In this article, we study and investigate the analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov-Kuznetsov (mKDV-ZK) equation. We have got new exact solutions of the fractional mKDV-ZK equation by using first integral method; we found new types of hyperbolic solutions and trigonometric solutions by symbolic computation.

1. Introduction

Partial differential equations have their both effective and active roles in describing natural phenomena and complicated phenomena in theoretical and applied physics. It has a vital role in engineering sciences, biology, viscoelasticity, fluid mechanics, population technology, and describing flowing motion since population growth passing the human behaviorism up to astronomy, so many phenomena can be explained by using PDEs [13].

Guy [4] has proposed modified Riemann-Liouville derivative; these derivatives have their wider area applications, particularly fractal calculus in both sides on genetic and nongenetic; the last side has its own application that most of them rotate on diseases and pandemics spreading; this gives a proper description in this area [5, 6]. It is pointed that the complicated physical systems are described by using comfortable fractal derivative. Fractal derivative can be changed to differential equation by using variable transformation; several methods have been proposed to obtain exact and approximate solution of fractional differential equation [718], such as the homotopy perturbation method [19], the homotopy analysis method [20], fractional subequation method [21], the Lagrange characteristic method [22], and so on [23, 24]. In this article, we study exact and approximate solution of some nonlinear time-fractional partial differential equations using the first integral method.

2. Introductions and Basic Definitions

Guy’s modified Riemann-Liouville derivative, of order , can be defined by the following expression:

Moreover, some properties for the modified Riemann-Liouville derivative can be given as follows:

Then, the time-fractional differential equation with independent variablesand a dependent variableis given as using the variable transformation where , and are constants. equation (3) is reduced to a nonlinear ordinary differential equation

We suppose that equation (5) has a solution in the form which introduces a new independent variable; we get a new system of ODEs: by using the division theorem for two variables in complex domain which is based on the Hilbert-Nullstellensatz theorem [25]. An exact solution to equation (3) is when we obtain a first integral to equation (7) which can applied to equation (5) to obtain a first-order ordinary differential equation. Now, we wish to quickly recall the division theory.

Theorem 1 (the division theorem). Assume that and are polynomials of two variables and in and is irreducible in . If vanishes at all points of , then there exists a polynomial in such that

3. The First Integral Method: Formula (1)

We need to apply the FIM to the following form: where is a polynomial in and and is a polynomial with real coefficients.

Now, choose and, so (10) changes. We get

Using (2.9) and (4), (10) is equivalent to the two-dimensional autonomous system:

Now, we are applying the division theorem to find the first integral to (12), supposing that and are the nontrivial solutions to (12) and , which is an irreducible polynomial in the complex domain ; thus, where are polynomials and ; equation (14) is called the first integral method. There exists a polynomial in the complex domain such that

Case 1. Suppose that. Then, (15) becomes

Setting all coefficients of , we take

Also, where

Then, are polynomials; from (14), we deduce that is a constant, , and taking ; then, from (17), we have

Compare the degrees of, and; we conclude that; suppose that; then, from (17), we find, as follows: where, and are constants. Substituting , and into (18) and comparing all coefficient of to be zero, we obtain

It follows from equation (14) with equation (23), then

Combining (24) with (12), we obtain the following exact solution of equation (11): where is arbitrary constant.

Case 2. Suppose that; equation (15), equating the coefficients, gives

We have where

Then,are polynomials; from (27), also takingis a constant,, and; then, from (27), we have

Remark 2. If , then deuce and from (30), supposing that and Then, we get

Substituting and into (28) and setting all coefficients to zero, then yield, and , we get

Then, we deduce

It follows from (26) and (33) that

Combining (34) with (12), we obtain the following exact solution of equation (11):

4. Application

4.1. The ()-Dimensional Space-Time-Fractional mKDV-ZK Equation

In the present subsection, we apply the current method to solve the ()-dimensional mKDV-ZK space-time-fractional equation of the form where , and are nonzero constants, is a parameter representing the order of the time-space-fractional derivative. When , equation (36) is defined the modified KDV-ZK equation as when , , and ; equation (36) is defined the modified KDV fractional equation as

The solutions of mKDV-ZK equation have been introduced by several papers for the importance of applying this equation in various physics fields (see references [26, 27]. Then, we obtain the soliton solutions to equation (36). Therefore, by (4), we utilize the transformations as follows: where

Equation (36) transforms to ordinary differential equation by substituting (39); we get where when integrating one time and equalizing the constant of integration to zero; we have then

We apply (7) to obtain the soliton solutions for equation (36); then equation (43) becomes

Now, to get the first integral to equation (44); suppose that , and are the nontrivial solutions to equation (44) and , an irreducible polynomial in the complex domain , such that and . According to the concept of division theorem and applying equation (15), there exist two polynomials in such that

In our study, we will discuss three cases, which are explained as follows:

Case 1. We start our study by assuming; equation (45) becomes

Equating the coefficients of , we have system of ODE,

According to equations (17), (18), and (19), we can write system (47a), (47b), and (47c) as follows: and where

Then,are polynomials; from (21), we deduce thatis a constant,, and taking; then from equation (48a) as equation (21), we have

Compare the degrees of , and ; we conclude that ; suppose that , then from (23), we find , as follows: where , and are arbitrary integration constants. Then, when solving a system of nonlinear algebraic equations which is obtained by substituting , and in equation (48b) and equating all coefficients of powers from both sides of equation (48b), we get

We obtain two solutions ofby substituting first and second sets of solutions in, then

By combining equations (53a) and (53b) with (43), respectively, we get the exact solutions of equation (44) as follows: where and are an arbitrary integration constants; then, the solution of equation (36) is as follows:

Case 2. When taking in (45) and , this implies on both sides of (56) by equating the coefficients of , and according to equations (27) and (28), we have where

Since are polynomials of , we deduce that is a constant from (57a), and . For simplicity, take as remark and balance the degrees of , and if ; then, deuce and ; therefore, we obtainonly, and suppose that and ; after that, we calculate , and From (57b), we have

Then, we get where and are constants. By substituting , , and in the last equation in (60), then system of nonlinear algebraic equations obtained by putting all the coefficients of power to be zero, and by solving this system, we get sets of values as follows:

The first set represents the trivial solution therefore neglected; by the second and third sets, we get two solutions and , respectively, as follows:

And combining equations (64a) and (64b) with (43), respectively, we will get two similar solutions of (44) as where is an arbitrary integration constant; then, the solution of (36) is

Case 3. When taking in (45), and , this implies

The same as the steps in Case 1 and Case 2, we get two solutions, andas follows:

And, combining equations (68a), (70b), and (68c) with (43), respectively, we will get three similar solutions of (44) as where, andare an arbitrary integration constants; then, the solution of (36) is as follows:

5. Conclusion

In this study, the equation of the space-time nonlinear fractional modified KDV-Zakharov-Kuznetsov (mKDV-ZK) equation has been discussed by using the first integral method, accuracy of finding the mentioned solutions, adding, verifying, and checking them by using symbolic computation. For the illustration of solutions, thus drawing diagrams of solutions have been created. It has been noticed that all solutions in all situations are fully similar and repeating; however, we have found that we can gain the same solution as considered to be the only solution to this equation whenever increases. This method is effective, direct, and much more accurate; it can be applied upon other dynamic and engineering models (Figure 1).

Data Availability

No data is available.

Conflicts of Interest

The authors declare that they have no conflicts of interest.