Direct method for solving nonlinear strain wave equation in microstructure solids

The modeling of wave propagation in microstructure materials should be able to account for the various scales of microstructure. In this paper, the extended trial equation method was modified to construct the traveling wave solutions of the strain wave equation in microstructure solid. Some new different kinds of traveling wave solutions was gotten as, hyperbolic functions, trigonometric functions, Jacobi elliptic functions and rational functional solutions for the nonlinear strain wave equation when the balance number is positive integer. The balance number of this method is not constant and changes by changing the trial equation. These methods allow us to obtain many types of the exact solutions. By using the Maple software package, it was noticed that all the solutions obtained satisfy the original nonlinear strain wave equation.

Recently, Gurefe et al. (2013) have presented a direct method, namely, the extended trial equation method for solving the nonlinear partial differential equations.Demiray et al. (2016;2015a;2015b); Demiray and Bulut (2015) and Bulut et al. (2014) have successively applied the extended trial method for solving the nonlinear partial differential equations.The governing nonlinear equation of the strain waves in microstructure solid is given by (Alam et al., 2014;Samsonov, 2001) where  accounts for elastic strains,  characterizes the ratio between the microstructure size and the wavelength,  characterizes the influence of dissipation .
Previous models were derived using the assumption of the homogeneity of microstructure.This is the case for the example of functionally graded materials which are made up of two or more material combined in solid state (Mahamood et al., 2012;Birman and Byrd, 2007).The main objective of this paper is to use the modified extended trial equation method to find a series of new analytical solutions to the strain wave Equation 2 for many different type of the roots of the trial equation.

DESCRIPTION OF THE EXTENDED TRIAL EQUATION METHOD
Suppose we have a nonlinear partial differential equation in the following form: and its partial derivatives, in which the highest order derivatives and nonlinear terms are involved.Let us now give the main steps for solving equation (3) using the extended trial equation method as (Gurefe et al., 2013;Demiray et al., 2016;2015a;2015b;Demiray and Bulut, 2015;Bulut et al., 2014;Ekici et al., 2013): Step 1.The traveling wave variable: where V is a nonzero constant, Equation 4 permits reducing equation (3) to the following ODE: where P is a polynomial of ) ( u and its total derivatives.
Step 2. Suppose the solution of Equation 5 takes the form: where j i   , are constants to be determined later.Using Equations 6 and 7, we have Step 3. Balancing the highest order derivative with the nonlinear terms, we can find the relations between Step 5. Setting the coefficients of this polynomial ) ( y  to be zero, we yield a set of algebraic equations: .,..., 0 , 0 Solve this system of algebraic equations to determine the values of , , , ,..., , Step 6. Reduce Equation 7 to the elementary integral form: where 0  is an arbitrary constant.Using a complete discrimination system for the polynomial to classify the roots of ), (Y  we solve Equation 11 with the help of software package such as Maple or Mathematica and classify the exact solutions to Equation 5.In addition, we can write the exact traveling wave solutions to Equation 3, respectively.
Remark 1.The difference between the modified trial expansion method, extended trial expansion method and modified extended trial method: (i) In the modified trial method, the trial equation is taking the following form: and the reduced elementary integral takes the following form: (ii) In the extended trial method, the trial equation is taking the following form: Gepreel et al. 123 and the reduced elementary integral takes the following form: (iii) In the modified extended trial expansion method, it seems to the reader as extended trial expansion method.But in the extended trial equation, there is no connection between the roots of the right side of Equation 11i  and the coefficients of the solutions i  and i  .Many papers have used the extended trial equation without making the connection between the root i  and the coefficients of the solutions i  and i  .So all the solutions in these papers does not satisfy the original equations.Then, this response was searched for, the authors which used the extended trial equation must be related between the roots of right side of Equation 11 and the solution coefficients i  and the trial equation coefficients i  .
For this, we call the modified extended trial expansion method.

MODIFIED EXTENDED TRIAL EQUATION METHOD FOR THE STRAIN WAVE EQUATION
Here, the modified extended trial equation method was used to find the traveling wave solutions to the following nonlinear strain wave differential equation: Porubov and Pastrone (2004) studied the propagation and attenuation or amplification of bell-shaped and kink-shaped waves, whose parameters are defined in an explicit form through the parameters of the microstructured medium.Also, Alam et al. (2014) used the generalized (G′/G)-expansion method to find an exact traveling wave solution of nonlinear strain wave differential equation.The traveling wave variable: where V is the speed of the traveling wave, permitting us to convert Equation 16 into the following ODE: Integrating Equation 18twice with respect to  , we have: where k is the integral constant.We suppose the traveling wave solution of the Equation 19 into the following form: where Y satisfies Equation 7 and 1  is an arbitrary positive integer.Balancing the highest order derivative u   with the nonlinear term 2 u in Equation 19, we have: Equation 21 has infinitely many solutions, consequently, we suppose some of these solutions as the following cases.
Case 1.In the special case, if Substituting equations ( 22) into Equation 19 we get a system of algebraic equations which can be solved by using the Maple software package to obtain the following results: , 2 . Now we will discuss the roots of the following equation: to integrate Equation 24 as the following families: From equating the coefficients of Y to both sides of Equation 26, we get a system of algebraic equations: , 0 We use the Maple software package to solve the system (equation 27) in  and 1  .We get the following results: Equations ( 27), ( 23) and ( 24) lead to: , 2 where 0  is an arbitrary constant and Substituting Equations 30, 28 and 27 into Equation 22, we get the traveling wave solution of nonlinear strain wave Equation 16 takes the following form: From equating the coefficients of Y to both sides of Equation 32, we get a system of algebraic equations in , 4 Equations 33, 23 and 24 lead to: , 2 where 3  is an arbitrary constant.In this family, the solution of Equation 24, when 1 2    takes the following form: Substituting Equations 36, 34 and 33 into Equation 22, we get the traveling wave solution of nonlinear strain wave Equation 16 taking the form: Also when 2 1    , the solution of Equation 24 has the form: Substituting Equations 38, 34 and 33 into Equation 22, we get the traveling wave solution of nonlinear strain wave Equation 16 takes the form:  25in the following form: From equating the coefficients of Y to both sides of Equation 40, we get a system of algebraic equations in which can be solved by using the Maple software package to get the following results: Equations 41, 23 and 24 lead to: , 2 where 3  is an arbitrary constant.In this family, the solution of Equation 24 has the form: Substituting Equations 44, 42 and 41 into Equation 22, we get the traveling wave solution of nonlinear strain wave Equation 16 takes the form: The behavior of the exact Solution 45 has been illustrated in Figure 1.
real numbers, consequently we can write Equation 25 in the following form: From equating the coefficients of Y to both sides of Equation 46, we get a system of algebraic equations in


