On Semianalytical Study of Fractional-Order Kawahara Partial Differential Equation with the Homotopy Perturbation Method

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China Department of Mathematics and Statistics, University of Swat, Swat, Khyber Pakhtunkhwa, Pakistan Department of Mathematics, University of Malakand, Chakdara Dir (L) 18000, Khyber Pakhtunkhwa, Pakistan College of Science, Mathematical Sciences, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia Department of Mathematical Sciences, UAE University, P.O. Box 15551, Al-Ain, UAE


Introduction
PDEs have important applications in physics, engineering, and other applied sciences. ey can describe different phenomena and processes of real-world problems. One of the important KPDE arises in the theory of magnetoacoustic and shallow-water waves. Furthermore, it arises in the theory of shallow-water waves with surface tension and magnetoacoustic waves in plasmas. erefore, several analytical and numerical methods have been established in literature to investigate the prosed problems of PDEs. For instance, [1] authors have used the comparison method for the solution of the famous Kawahara equation. In the same line, a procedure was developed in [2] for the exact solution of the said problem. Also, authors [3] have computed the solution of the Kawahara equation by using symbolic computation. In this study, we apply a semianalytic HPM to solve the fifth-order KPDEs. As in the last several decades' investigation, traveling-waves solutions for nonlinear equations played an important role in the study of the nonlinear physical phenomenon [4]. e mentioned method provides an efficient approach to solve a nonlinear problem.
e KPDE was first suggested by Kawahara [5] in 1972. Since these nonlinear equations need to be solved by using some approximate methods, researchers have solved several nonlinear problems by using HPM. is method was first proposed by He [6] and has been applied in [7] for the solution of differential equations and integral equations in both linear and nonlinear cases. e said method is a combination of topological homotopy and traditional perturbation methods. e advantage of this method is to provide an analytic approximate solution in applied sciences with a capacious range, and in this method, a small parameter is not necessary for an equation. is method is also applied to the system of the nonlinear system of equations as in [8] for the analytic approximate solution for the model of rabies transmission dynamics.
Because of the popularity of fractional calculus and applications in many fields of science and engineering [9], fluid mechanics [10], some more frequent applications in a diverse area of science by using fraction calculus have been investigated in [11,12]. e mentioned derivative extends order from integer to any real or complex number which provides a detailed explanation to physical problems. Fractional derivatives can produce a complete spectrum of the geometry which includes its integer counterpart as a special case. Motivated from the aforesaid work, we extend the given KPDE [13].
For the demonstration of our problem, we testified the example given by Albert [14] as with the initial condition by using fractional derivative in Caputo sense. We also present the solutions graphically and then, at the end, provide conclusion and discussion.

Preliminaries and Notations
Here, we recall some preliminaries and notations from [15].

A General Algorithm about HPM
Consider a general type problem given by with boundary condition as where A is a general differential operator, β is a boundary operator, f(r) is a known analytic function, and Γ is the boundary of the domain Ω. e operator A is divided into linear part L and nonlinear part N. erefore, (9) can be written as By HPM, we can construct a homotopy as v(r, p): satisfying which is also equivalent to where p ∈ [0; 1] is an embedding parameter, and μ 0 is the initial approximation of the given equation that satisfies the boundary conditions; we have Keeping these points, we construct the required solution to equation (11) as (16) Furthermore, by taking limit as p ⟶ 1 in the approximation equation (16), one has which yields (18) Equation (18) represents the semianalytic solution of the problem equation (9).

Approximate Solution to Considered Problem
Here, in view of HPM as discussed in previous section, we proceed as We assume the solution of equation (2) as follows: Now, using equation (20) in (19) and comparing the coefficients of p i , for i � 0, 1, 2, 3, . . ., we have From system equation (21), we get Zeroth-order problem Consider zeroth-order problem as Which yields First-order problem Second-order problem ird-order problem In the same way, one has Now, taking the limit as p ⟶ 1 in equation (20), we get Next, equations (23)-(28) imply that where Hence, (29) is a required solution of the fractional-order KPDE.

Fractional Temporal Numerical Example. Consider the fractional-order KPDE given by
with initial condition Using HPM, equation (21) yields that Using equations (20) and (33), we get the following comparison with respect to p: Zeroth-order problem From equation (34), we get the zeroth-order problem or u(x, 0): First-order problem From equation (35), which gives Journal of Mathematics If we use p ⟶ 1, then solution of equation (31) implies that (39) In Figures 1-4, we present graphical presentation of solutions.

(40)
In this equation, for the fractional spatial solution, we only consider the first fractional derivative with order β for the sake of eliminating long calculations. erefore, the firstorder problem is turn out to be as follows.
First-order problem 6 Journal of Mathematics implies that Hence, the solution at p ⟶ 1 becomes

Journal of Mathematics
which implies that (44)

Results and Discussion
In Figure 1, we have plotted the temporal solution of the fractional-order Kawahara partial differential equation against position x and time t based on equation (35), for different fractional-order α, the plot shows that with α amplitude of the solitary wave potential increases while its width squeeze in size slightly. In Figure 2, we have the comparison of the solitary wave temporal solution u(x) against x for t � 0.01 and with different fractional-order α; this simulation shows a more clear picture of amplitude and dispersion variation with α. Figure 3 is the contour plot of solitary wave propagation against x and t for order α � 0.6. In Figure 4, the plot is among solitary wave propagation u(x, t) against x and for different time t, which shows a very interesting situation of the solitary waves structure; while at the smaller time, we found a compressive type of solitary wave, but when we take the time t greater than 0.6, then we observe the refractive type of the solitary wave and that waves increase its amplitude with time t and also its dispersion property.
In Figure 5, we have the 3D spatial numerical solution u(x, t) of solitary wave propagation against x and t for differential spatial-order β based on equation (40); the simulation shows us that with spatial-order fluctuation, amplitude of the solitary wave change slightly, but the width of the solitary wave change dramatically in greater steps; likewise in Figure 6, we have shown the 2D cross-sectional wave of Figure 5 that demonstrates the amplitude and width of the solitary wave clearly with spatial-order β for t � 0.01.

Conclusion
Upon the use of the homotopy perturbation method (HPM), we have investigated the Kawahara fractional-order partial differential equation of fifth-order under fractional order. By using Caputo derivative of fractional order separately on temporal and spatial bases, obtained the semianalytical solution for the Kawahara frictional-order differential equation. We have then stimulated various parametric effects (such as x, t, α, and β) on the structure of the solitary wave propagation that demonstrates that the width, as well as the amplitude of the solitary wave potential clearly, can change with the change of these parameters. We have shown through our calculation and simulation that He's technique is very useful and power full for the solution of such a higher-order nonlinear partial differential equation. We can extend our calculation to other complex problems especially to the applied side such as astrophysics, plasma physics, and quantum mechanics to solve a complex theoretical calculation by that technique.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.