Abstract
The fractional partial differential equations stand for natural phenomena all over the world from science to engineering. When it comes to obtaining the solutions of these equations, there are many various techniques in the literature. Some of these give to us approximate solutions; others give to us analytical solutions. In this paper, we applied the modified trial equation method (MTEM) to the one-dimensional nonlinear fractional wave equation (FWE) and time fractional generalized Burgers equation. Then, we submitted 3D graphics for different value of .
1. Introduction
All over the world, a physical event may depend not only on the time but also on the previous process, which can be successfully formed by using the theory of derivatives and integrals of fractional order. These processes represent different physical problems in the manner of variable order. In this sense, the fractional differential equations have been used for the definition of nonlinear phenomena in applied science, physics, chemistry, engineering, and other areas of science. In order to solve these problems, a general method cannot be defined even in the most useful works. Also, a remarkable progress has been become in the construction of the approximate solutions for fractional nonlinear partial differential equations [1–3]. Several powerful methods [4–13] have been proposed to obtain approximate and exact solutions of fractional differential equations, such as the Sumudu transform method, the Homotopy analysis method, and the homotopy perturbation method.
Liu introduced a new approach called the complete discrimination system for a polynomial to classify the traveling wave solutions as nonlinear evolution equations and applied this idea to some nonlinear partial differential equations [14, 15]. So, to the best of our knowledge, the modified trial equation method has not been widely applied for studying the invariance properties of fractional PDEs. Furthermore, some authors [16, 17] used the trial equation method proposed by Liu. However, we established a new modified trial equation method to obtain 1-soliton, singular soliton, hyperbolic function solutions [18, 19], elliptic integral function and Jacobi elliptic function solutions, or the others to nonlinear partial differential equations with generalized evolution in [20–22].
In Section 2, primarily we give some definitions and properties of the fractional calculus and also produce a new modified trial equation method for fractional nonlinear evolution equations with higher order nonlinearity. The power of this steerable method showed that this method can be applied to different equations. In Section 3, as an application, we solve the nonlinear fractional partial differential equation such as one-dimensional nonlinear fractional wave equation [23]: where , , and are arbitrary constants and is a parameter describing the order of the fractional time derivative. We consider time fractional generalized Burgers equation [27] described as follows: where , which occur in different areas in mathematical physics; here the time fractional derivative leads to subdiffusion and subdispersion, respectively, and extend the Lie symmetry analysis to derive their infinitesimals [24]. In this research, we obtain the classification of the wave solutions to (1) and (2) and derive some new solutions. Using the modified trial equation method, we found some new exact solutions of the fractional nonlinear physical problem. The purpose of this paper is to obtain exact solutions of the one-dimensional nonlinear fractional wave equation by modified trial equation method.
2. Preliminaries
In this part of the paper, it would be helpful to give some definitions and properties of the fractional calculus theory. Here, we shortly review the modified Riemann-Liouville derivative from the recent fractional calculus proposed by Jumarie [25, 26]. Let be a continuous function and . The Jumarie modified fractional derivative of order and may be defined by the expression of the following [23]: In addition to this expression, we may give a summary of the fractional modified Riemann-Liouville derivative properties which are used further in this paper. Some of the useful formulas are given as [23]
In this paper, a new approach to the trial equation method will be given. In order to apply this method to fractional nonlinear partial differential equations, we consider the following steps.
Step 1. We consider time fractional partial differential equation in two variables and a dependent variable : and take the wave transformation where . Substituting (6) into (5) yields a nonlinear ordinary differential equation:
Step 2. Take trial equation as follows: where and are polynomials. Substituting above relations into (7) yields an equation of polynomial of : According to the balance principle, we can get a relation of and . We can compute some values of and .
Step 3. Letting the coefficients of all be zero will yield an algebraic equations system: By solving this system, we will specify the values of and .
Step 4. Reduce (8) to the elementary integral form Using a complete discrimination system for polynomial to classify the roots of , we solve (12) with the help of Mathematica7 and classify the exact solutions to (7). In addition, we can write the exact traveling wave solutions to (5), respectively. For a better interpretation of results obtained in this way, we plotted 3D surfaces of (27), (40), (52), and (60) in Figures 1, 2, 3, and 4 by taking into consideration suitable parameter.
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3. Applications
In this section, we applied the method to the one-dimensional nonlinear fractional wave equation and time fractional generalized Burgers equation.
Example 1. Firstly, we consider one-dimensional nonlinear fractional wave equation [23]. In the case of , (1) reduces to the classical nonlinear one-dimensional nonlinear wave equation. Many researchers have tried to get the exact solutions of this equation by using a variety of methods.
Let us consider the travelling wave solutions of (1), and then and we perform the transformation and where and are constants. Then, integrating this equation with respect to and setting the integration constant to zero. When it comes to converting fractional order differential equation into differential equation with integer order, we can perform the following:
so, when we use and in (1), we get ordinary differential equation as follows:
When we rearrange to (8) and (9) for balance principle, we obtain the following:
Balancing the highest order nonlinear terms of and in (14), we get balance term for suitability
This resolution procedure is applied, and we obtain results as follows.
Case 1. If we take and , then
and then
where and When we use and in (14), we get a system of algebraic equations for (14). Thus, we have a system of algebraic equations from the coefficients of the polynomial of . Solving the algebraic equation system (14) by using Mathematica programming yields the following coefficients:
By substituting these coefficients into (12), we have
Integrating (21) by using Mathematica programming, we obtain the solutions to (1), as follows, for different values of the roots of the polynomial equation:
where , and also and are the roots of the polynomial equation as follows:
Therefore, we find solutions
For simplicity, if we take , then the solutions equations (24) and (25) can reduce to rational and single kink solution, respectively,
where , , and . Here, and are the inverse width of the soliton. We can regulate (27) to rewrite in the hyperbolic form as follows:
If we consider the following equation for simplicity of (28):
then, we get
If it takes for (28), we get the hyperbolic function solution of (28):
where .
Remark 2. The solutions equations (26)-(27) obtained by using the extended trial equation method for (1) have been checked by Mathematica. To our knowledge, the rational function solution and single kink solution that we found in this paper are not shown in the previous literature. These results are new traveling wave solutions of (1).
Case 2. In the same way as in Case 1, If we take and , then where , . Respectively, solving the algebraic equation system (11) yields the following: Substituting these coefficients into (12), we have Integrating (35), we procure the solution to (1) as follows: where , , and . Also , , and are the roots of the polynomial equation Therefore, we find a solution from (36): For simplicity, if we take , then the solution equation (39) can reduce to rational solution
Remark 3. The solution equation (40) computed in Case 2 has been checked by Mathematica. We think that these solutions have not been found in the literature, and these results are new traveling wave solutions of (1).
Example 4. Secondly, we consider the time fractional generalized Burgers equation [24] as follows:
In the case of and , (41) reduces to the well-known classical nonlinear Burgers equation. Many researchers have tried to get the exact solutions of this equation by using a different method [28, 29].
Let us consider the travelling wave solutions of (41), and then we perform the transformation and where , are constants. Then, integrating this equation with respect to and setting the integration constant to zero, we get
When we conduct once more transformation
we get the following:
Substituting (8) into (44) and using balance principle yield the following:
This resolution procedure is applied, and we obtain results as follows.
Case 1. If we take and , then where and . Thus, we have a system of algebraic equations from the coefficients of the polynomial of . Solving the algebraic equation system (11) yields the following: By substituting these coefficients into (11), we have By integrating (48), we procure the solution to (41) as follows: where . By substituting the solutions equation (49) into (43), we found solutions of the following exact traveling wave solutions, such as rational function solution and single kink solution: For simplicity, if we take , then the solutions equation (50) can reduce to the following: where , , and .
Remark 5. The solutions equations (51) and (52) obtained by using the modified trial equation method for (41) have been checked by Mathematica. To our knowledge, the rational function solution and single kink solution that we found in this paper are new traveling wave solutions of (41).
Case 1. If we take and , then where , . Thus, we have a system of algebraic equations from the coefficients of the polynomial of . Solving the algebraic equation system (11) yields the following: By substituting these coefficients into (11), we have where . By integrating (55), we procure the solution to (41) as follows: where , , , and . Also , , and are the roots of the polynomial equation: By substituting the solution equation (56) into (43), we found solution of the following exact traveling wave solutions, such as rational function solution: For simplicity, if we take , then the solution equation (59) can reduce to rational solution:
Remark 6. The solution equation (60) computed in Case 2 has been checked by Mathematica. We think that these solutions have not been found in the literature, and these results are new traveling wave solutions of (41).
4. Conclusions
In this paper, the modified trial equation method has been applied to the one-dimensional nonlinear fractional wave equation and time fractional generalized Burgers equation. We used it to obtain some soliton and rational function solutions to the one-dimensional nonlinear fractional wave equation and time fractional generalized Burgers equation. This method is reliable and effective and gives several new solution functions such as rational function solutions and single kink solutions. We think that the proposed method can also be applied to other generalized fractional nonlinear differential equations. In our future studies, we will solve nonlinear fractional partial differential equations by this approach.