which can be solved by using the Maple software package to get the following results: Equations 47, 28 and 24 lead to: where 3  is an arbitrary constant.In this family, the integration of Equation 24 takes the following form: Substituting Equations 50, 48 and 47 into Equation 22, we get the traveling wave solution of nonlinear strain wave Equation 16 has the form: The behavior of the exact Solution 51 has been illustrated in Substituting Equation 51into Equation 19, we get a system of algebraic equations which can be solved to obtain the following results: . 48 where Substituting Equation 53 into Equations 7 and 11, we have: . Now we will discuss the roots of the following equation: To integrate Equation 54, we discuss the roots of Equation 55as the following families: From equating the coefficients of Y to both sides of Equation 56, we get a system of algebraic equations in which can be solved by using the Maple software package to get the following results: .
Equations 57, 53 and 54 lead to: . 4 where 4  is an arbitrary constant and Substituting Equations 60, 58 and 57 into Equation 52, we get the traveling wave solution of nonlinear strain wave Equation 16 taking the following form: The behavior of the exact Solution 61 has been illustrated in Figure 3 Family From equating the coefficients of Y to both sides of Equation ( 62), we get a system of algebraic equations in which can be solved by using the Maple software package to get the following results: Equations 63, 53 and 54 lead to:  61) for nonlinear strain wave Equation (Equation 16) when where 4  is an arbitrary constant and Vt x e e Y (66) Substituting Equations 66, 64 and 63 into Equation 52, we get the traveling wave solution of the strain wave Equation 16 takes the form: The behavior of the exact Solution 67 has been illustrated in Figure 4.
From equating the coefficients of Y to both sides of Equation 68, we get a system of algebraic equations in      67) for nonlinear strain wave Equation (Equation 16) at  52, we get the traveling wave solution of nonlinear strain wave Equation 16 takes the form: Family 8.If Equation 55 has four complex roots, are real numbers, consequently we can write Equation 55 in the following form: From equating the coefficients of Y to both sides of Equation 74, we get a system of algebraic equations in

RESULTS AND DISCUSSION
This method allowed the construction of many types of the traveling wave solutions in the hyperbolic functions, trigonometric functions, and Jacobian elliptic functions.The balance number of this method is not constant as in other methods but changes when the trial equation changes.This method has generalized the tanh-function method, Jacobian elliptic functions methods, and Exp function method.

Conclusion
In this paper, the modified extended trial equation method was used to construct series of some new analytic solutions for some nonlinear partial differential equations in mathematical physics when the balance numbers is positive integer.The exact solutions were constructed in many different functions such as hyperbolic function solutions, trigonometric function solutions and Jacobi elliptic functions solutions and rational solutions for nonlinear strain wave equation.The performance of this method is reliable, effective and powerful for solving more complicated nonlinear partial differential equations in mathematical physics.This method is more powerful than other method for solving the nonlinear partial differential equations.This method can be used to solve many nonlinear partial differential equations in mathematical physics.
then we have the nondissipative case and governed by the double dispersive Equation 45 and 46 as follows:

Family 1 .
If Equation 25 has three equal repeated roots 1  , consequently we can write Equation 25 in the following form:

Family 2 .
If Equation 25 has two equal repeated roots 1  and the third root is 2  and 2 1    , consequently we can write Equation 25 in the following form:

)
solved by using the Maple software package to get the following results:

Figure 2 .
Figure 2. The real part of the traveling wave solution (Equation 51) and its projection at 0  t when the parameters take special values , 25 .0 , 5 .0 , 2 2 1 1

Family 5 .
If Equation 55 has four equal repeated roots 1  , consequently we can write the Equation 55 in the following form: 6.If the Equation 55 has two equal repeated roots

Family
, consequently we can write Equation 55 in the following form: solved by using the Maple software package to get the following results:

Figure 4 .
Figure 4.The traveling wave solution (Equation67) for nonlinear strain wave Equation (Equation16) at 72, 70 and 69 into Equation solved by using the Maple software package to get the following results: Equations 75, 66 and 67 lead to get: