Computational Technique to Study Analytical Solutions to the Fractional Modified KDV-Zakharov-Kuznetsov Equation

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 [1–3]. 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 [7–18], 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.


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 [1][2][3].
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 [7][8][9][10][11][12][13][14][15][16][17][18], 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.

Introductions and Basic Definitions
Guy's modified Riemann-Liouville derivative, of order α, 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 variablesX = ðx 1 , x 2 , ⋯, x m , tÞand a dependent variableuis given as using the variable transformation where m, l i , and λ ≠ 0 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 variableYðζÞ = X ′ ðζÞ; we get a new system of ODEs: by using the division theorem for two variables in complex domain C½X, Y 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.
3. The First Integral Method: Formula (1) We need to apply the FIM to the following form: where Tðu, u′Þ is a polynomial in u and u′ and RðuÞ is a polynomial with real coefficients. Now, choose Tðu, u′Þ = 0 and RðuÞ = Au + Bu 3 , so (10) changes. We get Using (2.9) and (4), (10) is equivalent to the twodimensional autonomous system: Now, we are applying the division theorem to find the first integral to (12), supposing that X = XðζÞ and Y = YðζÞ are the nontrivial solutions to (12) and qðx, yÞ = ∑ N i=0 a i ðXÞY i = 0, which is an irreducible polynomial in the complex domain C ½X, Y; thus, where a i ðXÞði = 0, 1, 2:⋯ ⋯ ⋯ , NÞ are polynomials and a N ðXÞ ≠ 0; equation (14) is called the first integral method. There exists a polynomial δðXÞ + θðXÞY in the complex domain C½ x, y such that Case 1. Suppose that N = 1. Then, (15) becomes Setting all coefficients of Y i ði = 0, 1Þ, we take Also, where Then, a i ðxÞ ði = 0, 1Þ are polynomials; from (14), we deduce that a 1 ðxÞ is a constant, θðxÞ = 0, and taking a 1 ðxÞ = 1; then, from (17), we have Abstract and Applied Analysis Compare the degrees ofgðXÞ, a 1 ðXÞ, anda 0 ðXÞ; we conclude thatdeg δðxÞ = 1; suppose thatδðxÞ = A 1 x + B 0 ; then, from (17), we finda 0 ðxÞ, as follows: where A 1 , B 0 , and B 1 are constants. Substituting δðxÞ, a 1 ðxÞ, and a 0 ðxÞ into (18) and comparing all coefficient of x 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 ζ 0 is arbitrary constant.
Case 2. Suppose that N = 1; equation (15), equating the coefficients, gives We have where Then,a i ðXÞði = 0, 1, 2Þare polynomials; from (27), also takinga 2 ðXÞis a constant,θðXÞ = 0, anda 2 ðXÞ = 1; then, from (27), we have Remark 2. If deg δðxÞ = k > 0, then deuce deg a 1 ðxÞ = k + 1 and deg a 0 ðxÞ = 2ðk + 1Þ from (30), supposing that δðxÞ = A 1 x + B 0 and a 1 ðxÞ ≠ 0: Then, we get Substituting a 1 ðxÞ and a 0 ðxÞ into (28) and setting all coefficients x i ði = 5, 4, 3, 2, 1Þ to zero, then yield, and c 0 = 0, we get Then, we deduce 3 Abstract and Applied Analysis It follows from (26) and (33) that Combining (34) with (12), we obtain the following exact solution of equation (11): 4. Application In the present subsection, we apply the current method to solve the (3 + 1)-dimensional mKDV-ZK spacetime-fractional equation of the form where β, e, r, and s are nonzero constants, α is a parameter representing the order of the time-space-fractional derivative. When α = 1, equation (36) 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 λ ≠ 0 Equation (36) transforms to ordinary differential equation by substituting (39); we get where F ′ = dF/dξ when integrating one time and equalizing the constant of integration to zero; we have 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 X = XðξÞ, and Y = YðξÞ are the nontrivial solutions to equation (44) and qðX ðξ Þ, Y ðξ ÞÞ = ∑ N i=0 a i ðxÞy i = 0, an irreducible polynomial in the complex domain ℂ½X, Y, such that a i ðXÞ ði = 0, 1, 2, ⋯NÞ and a N ðXÞ ≠ 0. According to the concept of division theorem and applying equation (15), there exist two polynomials δðXÞ + θðXÞY in ℂ½X, Y such that In our study, we will discuss three cases, which are explained as follows: Case 1. We start our study by assumingN = 1; equation (45) becomes Equating the coefficients of Y i ði = 1, 0Þ, we have system of ODE, According to equations (17), (18), and (19), we can write system (47a), (47b), and (47c) as follows: Abstract and Applied Analysis and where Then,a i ðxÞ ði = 0, 1Þare polynomials; from (21), we deduce thata 1 ðxÞis a constant, θðxÞ = 0, and takinga 1 ðxÞ = 1 ; then from equation (48a) as equation (21), we have Compare the degrees of δðxÞ, a 1 ðXÞ, and a 0 ðXÞ; we conclude that deg δðxÞ = 1; suppose that δðxÞ = A 1 x + B 0 , then from (23), we find a 0 ðxÞ, as follows: where A, B, and C 0 are arbitrary integration constants. Then, when solving a system of nonlinear algebraic equations which is obtained by substituting a 1 ðXÞ, a0ðXÞ, and g ðXÞ in equation (48b) and equating all coefficients of powers X from both sides of equation (48b), we get We obtain two solutions ofYby substituting first and second sets of solutions inqðX ðξ Þ, Y ðξ ÞÞ = ∑ N i=0 a i ðxÞy i = 0, then By combining equations (53a) and (53b) with (43), respectively, we get the exact solutions of equation (44) as follows: where C 1 and C 2 are an arbitrary integration constants; then, the solution of equation (36) is as follows: Case 2. When taking N = 2 in (45) and qðX, YÞ = 0, this implies dq/dξ = 0, on both sides of (56) by equating the coefficients of Y i ði = 2, 1, 0Þ, and according to equations (27) and (28), we have where Since a i ðxÞði = 0, 1, 2Þ are polynomials of X, we deduce that a 2 ðXÞ is a constant from (57a), and θðxÞ = 0. For simplicity, take a 2 ðXÞ = 1 as remark and balance the degrees of δðXÞ, a 1 ðXÞ, and a 0 ðXÞ if deg δðxÞ = k > 0; then, deuce deg a 1 ðxÞ = k + 1 and deg a 0 ðxÞ = 2ðk + 1Þ; therefore, we obtaindeg δðXÞ = 1only, and suppose that δðXÞ = AX + B and A ≠ 0; after that, we calculate a 1 ðXÞ, and a 0 ðXÞ: From (57b), we have Then, we get where C 0 and d are constants. By substituting a 2 ðXÞ, a 1 ðXÞ, a 0 ðXÞ, and δðXÞ in the last equation in (60), then system of nonlinear algebraic equations obtained by putting all the coefficients of power X 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 Y 1 and Y 2 , respectively, as follows: 6 Abstract and Applied Analysis

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 N = 1, 2, 3 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 N 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.