Abstract

The phenomena, molecular path in a liquid or a gas, fluctuating price stoke, fission and fusion, quantum field theory, relativistic wave motion, etc., are modeled through the nonlinear time fractional clannish random Walker’s parabolic (CRWP) equation, nonlinear time fractional SharmaTassoOlver (STO) equation, and the nonlinear space-time fractional KleinGordon equation. The fractional derivative is described in the sense of conformable derivative. From there, the -expansion method is found to be ensuing, effective, and capable to provide functional solutions to nonlinear models concerning physical and engineering problems. In this study, an extension of the -expansion method has been introduced. This enhancement establishes broad-ranging and adequate fresh solutions. In addition, some existing solutions attainable in the literature also confirm the validity of the suggested extension. We believe that the extension might be added to the literature as a reliable and efficient technique to examine a wide variety of nonlinear fractional systems with parameters including solitary and periodic wave solutions to nonlinear FDEs.

1. Introduction

The subject FDEs can be considered as a generalization of the typical ordinary differential equations (ODEs). The advantages of the FDEs become apparent for us to understand real world problems. In fact, the FDEs have attracted significant attention to the past couple of decades due to their relatively much effectiveness than ODEs. There are several definitions of fractional derivatives, as for instance, the RiemannLiouville derivative, the Caputo derivative, and the conformal fractional derivative [1–3]. Recently, Khalil et al. [4] proposed a compatible definition of fractional derivative called the conformable fractional derivative. Therefore, several properties related to this new definition have been studied. Jarad et al. [5] used conformable type derivatives defined in [6] to generate new type of generalized fractional derivatives with memory effect. In [7, 8], the authors used conformable derivatives and integrals to formulate new generalized Liapunov-type inequality. The exact solutions of nonlinear FDEs play fundamental role in describing different qualitative and quantitative features of nonlinear complex physical phenomena. For this reason, researchers have proposed different methods to obtain exact solutions to nonlinear FDEs such as -expansion method [9–12], the -expansion method [13–19], auxiliary equation method [20, 21], -expansion method [22–27] the trial equation method [28, 29], fractional subequation method [30, 31], modified simple equation method [32], generalized Kudrayshov method [33, 34], and others [35–38].

In this study, we introduce an extension of the -expansion method for analyzing nonlinear FDEs in mathematical physics, engineering, and applied mathematics. To demonstrate the reliability and advantages of the suggested extension, the time-fractional CRWP equation, the time fractional STO equation, and the space-time fractional KleinGordon equation are examined and further broad-ranging and new families of exact wave solutions are established. This new extension can be applied to further nonlinear FDEs which can be done in forthcoming work.

2. Conformable Fractional Derivative and Its Important Properties

In the last years, Khalil et al. [4] introduced a simple, interesting, and compatible with typical definition of derivative named conformable fractional derivative, which can rectify the deficiencies of the other definitions. One can also find several useful studies related to this new definition in [5–8]. In [39], the geometrical and physical interpretations of this definition are investigated and the potential applications in science and engineering are pointer out. The conformable fractional derivative of a function of order is defined aswhere and . Some important properties of above definition are given below:(i)(ii)(iii), for each constant function (iv)

3. Methodology

In this section, we will suggest an extension of the -expansion method to ascertain the analytic solutions to nonlinear FDEs. We begin with considering the second-order linear ODE:

We choose

Thus, from equation (2) and (3), it is found

From the different general solutions of the linear ODE (2), we attain the following: Case 1: when , the hyperbolic function solution is and thus we obtain where and are arbitrary constants. Case 2: when , the trigonometric function solution is and corresponding relation is where and and are arbitrary constants. Case 3: when , the rational function solution is  and thus it is foundwhere and are arbitrary constants.

Let us consider a general nonlinear FDE:where and are the conformable fractional derivatives of the wave function with respect to spatial variables and the temporal variable , and is a polynomial of and its various partial derivatives.

The main steps of the new extension of the -expansion method to seek exact solutions of nonlinear FDEs are as follows: Step 1: we estimate the new form of the fractional wave variable: where are nonzero constants and is the wave velocity to be determined later. The complex wave transformation (12) translates equation (11) into an ODE as follows: Step 2: suppose that the general solution of nonlinear ODE (13) can be expressed by a polynomial in and as where satisfies the auxiliary linear ODE (2) and and are arbitrary constants to be determined later, and balancing the highest order derivative with the nonlinear terms in equation (13), we can find the positive integer . Step 3: inserting solution (14) into equation (13) and utilizing (4) and (6) (here case 1 is selected as an example), the left-hand side of equation (13) can be converted into a polynomial in and , where the degree of is not more than one. Equating all coefficients of this polynomial to zero yield a set of algebraic equations for , and . Step 4: the solution of the algebraic equations found in the step 3 can be found with the aid of Maple software package. Making use of the values of , and into (14), we might determine exact solutions expressed by hyperbolic functions of equation (13). Step 5: similar to step 3 and step 4, substituting (14) into equation (13) and utilizing equation (4) and (6) (or equation (4) and (10)), we can get the exact solutions of equation (13) expressed by the trigonometric functions (or expressed by the rational functions).

