Abstract

We investigate methods for obtaining exact solutions of the (3 + 1)-dimensional nonlinear space-time fractional Jimbo-Miwa equation in the sense of the modified Riemann-Liouville derivative. The methods employed to analytically solve the equation are the -expansion method and the novel -expansion method. To the best of our knowledge, there are no researchers who have applied these methods to obtain exact solutions of the equation. The application of the methods is simple, elegant, efficient, and trustworthy. In particular, applying the novel -expansion method to the equation, we obtain more exact solutions than using other existing methods such as the -expansion method and the -expansion method. The exact solutions of the equation, obtained using the two methods, can be categorized in terms of hyperbolic, trigonometric, and rational functions. Some of the results obtained by the two methods are new and reported here for the first time. In addition, the obtained exact explicit solutions of the equation characterize many physical meanings such as soliton solitary wave solutions, periodic wave solutions, and singular multiple-soliton solutions.

1. Introduction

Various phenomena such as shallow water waves and multicellular biological dynamics arising in the nonlinear physical sciences [1, 2], engineering [3, 4], and biology [5] can be modeled by a class of integrable nonlinear evolution equations which can be expressed in terms of nonlinear partial differential equations (NPDEs) of integer orders. Consequently, study of traveling wave solutions of NPDEs plays a significant role in the investigation of behaviors of nonlinear phenomena. Due to the efficiency, reliability, and easy use of symbolic software packages such as Maple or Mathematica, many powerful methods have been constructed and developed to analytically solve NPDEs with their aid. Over the last few decades, exact solutions, analytical approximate solutions, and numerical solutions of NPDEs have been successfully obtained. The methods for obtaining exact explicit solutions of NPDEs are, for example, the -expansion method [6], the tanh-function method [7, 8], the exp-function method [9, 10], the -expansion method [11], Hirota’s direct method [12, 13], Kudryashov method [14, 15], and so on. Examples of the methods for obtaining analytical approximate solutions to NPDEs are the variational iteration method [16, 17] (VIM), the Adomian decomposition method [18, 19] (ADM), and the homotopy perturbation method [20, 21] (HPM). In addition, the examples of useful methods for solving NPDEs numerically are the generalized finite difference method [22], the finite volume method [23], and the finite element method [24].

Since fractional derivatives [25] such as the Riemann-Liouville derivative and the Caputo derivative can describe the memory and hereditary properties of materials and processes which is different from ordinary derivatives, fractional differential equations (FDEs), which are associated with fractional derivatives and the generalization of the classical differential equations of integer orders, are expansively used to model various complex phenomena in many study fields such as physics [26], engineering [27], finance [28], and biology [29]. It has been found that the above-mentioned methods with their improvements (see, e.g., [30–33]) are also widely applicable to solve FDEs. Searching for exact explicit solutions to nonlinear fractional partial differential equations (NFPDEs) is a research field of active interest. Nowadays, many approaches with the help of symbolic software packages have been developed to efficiently provide exact solutions of NFPDEs, for example, the improved extended tanh-coth method [34], the improved generalized exp-function method [35], the fractional Riccati expansion method [36], the -expansion method [37–41], and the novel -expansion method [42–45]. The common idea of these mentioned methods is based on the homogeneous balance principle.

The (3 + 1)-dimensional Jimbo-Miwa equation, which was introduced by Jimbo and Miwa [46], is written as which is the second member of integrable systems of the well-known Kadomtsev-Petviashvili (KP) hierarchy [47, 48]. The Jimbo-Miwa equation in (1) is employed to describe particular interesting (3 + 1)-dimensional traveling waves in physics. According to the valuable literature obtaining exact solutions of (1) by several methods (see, e.g., [49–52]), the equation has a variety of solutions with distinct structures such as single-soliton solutions, multiple-soliton solutions, periodic wave solutions, and traveling wave solutions. The aim of this article is to apply the -expansion method and the novel -expansion method to solve the -dimensional space-time fractional Jimbo-Miwa equation in the sense of Jumarie’s modified Riemann-Liouville derivative. To the best of our knowledge, there are no researchers who have applied these methods to the fractional Jimbo-Miwa equation to obtain exact solutions. Some new exact solutions of the equation are reported for the first time. The rest of this paper is organized as follows. In Section 2, the definition of Jumarie’s modified Riemann-Liouville derivative and some of its properties are given. Additionally, the descriptions of the and the novel -expansion methods are provided. In Section 3, we illustrate the application of the two methods to the -dimensional space-time fractional Jimbo-Miwa equation. In Section 4, we provide graphs and physical explanations of some selected exact solutions of the equation obtained by the two methods. Some conclusions and discussions which are relevant to the obtained results using the mentioned methods are given in Section 5.

2. Mathematical Preliminaries

In this section, we will provide fundamental concepts required in this paper for obtaining exact explicit solutions of the -dimensional space-time fractional Jimbo-Miwa equation using the -expansion method and the novel -expansion method. We first give a definition and vital properties of the modified Riemann-Liouville derivative defined by Jumarie. Then we describe the -expansion method and the novel -expansion method to obtain exact analytical solutions of nonlinear FDEs associated with the mentioned fractional derivative.

2.1. Jumarie’s Modified Riemann-Liouville Derivative and Its Properties

The definition of the modified Riemann-Liouville derivative and its properties, which will be employed in this papers, are given as follows.

Definition 1. Assume that denote a continuous (but not necessarily first-order differentiable) function. Jumarie’s modified Riemann-Liouville derivative of order is defined by the following expression [53, 54]:in which is the gamma function defined by

Some important properties of Jumarie’s modified Riemann-Liouville derivative of order are summarized as follows:

Remark 2. Properties (7)–(9) are direct results [53–55] from using the fractional Leibniz rule, the fractional Barrow’s formula, and the relation . Property (7) requires only that the functions and are continuous (not necessarily differentiable) functions. The use of property (8) requires that the function is differentiable with respect to the function and the function is continuous (not necessarily differentiable). On the other hand, property (9) needs the function to be continuous (not necessarily differentiable) with respect to the function and the function differentiable with respect to . In particular, property (8) will be utilized in our work.

2.2. Descriptions of the -Expansion Method and the Novel -Expansion Method

In this section, the descriptions of the -expansion method and the novel -expansion method are concisely given. Consider a nonlinear fractional evolution partial differential equation in four independent variables , and as follows: where , , , and are Jumarie’s modified Riemann-Liouville derivatives of a dependent variable with respect to , and . is a polynomial of the unknown function and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved. The first common step of the two methods is to convert the NFPDE in (10) into an ordinary differential equation (ODE) with a fractional complex transformation [56–58] by using a traveling wave variable . We suppose that where is a nonzero arbitrary constant. With the transformation in (11) and an integration with respect to (if possible), (10) is reduced to an ODE for as follows: where is a function of and its various derivatives. The prime notation () denotes the derivative with respect to . Next we provide the main steps of the -expansion method and then the novel -expansion method for solving the ODE in (12) as demonstrated below.

2.2.1. The -Expansion Method

The following relevant concepts required before providing the main steps of the -expansion method are introduced below. Consider the following second-order linear ODE:where the prime notation () denotes the derivative with respect to and where , are constants. Next we set Equations (13) and (14) can be transformed into the system of two nonlinear ordinary differential equations as follows: The solutions of (13) can be categorized into the following three cases.

Case 1. If , then the general solution of (13) is of the form and we have where and are arbitrary constants and

Case 2. If , then the general solution of (13) can be given as and we have the following associated relation: where and are arbitrary constants and

Case 3. If , then the general solution of (13) can be written as and the corresponding relation is where and are arbitrary constants.

The main steps of the -expansion method [37–41] are as follows.

Step 1. Suppose that the solution to (12) can be expressed by a polynomial in the two variables and as follows: where , , and are constants to be determined later with and where the functions and are implicitly associated with (13) using the relations in (14).

Step 2. Determine the positive integer in (22) by inserting (22) into (12) and then using the homogeneous balance between the highest order derivatives and the nonlinear terms in (12). If the degree of is , then the degree of other terms will be expressed as follows: In particular, if the balance number of some nonlinear equations is not a positive integer then the special transformations (e.g., when is a fraction in the lowest terms, we set ) are applied for in (12) to have a new equation in terms of the new function with a positive integer balance number (see details in [39, 59]).

Step 3. Substituting the resulting equation of (22) into (12) with the aid of (15) and (17), the function in (12) can be converted into a polynomial in and , in which the degree of is not larger than one. Equating each coefficients of the resulting polynomial to zero, we obtain a system of algebraic equations, which can be solved using the symbolic computational packages such as Maple or Mathematica, for the following unknowns , ,   , , , <0, , and . The resulting traveling wave solutions generated by this step with the transformation in (11) are expressed by hyperbolic functions.

Step 4. In the same manner as Step , substituting the resulting equation of (22) into (12) with the aid of (15) and (19) for , we can obtain the exact solutions of (10) by using the transformation in (11). The obtained exact solutions are written as trigonometric functions.

Step 5. In the same manner as Step , substituting the resulting equation of (22) into (12) with the aid of (15) and (21) for , we can obtain the traveling wave solutions of (10) by using the transformation in (11). The resulting exact solutions are expressed by rational functions.

2.2.2. The Novel -Expansion Method

In the following, we give the main steps of the novel -expansion method [42–45].

Step 1. Suppose that the solution of (12) can be expressed in power of as follows: where The unknown constants or may be zero, but both of them cannot be zero simultaneously. The constants and are determined at a later step and the function satisfies the following nonlinear second-order ODE: where the prime notation () denotes the derivative with respect to and where , , and are real parameters.
The Cole-Hopf transformation reduces (26) into the following generalized Riccati equation: It is been discovered that (27) has thirty-nine solutions (see [60, 61] and Appendix for details).

Step 2. The value of the positive integer can be computed by balancing the highest order linear terms with nonlinear terms of the highest order occurring in (12). If the degree of is , then the degree of the other expressions will be expressed as in (23).

Step 3. Substituting (24) along with (25) and (26) into (12), we obtain polynomials in ( where is a positive integer). Collecting all coefficients of like-power of the resulting polynomials to zero, we yield an overdetermined set of algebraic equations for , , and

Step 4. Assuming that the unknown constants of the algebraic equations in Step can be obtained, we substitute the values of the constants together with the solutions of (27) into (24) to obtain exact traveling wave solutions of the nonlinear fractional evolution partial differential equation (10) when is set in (11).

Remark 3. It is worth noting that the novel -expansion method is a generalization of many types of -expansion methods. First, if we set in (25) and in (26) and negative indices of are zero in (24), then the method is reduced to the basic -expansion methods proposed by Wang et al. [62]. Secondly, if in (25) and in (26) then the method is identical to the improved -expansion methods introduced by Zhang et al. [63]. Thirdly, if , , and in (26) are replaced by , , and , respectively, then the novel -expansion method translates to the generalized and improved -expansion method presented by Akbar et al. [64]. Finally, if we let be functions of and instead of constants and in (26) then the method coincides with the generalized -expansion method introduced by Zhang et al. [65].

3. Application of the Two Methods to the (3 + 1)-Dimensional Space-Time Fractional Jimbo-Miwa Equation

Consider the -dimensional space-time fractional Jimbo-Miwa equation [66] where denotes Jumarie’s modified Riemann-Liouville derivative of a dependent variable with respect to with the fractional order and is a traveling wave solution of (28) which is an unknown function of four independent variables , and . Using the traveling wave transformation in second equation of (11) and the properties in (5) and (8), we obtain , , , , , , , , and . Hence, (28) is reduced into the ODE of the variable as where the prime notation () denotes the derivative with respect to . Integrating (29) with respect to and then choosing the constant of integration to be zero, we obtain the following ODE: Applying the formulas in (23) for balancing the highest order degree terms in and in (30), we have Consequently, the solution forms of (30) obtained using the -expansion method and the novel -expansion method are formed with by using (22) and (24), respectively. Next, we will apply the mentioned two methods to (30) to eventually produce exact solutions of the fractional Jimbo-Miwa equation in (28) as follows.

3.1. Using the -Expansion Method

From the balance which is done above, (22) is reduced to where the constant coefficients , , and are determined at a later step, subject to the inequality . There are three cases of the function associated with the functions and of the solution in (31) depending on the sign of described above.

Case 1 (hyperbolic function solutions ()). If , we substitute (31) into (30) along with the use of (15) and (17). Then the left-hand side of (30) turns out to be a polynomial in and . Setting all of the coefficients of this resulting polynomial to be zero, we obtain the following system of nonlinear algebraic equations in : It is required that the denominator of in (17) is not zero, so system (32) is valid if . By solving the above algebraic system using the Maple package program, we have the following results.
Result 1From (16), (31), and (33), we obtain the traveling wave solution of (28) as follows: where is defined in (11) with shown in (33) and , are arbitrary constants.
Result 2From (16), (31), and (35), we obtain the traveling wave solution of (28) as follows: where is defined in (11) with shown in (35) and , are arbitrary constants.