Abstract
A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the -dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.
1. Introduction
Fractional partial differential equations (fPDEs) are the generalized form of the integer order differential equations. fPDEs can more accurately describe the complex physical phenomena occurring in fluid dynamics, high-energy physics, plasma physics, elastic media, optical fibers, chemical kinematics, chemical physics, acoustic waves, biomathematics, and many other areas [1, 2]. In recent years, many researchers have shown great interest in the search for exact solutions to nonlinear fPDEs. At present, several methods for finding the exact solutions of fPDEs have been presented, for example, the Adomian decomposition method [3–6], variational iteration method [7–9], homotopy perturbation method [10–13], homotopy analysis method [14, 15], differential transform method [16], spectral methods [17, 18], discontinuous Galerkin method [19], Kansa method [20], fractional subequation method [21], generalized fractional subequation method [22], fractional projective Riccati expansion method [23], exp-function method [24, 25], -expansion method [26–28], functional variable method [29, 30], and first integral method [31, 32].
However, the above methods either are relatively complicated or have large computational cost. We propose a table lookup method in this paper. This method is straightforward and has small computational cost. We apply it to solve nonlinear fractional order partial differential equations with using the fractional complex transform and the modified Riemann-Liouville derivative defined by Jumarie [33]. Jumarie’s modified Riemann-Liouville derivative of order is defined by the following expression [34]:where denotes a continuous function and is the Gamma function.
Moreover, the modified Riemann-Liouville derivative has various useful properties, including the following:
The rest of this paper is organized as follows. In Section 2, the basic ideas and main steps of the table lookup method are given. In Section 3, the exact analytical solutions of the time fractional simplified MCH equation, the space-time combined KdV-mKdV equation, the -dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation are constructed using the proposed method. In Section 4, some conclusions are obtained.
2. Basic Idea of the Table Lookup Method
In this section, we outline the main steps of the table lookup method for solving nonlinear fPDEs. Let us consider a fractional order partial differential equation in the following form: where , , and denote modified Riemann-Liouville derivatives of , represents a polynomial in and its various partial derivatives, and , , and are variables. In the following, we give the main steps of our proposed method.
Step 1. First, we use the fractional complex transformation as follows: where is a nonzero constant. Thus, (6) can be transformed into the following nonlinear ordinary differential equation (ODE) of integer order with respect to in sense of the properties of Jumarie’s modified Riemann-Liouville derivative given in (2)–(5):where is a polynomial in and its various derivatives and .
Step 2. Integrating (8) once or several times with respect to , setting the integration constant to zero if possible, multiplying both sides of the equation by , and then integrating once again, we obtain different types of the auxiliary equation , where () are constant coefficients that can be determined in this step.
Step 3. According to the equation type obtained in Step 2, we create the tables of solutions to the corresponding type of auxiliary equation.
Step 4. Looking up the tables and determining the corresponding coefficients and existence conditions of the exact solutions, we successfully obtain the exact analytical solutions to (6).
3. Applications of the Proposed Method
In this section, we apply the table lookup method to construct exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time combined KdV-MKdV equation, the -dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation, which are very important nonlinear fPDEs in mathematical physics and have received much attention from researchers.
3.1. The Time Fractional Simplified Modified Camassa-Holm (MCH) Equation
We now consider the following time fractional simplified modified Camassa-Holm (MCH) equation:where , , and . Equation (9) is a variation of the following equation:
Using the fractional complex transformation , with (9), we can obtain the following nonlinear ODE:
Integrating (11) once with respect to and setting the integral constant to zero, we obtain
Multiplying (12) by and then integrating once again, we obtain where , , and is an integration constant.
According to the solutions of the auxiliary equation presented in [35], the exact solutions of (14) are listed in Table 1:
Equations (13) and (14) have the same form. Thus, looking up Table 1 and determining the coefficients and the existence condition of the exact solutions, we can obtain the following solutions of (9):(1)When , , and , we obtain (2)When , , and , we obtain (3)When , , and , we obtain (4)When , , and , we obtain (5)When , , and , we obtain (6)When , , and , we obtain
The evolution of exact solution for (15)–(24) is shown in Figures 1–6.
It can be observed from (15)–(24) and Figures 1–6 that triangular periodic solution, bell-shaped solitary wave solution, kink-shaped solitary wave solution, and Jacobi elliptic function solution of the time fractional simplified MCH are obtained.
When , (9) becomes (10), namely, simplified MCH equation, and , obviously, (10) still have triangular periodic solution, bell-shaped solitary wave solution, kink-shaped solitary wave solution, and Jacobi elliptic function four types of solutions. Reference [36] only obtained triangular periodic solutions, kink-shaped solitary wave solution, and singular solution of simplified MCH equation. Thus, our table lookup method is more powerful.
3.2. The Space-Time Fractional Combined KdV-mKdV Equation
In this section, we will apply the table lookup method to the following space-time fractional combined KdV-mKdV equation [37]: where and are nonzero constants. This equation may describe the wave propagation of a bound particle, sound wave, or thermal pulse.
Now, we use the fractional complex transformation , with (25), which is reduced to the following ODE of integer order:
Integrating (26) once with the integral constant of zero yields
Multiplying (27) by , integrating once again, and setting the integral constant to zero, we obtainwhere , , and .
By virtue of the solutions of the auxiliary equation given in [38], the exact solutions of (29) are shown in Table 2:
Equations (28) and (29) have the same form. Thus, looking up Table 2 and determining the coefficients and existence conditions of the exact solutions, we can obtain the following solutions of (25):(1)If and , we obtain (2)If , , and , we obtain (3)If , , and , we obtain (4)If , , and , we obtain (5)If , , and , we obtain (6)If , , and , we obtain (7)If , , and , we obtain
The evolution of exact solution for (30)–(41) is described in Figures 7–13.
It can be observed from (30)–(41) that we have successfully obtained twelve exact analytical solutions of the space-time fractional combined KdV-mKdV equation. In comparison, [37] using the fractional mapping method only obtained five analytical solutions; thus, the proposed lookup table method is more concise and more effective.
3.3. The -Dimensional Time Fractional Zoomeron Equation
Now, we use the proposed methods to construct the exact solutions of the following nonlinear -dimensional time fractional Zoomeron equation [39]:where is the amplitude of the relative mode.
We perform the fractional complex transformation:
Substituting (43) into (42) and utilizing (2), we can reduce (42) into an ODE of integer order as follows:
Integrating (44) twice with respect to , we obtainwhere is the first integral constant () and the second integral constant is set to zero.
Multiplying (45) by and then integrating once again, we obtainwhere , , and is an integral constant.
According to the characteristics of (46), looking up Table 1 and determining the corresponding coefficients and existence conditions of the exact solutions, we obtain several types of solutions to (42) as follows:(1)When , , , and , we obtain (2)When , , , and , we obtain (3)When , , , and , we obtain (4)When , , , and , we obtain (5)When , , , and , we obtain (6)When , , , and , we obtain
By the way, when taking , (42) and (9) have the same type solutions, so that we no longer give figures of the obtained solutions of (42) in this section.
If we take , , , and , then the expression of is the same as that of for Eq. in [39], the expression of is the same as that of for Eq. in [39], and they can be, respectively, expressed as follows:
If we take , , , and , then the expression of is the same as that of for Eq. in [39], the expression of is the same as that of for Eq. in [39], and they can be, respectively, expressed as follows:
However, we also obtain the bell-shaped solitary wave solution and Jacobi elliptic function solution of the -dimension time fractional Zoomeron equation.
3.4. The Space-Time Fractional ZKBBM Equation
Next, we solve the exact solution of the following space-time fractional nonlinear Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation [40]:where and are arbitrary nonzero constants.
We first use the fractional complex transform , , and (55) becomes an ODE of integer order:
Integrating (56) once with an integral constant of zero yields
Multiplying (57) by and integrating once again with an integral constant of zero, we obtainwhere and .
According to the solutions of the auxiliary equation presented in [35], the exact solutions of the (59) are listed in Table 3:
Equations (58) and (59) have the same form. Thus, looking up Table 3 and determining the coefficients and existence conditions, we can obtain the following solutions of (55):(1)If and , we obtain (2)If and , we obtain(3)If , we obtain
The asymptotic behavior for the obtained exact solution (60)–(62) is shown in Figures 14–16.
(a)
(b)
It can be observed from Figures 14–16 that we obtained single solitary wave and multiple solitary wave solutions of fractional ZKBBM equation, and when , (62) is singular wave solution, and when , (62) is multiple solitary wave solution. Consequently, the exact solution of fractional differential equation can better describe the complex wave propagation. Reference [40] only obtained Jacobi elliptic doubly periodic solution of fractional ZKBBM equation, but we obtain three types of solutions using our proposed method.
4. Conclusions
In this paper, by looking up the tables of solutions of corresponding auxiliary equations, the exact analytical solutions of the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the -dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation have been constructed directly and successfully. Compared with previous results, solutions of more types are obtained, some of which are new. Furthermore, the solving process is more concise and straightforward. Consequently, the table lookup method is concluded to be effective method and would be a powerful mathematical tool for obtaining more exact analytical solutions of a large number of nonlinear fPDEs.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this article and regarding the funding that they have received.
Acknowledgments
The work was supported by the National Natural Science Foundation of China (61673222, 51607004), the Major Project of Nature Science Foundation of Higher Education Institution of Jiangsu Province, China (13KJA510001), the Project of Department of Education of Anhui Province, China (AQKJ2015B015), Research Innovation Program for College Graduates of Jiangsu Province, China (KYLX15 _0873), Anhui Provincial Natural Science Foundation, China (1608085QF157), and Key Projects of Anhui Province University Outstanding Youth Talent Support Program, China (gxyqZD2016207).