4. Determination of Solutions

In this paragraph, we will search out solutions to three conformable FDEs as appertain of the new extension of the -expansion method.

4.1. Time Fractional CRWP Equation

First, we consider the time fractional CRWP equation [22, 40]:

The ensuing wave transformation iswhere is the wave number and is the velocity; reducing the equation (15) into the subsequent ODE, we get

Integrating (17) once, we findwhere is a constant of integration. It is clear that the homogeneous balance between and present in equation (18) gives . Therefore, the shape of the exact solution of equation (18) is

There are three cases to be considered: Case 1: when , substituting solution (19) into equation (18) and utilizing (4) and (6), equation (18) can be transmuted to a polynomial in and . Equalizing its coefficients to zero yields a set of algebraic equations in , , , , , , , , and : Resolving these algebraic equations by Maple software package, we attain the following values: where and are free constants. Inserting the values of the parameters assembled in (21) into solution (19) along with (6) and (16), the following wide-ranging hyperbolic function solution is ascertained: In particular, if we set , , and (or , and ) into the solution (23), we obtain the successive singular kink and kink soliton solutions to the CRWP equation (15), respectively: where . Furthermore, by means of the parameter values gathered in (22), we derive Since and are integral constants, one might select their values spontaneously. Thus, if we select , , and (or , , and ) into solution (26), we gain the following solitary wave solutions to the CRWP equation, respectively: where . Case 2: when , embedding solution (19) into (18) and putting in use (4) and (8), equation (18) becomes polynomial in and . Computing the action similar to case 1 and after resolving the algebraic equations, we ascertain the following values: where and are arbitrary constants. Substituting the values scheduled in (29) into solution (19) along with (8) and (16), the succeeding trigonometric solution to the CRWP equation (15) is established: Since and are free constants, we may set , and (or , and ) into solution (31), we derive the under mentioned periodic wave solutions to the CRWP equation: where . By means of the values organized in (30), we extract the following solution: Now, if we set , and (or , and ) into solution (34), we obtain other periodic wave solutions to the CRWP equation: where . Case 3: when , introducing (19) into equation (18) and executing (4) and (10), equation (18) becomes a polynomial in and . Vanishing the coefficients yields a set of algebraic equations, and solving this system the following results are obtained:

Setting the values gathered in (37) into solution (19) along with (10) and (16), we carry out the next rational function solution to the CRWP equation:where .

The obtained solutions can be compared with the solutions accessible in the literature. We detect that, by setting and , the obtained solutions (24) and (25) fully agree with the corresponding solutions (3.19) and (3.20) established in [22], while the other solutions are different.

4.2. General Time Fractional STO Equation

The conformable general time fractional STO equation [41–43] iswhere is nonzero constant. The wave transformation is as follows:where is the velocity of the travelling wave; converting equation (39) into an ODE, we get

Integrating equation (41) once, we find

Balancing and obtained from equation (42), we get Thus, the solution structure of equation (42) is same of solution (19). Three cases as described in Section 3 will be discussed further: Case 1: when , placing the solution (19) into (42) and utilizing (4) and (6), equation (42) modifies to a polynomial in and . Setting each coefficient of the polynomial to zero yields a set of algebraic equations in , , , , , , , and . From these equations with the help of Maple algebra software, we find the ensuing results of constants: By means of parameter values sorted out in (43), along with (6) and (40), the following hyperbolic function solutions of the general time fractional STO equation are obtained: In solution (49), and are free parameters; therefore, we freely can choose our values. Thus, if we choose , , and or , , and , from solution (49), we obtain the under mentioned soliton solutions to the general time fractional STO equation: where . Furthermore, by means of the values scheduled in (44), we establish  Now, if we use , , and or , , and , in (52), we attain the next soliton solutions to the general time fractional STO equation:  where .  Moreover, by means of the values organized in (45), we carry out  If we consider , and or , and , from solution (55), we derive the solitary wave solutions given underneath:  where .  Similarly, making use of the values accumulated in (46), we determine  For , , and or , , and , the solution (58) generates the kink and singular kink solutions given underneath:  where .  Likewise, with the help of the parametric values amassed in (47), we achieve  Choosing , , and or , , and , solution (61) turns into  where .  For the values gathered in (48), we found  In particular, if we set , and (or , and ), the solution (64) reduces to  where .  Case 2: when , we established by completing the parallel course of algorithms to case 1 and the following values for the constants: