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

Authors: Muhammad Sinan; Kamal Shah; Zareen A. Khan; Qasem Al-Mdallal; Fathalla Rihan
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such asx, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.


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## Body

## 1. Introduction

PDEs have important applications in physics, engineering, and other applied sciences. They 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. Therefore, 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]. The mentioned method provides an efficient approach to solve a nonlinear problem. The 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. This 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. The said method is a combination of topological homotopy and traditional perturbation methods. The 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. This 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]. The 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].(1)∂u∂t+ρu∂u∂x+σ∂3u∂x3−γ∂5u∂x5=0,ux,0=gx,where g is a continuous function, while ρ,σ,andγ are the nonzero arbitrary constants to fractional order as(2)∂αu∂tα+ρu∂βu∂xβ+σ∂κu∂xκ−γ∂δu∂xδ=0,0<α≤1,(3)ux,0=gx,while 0<β≤1,2<κ≤3,4<δ≤5, and g:R⟶R is continuous function.For the demonstration of our problem, we testified the example given by Albert [14] as(4)∂αu∂tα+ρu∂βu∂xβ+σ∂κu∂xκ−γ∂δu∂xδ=0,with the initial condition(5)ux,0=105169sech4x213,by using fractional derivative in Caputo sense. We also present the solutions graphically and then, at the end, provide conclusion and discussion.


## 2. Preliminaries and Notations

Here, we recall some preliminaries and notations from [15].Definition 1.
Caputo fractional-order derivative of a functionϕ on the interval 0,∞ is defined as(6)Dc0+αϕt=1Γn−α∫0tt−sn−α−1dndsnϕsds,where n=α+1 and α represent the integral part of α.Definition 2.
Ifα>0, the Caputo fractional integral is defined as(7)Iαϕt=1Γα∫0tt−sα−1ϕsds,where α∈0,∞.Lemma 1.
The following result holds:(8)InCD0+nϕt=ϕt+a0+a1t+a2t2+⋯+an−1tn−1,for arbitrary ai∈R, i=0,1,2,3,…,n−1, where n=η+1 and η represent the integral part of η.


## 3. A General Algorithm about HPM

Consider a general type problem given by(9)Aμ−fr=0,r∈Ω,with boundary condition as(10)βμ,∂μ∂n=0,r∈Γ,where A is a general differential operator, β is a boundary operator, fr is a known analytic function, and Γ is the boundary of the domain Ω. The operator A is divided into linear part L and nonlinear part N. Therefore, (9) can be written as(11)Lμ+Nu−fr=0.By HPM, we can construct a homotopy as(12)vr,p:Ω×0,1⟶R,satisfying(13)Hv,p=1−pLv−Lμ+pAv−fr=0,which is also equivalent to(14)Hv,p=Lv−Lμ0+pLv0+pNv−fr=0,where p∈0;1 is an embedding parameter, and μ0 is the initial approximation of the given equation that satisfies the boundary conditions; we have(15)Hv,0=Lv−Lμ0=0,Hv,1=Av−fr=0.Keeping these points, we construct the required solution to equation (11) as(16)v=v0+p1v1+p2v2+p3v3+⋯.Furthermore, by taking limit asp⟶1 in the approximation equation (16), one has(17)limp⟶1v=limp⟶1v0+p1v1+p2v2+p3v3+⋯,which yields(18)v=v0+v1+v2+v3+⋯.Equation (18) represents the semianalytic solution of the problem equation (9).


