Abstract
Based on the homogenous balanced principle and subequation method, an improved separation variables function-expansion method is proposed to seek exact solutions of time-fractional nonlinear PDEs. This method is novel and meaningful without using Leibniz rule and chain rule of fractional derivative which have been proved to be incorrect. By using this method, we studied a nonlinear time-fractional PDE with diffusion term. Some general solutions are obtained which contain many arbitrary parameters. Solutions given in related reference are just our especial case. And we also obtained some new type of solutions.
1. Introduction
It is well known that fractional-order models are more adequate than previously used integer-order models due to an exact description of nonlinear phenomena. Therefore, nonlinear fractional partial differential equations (nfPDEs) have attracted more and more attention. Most recently, FPDEs are increasingly used in mathematical modeling of fluid mechanics, biological and chemical processes, signal processing, and control systems, and they are also used in fractal and differential geometry, and so on(see [1–8] and their references cited). Many natural phenomena associated with real-time problems which depend on both time instant and the previous time history, especially, can be successfully modeled by time-fractional nonlinear partial differential equations. Of course, investigating solutions of nfPDEs plays an important role in a large number of research fields. Until now, there is no general method. Nevertheless, many powerful methods were used for solving fractional differential equations, such as adomian decomposition method [9], symmetry method [10–12], numerical method [13, 14], first integral method [15], the homotopy perturbation method [16], invariant subspace method [17], and fractional variational iteration method [18], and have achieved significant progress.
There are many definitions of fractional derivative. The fractional differential equations can be expressed in terms of different differential operators defined by Riemann-Liouville, Caputo, Weyl, and many others. Among these definitions, Caputo definition is most frequently used. Here, let us briefly review this definition of fractional derivative.
The Caputo fractional derivative of order is defined by the following expression: where . In this paper we will adopt Caputo fractional derivative definition to investigate exact solution of the following type of nonlinear time-fractional partial differential equation: where is Caputo differential operator, , and , , . Equation (2) denotes a series of nonlinear time-fractional PDEs; these models defined by (2) can be used to accurately describe nonlinear phenomena in connection with real-time problems, which not only depend on time instant but also depend on the previous time history.
Recently, Feng [19] introduced a fractional method for seeking traveling wave solutions of space-time-fractional partial differential equations under the following modified Riemann-Liouville derivative definition [20]:
The following properties for the modified Riemann-Liouville derivative are usually used:
Let us briefly review this method. Using traveling wave transformation and (4) and (5), they reduced the following fractional partial differential equation: which can be converted into the following fractional ordinary differential equation with respect to the variable :
Then, they suppose that the solution of (7) can be expressed by a polynomial in as follows: where satisfies the following fractional ordinary differential equation:
In order to find exact solutions of (9), a nonlinear fractional complex transformation is used. Then, (9) can be turned into the following second ordinary differential equation: The exact solutions of (10) are known.
Finally, substituting (8) into (7), equating each coefficient of this polynomial on to zero, they can obtain a large number of exact solutions of space-time fractional partial differential equations (6).
We noticed two problems as follows. At first, in a short communication [21], Tarasov proved that formula (4) and the chain rule (5) cannot be performed together for fractional derivatives of noninteger orders . Therefore, using traveling wave transformation , space-time fractional partial differential equations (6) cannot be converted into fractional ordinary differential equation (7). And, (9) cannot convert to (10) under transformation .
In addition, it should be noted that the Leibniz rule in the form cannot hold for fractional derivatives of order for sets of differentiable and nondifferentiable functions [22]. It can be proved easily by using a counterexample. Let where . Obviously we have Then, one has
As a result, we have the condition which must be performed if the Leibniz rule (11) holds. For example, if we make and take into account , then condition (15) can be represented in the formwhere . It is not difficult to find that (16) holds only if . That is to say, the Leibniz rule (11) does not hold with . Therefore, the following formula used in references is incorrect:
Therefore, when substituting (8) into (7), the left-hand side of (7) cannot be expressed by the polynomial in .
So, this fractional method is not reliable and the obtained results are incorrect. By the way, we should be more cautious when using this kind of methods based on chain rule (5) and Leibniz rule (11), like the subequation method, the -expansion method, the exp-function method, the functional variable method, the trial equation method, the simple equation method, and so forth. We cannot obtain the exact solutions of compound function type of time-fractional PDEs (2) as in the references. Encouraged by Rui’s work [23], we shall introduce an improved method based on the homogenous balanced principle; by using this improved method we shall investigate exact solutions of a series of nonlinear time-fractional PDEs formed as (2).
The rest of this paper is organized as follows. In Section 2, we will introduce the improved separation variable expansion method based on the homogenous balanced principle. In Section 3, by using this new method, we will investigate exact solutions of a nonlinear time-fractional PDE with diffusion term discussed in (2).
2. Introduction of Improved Separation Variable Function-Expansion Method
Although the fractional chain rule (5) and Leibniz rule (11) do not hold, they do not affect the investigation of the exact solutions of nonlinear time-fractional PDE (2) since formulas (4) still hold.
Remark 1. In reference [23], Rui points out that Leibniz rule (11) still holds, which is incorrect. Actually, in our method, it does not matter that Leibniz rule (11) does not hold. Because we just need the following formula: which is easy to be proved by definition for Caputo fractional derivative (1) and for the modified Riemann-Liouville derivative (3).
In the following, we introduce main steps of improved function-expansion method of separation variable type as follows.
Step 1. According to the formulas (4) and (18) and characters of nonlinear time-fractional partial differential equation (2), we suppose that (2) has the following exact solutions of the separation variable type: where the function can be taken as power function , Mittag-Leffler function , or , is a positive integer, and are constants to be determined later. The function satisfies the following subequation: where are constants. Some solutions of (20) are listed as follows.
When , where is an arbitrary integral constant.
When , When ,
Taking integral constant , under some conditions, we can obtain many special solutions of (20) which are listed in Table 1.
Step 2. In order to determine the value of the function , we balance power of between the term of the highest order in the right-hand side of (2) and the highest order term in the left-hand side of (2). By the way, the highest order term in the left-hand side of (2) is still because we only make derivation for in the left-hand side of (2). Once the value has been determined, the expansion expression (19) can be fixed correspondingly.
Step 3. By using specific expansion expression obtained in the Step 2, we substitute it into (2) (make fractional-order derivations for and make integer-order derivations for ); then we balance the power of (such as , ); thus the values of , can be obtained.
Step 4. In two sides of equation obtained by Step 2, we let coefficients of the same order for every term in two sides of the equation be equal; thus we can determine values of all the coefficients .
Step 5. Finally, substituting the values of all the parameters obtained in above steps into (19), we will obtain exact solutions of the nonlinear time-fractional PDE (2).
Remark 2. In [23], is taken as one of , and , which is just a special solution of our (20). So our method is more general than Rui’s method [23]. In addition, our method can test a series of functions at one time which satisfy (20). The improved method is more efficient and simple.
3. Exact Solutions of a Nonlinear Time-Fractional PDE with Diffusion Term
In this subsection, we will study a nonlinear time-fractional PDE with diffusion term as follows: where ; the is diffusion term. When and , (24) can be rewritten as the following nonlinear diffusion PDE: which have appeared in problems related to plasma and solid state physics; see [24] and references cited therein. In [23], Rui studied the situation of . Equation (25) becomes the following time-fractional PDE with diffusion term: And exact solutions of (26) with parameters and are given. Here we will investigate more exact solutions by improved method introduced in Section 2.
When , taking , we suppose that (26) has an exact solution formed as . We find that the highest order of in the term is just . In the right-hand side of (26), the highest order of in the nonlinear terms and is . Considering the relation between parameters and , we can let be an arbitrary positive integer to test (some terms may be counteracted under certain conditions). Here, for simplicity, we just discuss the situations of and .
3.1. Situation of
Taking , we first suppose that (26) has an exact solution as the following form: where satisfies (20) and are undetermined constants that can be determined later. Substituting (27) into (26), using (20), it can be reduced to According to homogenous balanced principle, we let Solving (29) yields Substituting (30) into (28), it can be reduced to
3.1.1.
Equation (31) can be simplified to According to homogenous balanced principle, we let Solving (33) yields Substituting (34) into (32), it can be reduced to Balancing the power of , one has Solving (36), we obtain the following result: Substituting (37) into (27), using solution (23) of subequation, we can obtain exact solution of (26) with as follows: where and is the root of .
3.1.2.
Equation (31) can be simplified to According to homogenous balanced principle, we let Solving (40) yields Substituting (41) into (39), it can be reduced to Balancing the power of , one has Solving (43), we obtain the following results: where . Substituting (44) into (27), using solution (23) of subequation, we can obtain exact solution of (26) with as follows:
By using Table 1, taking parameters as some particular values, many specific exact solutions of (26) can be got, parts of which are listed as follows.
When , , then , . Equation (45) can be reduced to where is positive and is an arbitrary constant.
When , , then , . Equation (45) can be reduced to where , is positive, and is an arbitrary constant.
If one lets or , then , or . Therefore, solutions (46) and (47) are reduced to
Remark 3. Equations (48) are just the solutions (3.37) and (3.39) given in reference [23], so we can say that our solutions are general, including many unreported solutions.
When , , then , . Equation (45) can be reduced to where is positive and is an arbitrary constant.
When , , then , . Equation (45) can be reduced towhere , is positive, and is an arbitrary constant.
3.1.3. ,
Equation (31) can be simplified to
Case 1 (). According to homogenous balanced principle, we let Solving (52) yields Substituting (53) into (51), it can be reduced to Balancing the power of , one has Solving (55), we obtain the following results: Substituting (56) into (27), using solution (22) of subequation, we can obtain exact solution of (26) with as follows: where .
Case 2 (). Equation (51) becomes According to homogenous balanced principle, we let Solving (59) yields Substituting (60) into (58), it can be reduced to Balancing the power of , one has Solving (62), we obtain the following result: Substituting (63) into (27), using solution (22) of subequation, we can obtain exact solution of (26) with as follows: where and is the root of .
3.1.4. ,
According to (31) and homogenous balanced principle, we let Solving (65) yields Substituting (66) into (31), it can be reduced to Balancing the power of , one has Solving (68) we obtain the following results: where , . Substituting (69) into (27), using solution (23) of subequation, we can obtain exact solution of (26) with as follows: where .
When , then . Equation (70) can be reduced to where .
When , then . Equation (70) can be reduced to where .
If , substituting (27) into (26), using (20), we can obtain the following five families of exact solutions of (26).
Family 1. where .
Family 2. where , is an arbitrary constant.
Family 3. where . Taking parameters as suitable values, we can get the following special exact solutions of (26): where , is positive, and is an arbitrary constant. where , is positive, and is an arbitrary constant.
Family 4. where and is an arbitrary constant.
Family 5. where and is an arbitrary constant.
3.2. Situation of
Taking , we suppose that (26) has an exact solution as the following form: where satisfies (20) and are constants that can be determined later. Substituting (80) into (26), using (20), balancing the power on of the reduced equation, we have Substituting (81) and (80) into (26), balancing the power of , yields Solving the above algebraic equations, we have the following results.
Case 1 (, , ). Substituting (83) into (27), using solution (21) of subequation, we can obtain exact solution of (26) with as follows:
Case 2 (, ). where . Substituting (85) into (27), using solution (23) of subequation, we can obtain the exact solution of (26) with as follows: When and , then . Equation (86) can be reduced to where is positive and is an arbitrary constant.
When and , then . Equation (86) can be reduced to where is positive and is an arbitrary constant.
If , substituting (80) into (26), using (20), we can obtain the following exact solutions of (26): When and , then . Equation (89) can be reduced to where is positive and is an arbitrary constant.
When and , then . Equation (89) can be reduced to where is positive and is an arbitrary constant.
4. Conclusions
In this work, we proved that the fractional Leibniz rule that appeared in many references does not hold under Riemann-Liouville definition and Caputo definition of fractional derivative. Based on the homogenous balanced principle, we introduced a general method for investigating exact solution of nonlinear time-fractional PDEs. By using this method called improved separation variable function-expansion method, we studied a nonlinear time-fractional PDE with diffusion term. Some new results are obtained.
Firstly, compared with Rui’s method [23], it is easy to find that our method is more general. All solutions given in reference [23] can be obtained by taking special parameters in our results. For example, taking , our solutions (46) and (47) become solutions (3.40) and (3.41) in [23], respectively. Taking , our solutions (38) become solution (3.47) in [23]. Taking , our solutions (84) become solution (3.60) in [23]. Other solutions obtained in our work are new, such as solutions (49), (50), (57), (64), (71), (72), (87), and (88) which are not reported in related references. In addition, we should adopt different subequation for other time-fractional PDEs, such as . Finally, our method is simple and efficient for application without any skill.
According to symmetrical characteristic, this method also can be used to investigate exact solutions of space fractional PDEs which are formed as follows:
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This research is supported by the Natural Science Foundation of China (nos. 11461022, 11361023), Science Foundation of Yunnan Province (2014FA037), and Middle-Aged Academic Backbone of Honghe University (no. 2014GG0105).