Computational Technique to Study Analytical Solutions to the Fractional Modified KDV-Zakharov-Kuznetsov Equation
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.
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  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 , the homotopy analysis method , fractional subequation method , the Lagrange characteristic method , 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 . 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
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):
4.1. 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:
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).
No data is available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
A. W. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering, Springer Science & Business Media, vol. 49, 2013.
T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser Boston, vol. 153, 2013.View at: Publisher Site
L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Springer Science & Business Media, 2012.
G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.View at: Publisher Site | Google Scholar
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, 1993.
H. KOGA and M. NAKAGAWA, “An Evaluation Method of Fractal Dimensions in Terms of Fractional Integro-Differential Equations,” IEICE transactions on fundamentals of electronics, communications and computer sciences, vol. 86, no. 4, p. 974, 2003.View at: Google Scholar
Z. Feng, “Exact solutions to the Lienard equation and its applications,” Chaos, Solitons & Fractals, vol. 21, no. 2, pp. 343–348, 2004.View at: Publisher Site | Google Scholar
A. K. Golmankhaneh and D. Baleanu, “On wave nonlinear fractional Klein-Gordon equation,” Sigal Process, vol. 91, pp. 446–451, 2011.View at: Publisher Site | Google Scholar
Z. S. Feng, “The first integer method to study the Burgers –Korteweg –devries equation,” Journal of Physics A: Mathematical and General, vol. 35, p. 343, 2002.View at: Google Scholar
M. A. Abdoon, “First integral method: a general formula for nonlinear fractional Klein-Gordon equation using advanced computing language,” Journal of Computational Mathematics, vol. 5, no. 2, pp. 127–134, 2015.View at: Publisher Site | Google Scholar
A. H. Ahmed Ali and K. Raslan, “New solutions for some important partial differential equations,” International Journal of Nonlinear Science, vol. 4, no. 2, pp. 109–117, 2007.View at: Google Scholar
F. L. Hasan and M. A. Abdoon, “The generalized (2+ 1) and (3+ 1)-dimensional with advanced analytical wave solutions via computational applications,” International Journal of Nonlinear Analysis and Applications, vol. 12, no. 2, pp. 1213–1241, 2021.View at: Google Scholar
F. L. Hasan, “First integral method for constructing new exact solutions of the important nonlinear evolution equations in physics,” Journal of Physics: Conference Series, vol. 1530, no. 1, article 012109, 2020.View at: Publisher Site | Google Scholar
V. P. Dubey, J. Singh, A. M. Alshehri, S. Dubey, and D. Kumar, “Numerical investigation of fractional model of phytoplankton–toxic phytoplankton–zooplankton system with convergence analysis,” International Journal of Biomathematics, vol. 15, no. 4, 2022.View at: Publisher Site | Google Scholar
V. P. Dubey, R. Kumar, J. Singh, and D. Kumar, “An efficient computational technique for time-fractional modified Degasperis- Procesi equation arising in propagation of nonlinear dispersive waves,” Engineering and Science, vol. 6, no. 1, pp. 30–39, 2021.View at: Publisher Site | Google Scholar
V. P. Dubey, R. Kumar, and D. Kumar, “Approximate analytical solution of fractional order biochemical reaction model and its stability analysis,” International Journal of Biomathematics, vol. 12, no. 5, article 1950059, 2019.View at: Publisher Site | Google Scholar
D. Kumar and J. Singh, Eds., Fractional Calculus in Medical and Health Science, CRC Press, 2020.View at: Publisher Site
V. P. Dubey, D. Kumar, and S. Dubey, “A modified computational scheme and convergence analysis for fractional order hepatitis E virus model,” Advanced Numerical Methods for Differential Equations, CRC Press, pp. 279–312, 2021.View at: Publisher Site | Google Scholar
S. Abbasbandy, “Homotopy analysis method for heat radiation equations,” International Communications in Heat and Mass Transfer, vol. 34, no. 3, pp. 380–387, 2007.View at: Publisher Site | Google Scholar
S. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004.View at: Publisher Site | Google Scholar
S. Zhang and H.-Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, 2011.View at: Publisher Site | Google Scholar
J. Guy, “Lagrange characteristic method for solving a class of nonlinear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 19, no. 9, pp. 873–880, 2006.View at: Publisher Site | Google Scholar
J.-H. He, “Variational iteration method for autonomous ordinary differential systems,” Applied Mathematics and Computation, vol. 114, no. 2-3, pp. 115–123, 2000.View at: Publisher Site | Google Scholar
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 1994.View at: Publisher Site
Y. B. Zhou, M. L. Wang, and Y. M. Wang, “Periodic wave solutions to a coupled KdV equations with variable coefficients,” Physics Letters A, vol. 308, pp. 31–36, 2003.View at: Publisher Site | Google Scholar
K. S. Al-Ghafri and H. Rezazadeh, “Solitons and other solutions of (3+ 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation,” Applied Mathematics and Nonlinear Sciences, vol. 4, no. 2, pp. 289–304, 2019.View at: Publisher Site | Google Scholar
O. Guner, E. Aksoy, A. Bekir, and A. C. Cevikel, “Different methods for (3+ 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation,” Computers & Mathematics with Applications, vol. 71, no. 6, pp. 1259–1269, 2016.View at: Publisher Site | Google Scholar