## 4. Approximate Solution to Considered Problem

Here, in view of HPM as discussed in previous section, we proceed as(19)Hu,p=∂αu∂tα−∂α∂tαgx+p∂α∂tαgx+pαu∂βu∂xβ+σ∂κu∂xκ−γ∂δu∂xδ=0.We assume the solution of equation (2) as follows:(20)ux,t=∑i=0∞piuix,t.Now, using equation (20) in (19) and comparing the coefficients of pi, for  i=0,1,2,3,…, we have(21)p0≔∂αu0∂tα−∂α∂tαgx=0,p1≔∂αu1∂tα+ρu0∂β∂xβu0+σ∂κ∂xκu0−γ∂δ∂xδu0=0,p2≔∂αu2∂tα+ρu1∂β∂xβu1+σ∂κ∂xκu1−γ∂δ∂xδu1=0,p3≔∂αu3∂tα+ρu2∂β∂xβu2+σ∂κ∂xκu2−γ∂δ∂xδu2=0,p4≔∂αu4∂tα+ρu3∂β∂xβu3+σ∂κ∂xκu3−γ∂δ∂xδu3=0,⋮From system equation (21), we getZeroth-order problemConsider zeroth-order problem as(22)∂αu0∂tα=∂αgx∂tα.Which yields(23)u0=gx.First-order problem(24)∂αu1∂tα==∂αgx∂tα+γ∂δ∂xδu0−ρu0∂β∂xβu0−σ∂κ∂xκu0,u1=γ∂δ∂xδu0−ρu0∂β∂xβu0−σ∂κ∂xκu0tαΓα+1.Second-order problem(25)∂αu2∂tα=γtαΓα+1∂δ∂xδξ−ρξt2αΓα+12∂β∂xβξ−σtαΓα+1∂κ∂xκξ,u2=γt2αΓα+2∂δ∂xδξ−ρξt3αΓα+22Γα+3∂β∂xβξ−σt2αΓα+2∂κ∂xκξ.Where(26)ξ=γ∂δ∂xδu0−ρu0∂β∂xβu0−σ∂κ∂xκu0.Third-order problemIn the same way, one has(27)u3=γ∂δ∂xδγt3αΓα+3∂δ∂xδξ−ρξt4αΓα+12Γα+4∂β∂xβξ−σt3αΓα+3∂κ∂xκξ−αγt2αΓα+2∂δ∂xδξ−ρξt3αΓα+12Γα+3∂β∂xβξ−σt2αΓα+2∂κ∂xκξ×∂β∂xβγt3αΓα+3∂δ∂xδξ−ρξt4αΓα+12Γα+4∂β∂xβξ−σt3αΓα+3∂κ∂xκξ−β∂κ∂xκγt3αΓα+3∂δ∂xδξ−ρξt4αΓα+12Γα+4∂β∂xβξ−σt3αΓα+3∂κ∂xκξ.Now, taking the limit asp⟶1 in equation (20), we get(28)ux,t=∑i=0∞uix,t.Next, equations (23)–(28) imply that(29)ux,t=gx+γ∂δ∂xδu0−ρu0∂β∂xβu0−σ∂κ∂xκu0tαΓα+1+γt2αΓα+2∂δ∂xδξ−ρξt3αΓα+12Γα+3∂β∂xβξ−σt2αΓα+2∂κ∂xκξ+γ∂δ∂xδγt3αΓα+3∂δ∂xδξ−ρξt4αΓα+12Γα+4∂β∂xβξ−σt3αΓα+3∂κ∂xκξ−αγt2αΓα+2∂δ∂xδξ−ρξt3αΓα+12Γα+3∂β∂xβξ−σt2αΓα+2∂κ∂xκξ×∂β∂xβγt3αΓα+3∂δ∂xδξ−ρξt4αΓα+12Γα+4∂β∂xβξ−σt3αΓα+3∂κ∂xκξ−β∂κ∂xκγt3αΓα+3∂δ∂xδξ−ρξt4αΓα+12Γα+4∂β∂xβξ−σt3αΓα+3∂κ∂xκξ+⋯,where(30)ξ=γ∂δ∂xδu0−ρu0∂β∂xβu0−σ∂κ∂xκξ.Hence, (29) is a required solution of the fractional-order KPDE.


### 4.1. Fractional Temporal Numerical Example

Consider the fractional-order KPDE given by(31)∂αu∂tα+u∂βu∂xβ+∂κu∂xκ−∂δu∂xδ=0,0<α≤1,0<β≤1,2<κ≤3,4<δ≤5,with initial condition(32)ux,0=105169sech4x213.Using HPM, equation (21) yields that(33)Hu,p≔∂α∂tαu−105169∂α∂tαsech4x213+105169p∂α∂tαsech4x213+pu∂β∂xβu+∂κ∂xκu−∂δ∂xδu.Using equations (20) and (33), we get the following comparison with respect to p:(34)p0≔∂α∂tαu0−105169∂α∂tαsech4x213=0,p1≔∂α∂tαu1+1051692sech4x213∂β∂xβsech4x213.(35)+105169∂κ∂xκsech4x213−105169∂δ∂xδsech4x213=0.⋮Zeroth-order problemFrom equation (34), we get the zeroth-order problem or ux,0:(36)u0=105169sech4x213,ifux,0=105169sech4x213.First-order problemFrom equation (35),(37)u1x,t=11025tα28561Γα+1sech4x213∂∂xsech4x213+105tα169Γα+1∂3∂x3sech4x213−105tα169Γα+1∂5∂x5sech4x213=0,which gives(38)u1x,t=−2205013tα371293Γα+1sech4x213sechx13264tanhx1326+73513tα28561Γα+1sechx13264×tanhx13261−tanhx13262−84013tα28561Γα+1sechx13264tanhx13263+1375513tα371293Γα+1sechx13264tanhx132631−tanhx13262−336013tα371293Γα+1sechx13264×tanhx13265−493513tα371293Γα+1sechx13264tanhx13261−tanhx132622.If we usep⟶1, then solution of equation (31) implies that(39)ux,t=u0+u1x,t+u2x,t+⋯ux,t=105169sech4x213−2205013tα371293Γα+1sech4x213sechx13264tanhx1326+73513tα28561Γα+1sechx13264tanhx13261−tanhx13262−84013tα28561Γα+1×sechx13264tanhx13263+1375513tα371293Γα+1sechx13264tanhx132631−tanhx13262−336013tα371293Γα+1sechx13264tanhx13265−493513tα371293Γα+1sechx13264tanhx1326×1−tanhx132622+⋯.In Figures1–4, we present graphical presentation of solutions.Figure 1
3D fractional temporal numerical solutionux,t for order α∈0,1, κ=4, and δ=5.Figure 2
2D fractional temporal numerical solutionux for order t=0.01, κ=4, and δ=5.Figure 3
Contour of fractional temporal numerical solutionux for order α=0.6, κ=4, and δ=5.Figure 4
2D fractional temporal numerical solutionux for order α=0.6, κ=4, and δ=5.


## 4.1. Fractional Temporal Numerical Example

Consider the fractional-order KPDE given by(31)∂αu∂tα+u∂βu∂xβ+∂κu∂xκ−∂δu∂xδ=0,0<α≤1,0<β≤1,2<κ≤3,4<δ≤5,with initial condition(32)ux,0=105169sech4x213.Using HPM, equation (21) yields that(33)Hu,p≔∂α∂tαu−105169∂α∂tαsech4x213+105169p∂α∂tαsech4x213+pu∂β∂xβu+∂κ∂xκu−∂δ∂xδu.Using equations (20) and (33), we get the following comparison with respect to p:(34)p0≔∂α∂tαu0−105169∂α∂tαsech4x213=0,p1≔∂α∂tαu1+1051692sech4x213∂β∂xβsech4x213.(35)+105169∂κ∂xκsech4x213−105169∂δ∂xδsech4x213=0.⋮Zeroth-order problemFrom equation (34), we get the zeroth-order problem or ux,0:(36)u0=105169sech4x213,ifux,0=105169sech4x213.First-order problemFrom equation (35),(37)u1x,t=11025tα28561Γα+1sech4x213∂∂xsech4x213+105tα169Γα+1∂3∂x3sech4x213−105tα169Γα+1∂5∂x5sech4x213=0,which gives(38)u1x,t=−2205013tα371293Γα+1sech4x213sechx13264tanhx1326+73513tα28561Γα+1sechx13264×tanhx13261−tanhx13262−84013tα28561Γα+1sechx13264tanhx13263+1375513tα371293Γα+1sechx13264tanhx132631−tanhx13262−336013tα371293Γα+1sechx13264×tanhx13265−493513tα371293Γα+1sechx13264tanhx13261−tanhx132622.If we usep⟶1, then solution of equation (31) implies that(39)ux,t=u0+u1x,t+u2x,t+⋯ux,t=105169sech4x213−2205013tα371293Γα+1sech4x213sechx13264tanhx1326+73513tα28561Γα+1sechx13264tanhx13261−tanhx13262−84013tα28561Γα+1×sechx13264tanhx13263+1375513tα371293Γα+1sechx13264tanhx132631−tanhx13262−336013tα371293Γα+1sechx13264tanhx13265−493513tα371293Γα+1sechx13264tanhx1326×1−tanhx132622+⋯.In Figures1–4, we present graphical presentation of solutions.Figure 1
3D fractional temporal numerical solutionux,t for order α∈0,1, κ=4, and δ=5.Figure 2
2D fractional temporal numerical solutionux for order t=0.01, κ=4, and δ=5.Figure 3
Contour of fractional temporal numerical solutionux for order α=0.6, κ=4, and δ=5.Figure 4
2D fractional temporal numerical solutionux for order α=0.6, κ=4, and δ=5.


