–Petviashvili equations [20]. The readers are also referred to see [21–34]. Therefore, this study is focused on constructing the exact solutions of the strain wave equation in microstructured solids as follows [35, 36]

utt−uxx−εα1u2xx−α3uxxxx+α4uxxtt=0,(1)

by using the Kudryashov and functional variable methods. Recently in [35, 36], the abundant solutions of the strain wave equation, including the solitary wave solutions of topological kink, singular kink, nontopological bell type solutions, solitons, compacton, cuspon, periodic solutions, and solitary wave solutions of rational functions have been obtained by a new generalized (G′/G) expansion method and a new exponential expansion method. The rest of the article is as follows: In Section 2, a brief description of Kudryashov and functional variable methods are presented. In Section 3, Kudryashov and functional variable methods are adopted to extract the exact solutions of the strain wave equation, and finally, conclusions are given in Section 4.

2 Overview of the methods

In this section, a brief description of Kudryashov and functional variable methods will be presented.

2.1 Kudryashov method

Let’s consider a nonlinear partial differential equation in the following form

Fu,ut,ux,utt,uxx,...=0.(2)

By using the transformation u(x, t) = f(η) which η = x − kt, Eq. (2) can be converted into the following nonlinear ordinary differential equation

Gf,f′,f″,...=0,(3)

where prime indicates the derivative with respect to η. We look for the solution of Eq. (3) as a truncated series of the form

fη=∑n=0NanQnη,(4)

where a_n, n = 0, 1, …, N (a_N ≠ 0) are unknown constants and the function Q(η) = 1 / (1 + d exp (η)) is the solution of the following equation

Q′η+Qη=Q2η.

It should be noted that the value of N is usually computed by balancing the linear and nonlinear terms of highest orders in Eq. (3). Substituting Eq. (4) and its necessary derivatives, for example

f′=∑n=1NannQnQ−1,f″=∑n=1NnQ−1Qn1+nQ−nan,

into Eq. (3) yields P(Q(η)) = 0, where P(Q(η )) is a polynomial of the function Q(η). By equating the coefficient of each power of Q(η) in P(Q(η)) to zero, a system of algebraic equations will be obtained, for determining a_n’s and k. Replacing them into (4), finally results in the exact solutions of Eq. (2).

2.2 Functional variable method

Consider the following nonlinear partial differential equation

Fu,ut,ux,utt,uxx,...=0.(5)

By means of the transformation u(x, t) = f(η) which η = x − kt, Eq.(5) is converted to a nonlinear ordinary differential equation as follows

Gf,f′,f″,...=0,(6)

where ′ = d / dη. We introduce a transformation in which the unknown function f is considered as a functional variable of the following form

fη=Pf.(7)

The first three derivatives of f_η are

fηη=12P2′,fηηη=12P2″P2,fηηηη=12P2(3)P2+P2″P2′.

The Eq. (6) is now reduced in terms of f, P and its derivatives as

Qf,P,P′,P″,...=0.(8)

The above form is of special interest, because it yields the exact solutions of a wide variety of nonlinear wave type equations. After integration of Eq. (8), an expression for P will be derived which along with (7) provide the exact solutions of Eq. (5).

3 Application

This section deals with the application of the Kudryashov and functional variable methods to extract the exact solutions of the strain wave equation in microstructured solids. By using the transformation u(x, t) = f(η) which η = x − kt, the Eq. (1) is converted into the following nonlinear ordinary differential equation

k2−1f″−εα1f2″+εα3−α4k2f4=0.(9)

Integrating (9) twice with respect toη, gives

εα3−α4k2f″+k2−1f−εα1f2=0,(10)

where the integration constants are considered to be zero.

3.1 Kudryashov method

Balancing f″ and f² in Eq. (10) gives

N+2=2N,

and therefore N = 2. This suggests a truncated series as follows

f(η)=a0+a1Qη+a2Q2η.(11)

Inserting Eq. (11) into Eq. (10) and equating the coefficient of each power of Q(η) to zero, a system of algebraic equations will be derived in the following form

−εa02α1+k2a0−a0=0,−k2εa1α4−2εa0a1α1+k2a1+εa1α3−a1=0,3k2εa1α4−4k2εa2α4−2εa0a2α1−εa12α1+k2a2−3εa1α3+4εa2α3−a2=0,−2k2εa1α4+10k2εa2α4−2εa1a2α1+2εa1α3−10εa2α3=0,−6k2εa2α4−εa22α1+6εa2α3=0.

By solving this system, we get

case 1.

a0=0,a1=6α3−α4α1εα4−1,a2=−6α3−α4α1εα4−1,k=±εα3−1εα4−1.(12)

Now, substituting (12) into (11) results in

f(η)=6α3−α4α1εα4−1Qη−6α3−α4α1εα4−1Q2η,

and so, the exact solutions of the strain wave equation are obtained as follows

u1,2x,t=6α3−α4α1εα4−111+dexpx−kt−11+dexpx−kt2,

where k = ± εα3−1/εα4−1 and d is a constant.

case2.

a0=α3−α4α1εα4+1,a1=−6α3−α4α1εα4+1,a2=6α3−α4α1εα4+1,k=±εα3+1εα4+1.(13)

Now, setting (13) in (11) yields

f(η)=α3−α4α1εα4+1−6α3−α4α1εα4+1Qη+6α3−α4α1εα4+1Q2η,

and therefore, the exact solutions of the strain wave equation are derived as the following

u3,4x,t=6α3−α4α1εα4+116−11+dexpx−kt+11+dexpx−kt2,

where k = ± εα3+1/εα4+1 and d is a constant. The 3D graph of u₃ (x, t) is demonstrated in Figure 1.

Figure 1

3D graph of u₃(x, t)for ε = 0.5, α₁ = 1, α₃ = 0.5, α₄ = 1 and d = 1.

3.2 Functional variable method

By means of the transformation f_η = P(f), Eq. (10) is reduced as

12P2f′=k2−1εα4k2−α3f+α1α3−α4k2f2,

and therefore

Pf=33εα4k2−α3fεα4k2−α3−2εα1f+3k2−3.(14)

Now, from f_η = P(f) and Eq. (14), we find that

∫dffεα4k2−α3−2εα1f+3k2−3=33εα4k2−α3η+ε0,

where ε₀ is an integrating constant. After integration and considering the constant of integration to zero, the exact solution of Eq.(10) is obtained as the following form

fη=3k2−12εα1sec⁡h2εk2−1k2α4−α32εk2α4−α3η+ε0.

Now, there are two cases.

Case 1. If ε(k² − 1)(k²α₄ − α₃) > 0, then two hyperbolic solutions to the strain wave equation in microstructured solids are derived as

u1x,t=3k2−12εα1sec⁡h2εk2−1k2α4−α32εk2α4−α3x−kt+ε0,

u2x,t=−3k2−12εα1csc⁡h2εk2−1k2α4−α32εk2α4−α3x−kt+ε0.

Case 2. If ε(k² − 1)(k²α₄ − α₃) < 0, then two periodic solutions to the strain wave equation in microstructured solids are obtained as

u3x,t=3k2−12εα1sec2−εk2−1k2α4−α32εk2α4−α3x−kt+ε0,

u4x,t=3k2−12εα1csc2−εk2−1k2α4−α32εk2α4−α3x−kt+ε0.

The 3D graph of u₁ (x, t) is illustrated in Figure 2.

Figure 2

3D graph of u₁(x, t) for k = 0.5, ε =0.5, α₁ = 1, α₃ = 0.5, α₄ = 1 and ε₀ = 0.

Remark

Each exact solution was put into the strain wave equation, using Maple package, and their satisfactions demonstrate the validity of the solutions in this study.

4 Conclusion

The Kudryashov and functional variable methods along with the symbolic computation system have been successfully used to study the strain wave equation in microstructured solids. Some exact solutions of this equation have formally extracted, which are in agreement with those previously reported in the literature. The methods clearly have some advantages: they are remarkably simple to use; and can generate abundant exact solutions. Besides, the methods can be applied to a wide range of nonlinear wave equations; specially, the Kudryashov method which can be utilized to handle high order nonlinear partial differential equations. The computations associated with the methods have been performed by Maple 17.

Acknowledgement

The authors are very grateful to the anonymous reviewers whose insightful and constructive comments enhance the quality of the work undertaken here.

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Received: 2016-4-28

Accepted: 2016-12-3

Published Online: 2017-1-27

Published in Print: 2017-3-1

Articles in the same Issue

https://doi.org/10.1515/nleng-2016-0020

Keywords for this article

Strain wave equation; Kudryashov method; Functional variable method; Exact solutions