## 5. Fractional Spatial Numerical Solution

Consider the equation with orders0<β≤1,2<κ≤3,4<δ≤5 for fractional spatial numerical solution, that is,(40)∂αu∂tα+u∂βu∂xβ+∂κu∂xκ−∂δu∂xδ=0,0<α≤1.In this equation, for the fractional spatial solution, we only consider the first fractional derivative with orderβ for the sake of eliminating long calculations. Therefore, the first-order problem is turn out to be as follows.First-order problem(41)u1x,t=11025tα28561Γα+1sech4x213∂β∂xβsech4x213+105tα169Γα+1∂3∂x3sech4x213−105tα169Γα+1∂5∂x5sech4x213=0,implies that(42)u1x,t=105tα169Γα+1sech4x2135.93227042×10−10Γ2−βx1−β+0.04779241642Γ3−βx2−β+1.104877850×10−6Γ4−βx3−β+73513tα28561Γα+1sechx13264tanhx1326×1−tanhx13262−84013tα28561Γα+1sechx13264tanhx13263+1375513tα371293Γα+1sechx13264tanhx132631−tanhx13262−336013tα371293Γα+1sechx13264tanhx13265−493513tα371293Γα+1sechx13264×tanhx13261−tanhx132622.Hence, the solution atp⟶1 becomes(43)ux,t=∑i=0∞uix,t,which implies that(44)ux,t=u0+u1x,t+u2x,t+⋯ux,t=105169sech4x213+105tα169Γα+1sech4x2135.93227042×10−10Γ2−βx1−β+105tα169Γα+1sech4x2130.04779241642Γ3−βx2−β+1.104877850×10−6Γ4−βx3−β+73513tα28561Γα+1sechx13264tanhx13261−tanhx13262−84013tα28561Γα+1×sechx13264tanhx13263+1375513tα371293Γα+1sechx13264tanhx13263×1−tanhx13262−336013tα371293Γα+1sechx13264tanhx13265−493513tα371293Γα+1sechx13264tanhx13261−tanhx132622+⋯.


## 6. Results and Discussion

In Figure1, 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 ux 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 ux,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 Figure5, we have the 3D spatial numerical solution ux,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. Figure 7 is the contour plot of solitary wave spatial solution against x and t.Figure 5
3D fractional spatial numerical solutionux,t at order β∈0,1,κ=3,δ=5.Figure 6
2D fractional spatial numerical solutionux at orders α=0.9,β=1,κ=3,δ=5, and time t=0.01.Figure 7
Contour of fractional spatial numerical solutionux,t at orders α=β=0.6,κ=3,δ=5.


## 7. 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 asx, 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.



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