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Analytical approach for travelling wave solution of non-linear fifth-order time-fractional Korteweg–De Vries equation

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Analytical approach for travelling wave solution of non-linear fifth-order time-fractional Korteweg–De Vries equation

DELMAR SHERRIFFE1,2and DIPTIRANJAN BEHERA1,∗

1Department of Mathematics, The University of the West Indies, Mona Campus, Kingston 7, Jamaica

2Department of Mathematics, The Mico University College, Kingston 5, Jamaica

Corresponding author. E-mail: diptiranjanb@gmail.com; diptiranjan.behera@uwimona.edu.jm MS received 15 April 2021; revised 4 October 2021; accepted 18 November 2021

Abstract. In this paper, we have studied analytical travelling wave solution of a non-linear fifth-order time- fractional Korteweg–De Vries (KdV) equation under the conformal fractional derivative. This equation is very important as it has many applications in various fields such as fluid dynamics, plasma physics, shallow water waves etc. Also, most importantly, the considered fractional-order derivative plays a vital role as it can be varied to obtain different waves. Here, a new form of exact travelling wave solution is derived using the powerful sine–cosine method. To understand the physical phenomena, some visual representations of the solution by varying different parameters are given. Accordingly, it has been observed that the obtained wave solutions are solitons in nature. Also, from the results, one can conclude that the corresponding wave of the solution will translate from left to right by increasing the fractional orderα. Furthermore, extending the range ofxit can be noticed that there is a reduction in the heights of the waves. Also similar observations have been made for a particular time interval and by increasing the values ofx. Also, it can be observed that the present method is straightforward as well as computationally efficient compared to the existing methods. The obtained solution has been verified using Maple software and the results are validated.

Keywords. Sine–cosine method; travelling wave solution; conformal fractional derivative.

PACS Nos 02.70.Wz; 03.65.Ge; 04.20.Jb; 05.45.Yv; 47.35.Fg

1. Introduction

Nonlinear physical phenomena have been successfully described by models based on the fractional derivatives.

Due to its applicability, there is continued interest in the field of fractional calculus. Fractional calculus is extremely beneficial in areas such as physics, mechan- ics, chemistry, biology and so on [1–5]. Recently, Singh [6] employed the Sumudu transform algorithm to anal- yse the concentration of alcohol in the stomach and the concentration of alcohol in the blood. Phuonget al [7] examined an initial value problem for the Caputo time-fractional pseudoparabolic equations with frac- tional Laplace operator of order 0 < v < 1 and the nonlinear memory source term. As a result of its applica- bility to solve real-life problems, there is the continuous effort to develop the area. Singh et al [8] proposed a numerical approach for fractional multidimensional dif- fusion equations with exponential memory. In addition,

an efficient numerical technique for solving the time- fractional generalised Fisher’s equation was proposed by Majeedet al[9]. Using the Von Neumann stability formula, the proposed scheme is shown to be uncondi- tionally stable. Furthermore, Singhet al[10] proposed a numerical approach for studying a nonlinear model of fractional optimal control problems (FOCPs). The pro- posed method is a grouping of operational matrices of integrations for Jacobi polynomials and the Ritz method.

Moreover, there have also been numerous papers devoted to fractional KdV-type equations. Khader et al[11] used a spectral collocation method to solve the fractional KdV and KdV–Burgers equations. In addi- tion, Yavuzet al[12] examined the Schrödinger–KdV equation of fractional order and Liu et al [13] exam- ined the integrability of the higher-dimensional time- fractional KdV-type equation. Sainet al[14] obtained solitary wave solutions for the KdV-type equations in plasma by applying a new approach with the Kudryashov

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64 Page 2 of 8 Pramana – J. Phys. (2022) 96:64 function. Also, Luet al[15] derived numerical solutions

of coupled nonlinear fractional KdV equations using He’s fractional calculus. Veeresha et al [16] derived solutions for fractional potential KdV and Benjamin equations using a novel technique. Singla and Gupta [17] obtained series solutions of time-fractional three- coupled KdV system. Finally, Jafariet al[18] obtained solutions for a system of coupled KdV equations with fractional derivatives.

The exact solutions for various forms of fifth-order KdV equation have been studied extensively for both integer and fractional-order cases due to its importance.

For example, Wanget al[19] used Lie symmetry anal- ysis to obtain explicit solutions of the time-fractional fifth-order KdV Equation. Liu [20] looked at the com- plete group classifications and symmetry reductions of the fractional fifth-order KdV types of equations.

Also, Lu et al [21] solved the space–time fractional generalised fifth-order KdV equation employing the generalised exp(−φ(ξ))-expansion method using the Jumarie’s-modified Riemann–Liouville derivatives. In addition, Liu [22] obtained exact solutions to time- fractional fifth-order KdV equation by the trial equation method based on symmetry while employing the con- formal derivatives.

Because of the applicability of different types of KdV equations in fields of plasma physics, optics, fluid dynamics etc. the researchers were motivated to obtain the solution of a non-linear fifth-order time-fractional KdV equation under conformal derivative [22] given as Dtαu+au2ux+bu3x+cu5x =0, (1) where a,b,c are model parameters and α is the frac- tional order with 0< α≤1.

In this paper, we employ the sine–cosine method [23,24]. This method has been applied to nonlinear prob- lems of both integer order and fractional order [23,24].

We reduce the time-fractional fifth-order KdV equa- tion to a lesser order equation, and then we apply the method to obtain exact solutions. The derivative is taken in the conformal sense due to the reliability. Also, very recently in [25], exact solution of time-fractional cou- pled Whitham–Broer–Kaup (WBK) equations under the conformal fractional derivative has been studied. In the analysis, they have used modified extended tanh- function method. In addition, one can see in [26,27] that exact solutions have been obtained for other non-linear differential equations such as sine-Gordon equation, Schrödinger equation etc.

The paper is organised as follows. The conformal fractional derivative is explored in §2. In §3, we give the outline of sine–cosine method. In §4, application of the method for the new exact travelling wave solution of fifth-order time-fractional KdV equation has been

investigated. Section5gives the results and discussion for various parameters. Graphical representations of the obtained results are also given in this section. Finally,

§6gives conclusions.

2. Preliminaries of conformal fractional derivative In this section, an introduction to the idea of the confor- mal fractional derivative has been presented [22].

DEFINITION

Let g be a real-valued function defined in the domain [0,∞),that means g : [0,∞) → R.Then the con- formal fractional derivative of order α(0,1] in the half-spacet>0 is defined as

Dα·(g(t))= lim

h0

g(t+ht1−α)g(t)

h .

Some properties of the conformal fractional derivative have been presented below.

Theorem [22]. Let us consider α(0,1] and θ = θ(t), ν =ν(t)is anαdifferentiable function at a point t >0. Then we have

(i) Dtα(kθ+mν)=kDαt θ+mDtαν,for allk,m∈R.

(ii) Dtα(θν)=θDαt(ν)+νDαt (θ).

(iii) Dtα(tσ)=σtα−σ,for allσ ∈R.

(iv) For all constant functionθ(t)=η, Dαt(η)=0.

(v) Dtα(θ/ν)=(νDαt θθDtαν)/ν2 whereν=0.

(vi) Ifθ is differentiable, thenDtα(θ)(t) = t1−α(dθ/

dt).

(vii) Chain rule of conformable differentiable function:

Dαtν)(t)=t1−αdν

dtθ (ν(t)).

3. The sine–cosine function method

In this section, the sine–cosine method [23,24] has been described as follows. First, consider a nonlinear fractional partial differential equation under conformal fractional derivative

P(u,Dtαu,ut,Dαxuux,uxt,ux x,utt, . . .)=0, (2) whereu(x,t)is a travelling wave solution of eq. (2). We use the fractional travelling wave transformation

u(x,t)= f(ξ), (3)

where ξ =xωtα

α.

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Figure 1. 3D graphs (a,c,e,g) of eq. (18) and their corresponding contour plots (b,d,f,h) fora = −1,b = 1,c =1,

−3x3, 0t 5 and various values ofα.

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64 Page 4 of 8 Pramana – J. Phys. (2022) 96:64

Figure 2. 2D graphs of eq. (18) for (a)t =1 and (b)t =2 witha = −1,b=1,c=1,3x3 and various values ofα.

This allows us to make the following change:

∂t = −ω d dξ, 2

∂t2 =ω2 d2 dξ2,

∂x = d dξ, 2

∂x2 = d2

dξ2, . . . . (4)

Using eq. (4), we transform the fractional nonlinear par- tial differential equation (eq. (2)) to nonlinear ordinary differential equation (ODE)

Q(f, f , f , f , . . .). (5) In the sine–cosine method, the solution will be in the form

f(ξ)=φsinβ(μξ), |ξ| π

μ (6)

or

f(ξ)=φcosβ(μξ), |ξ| π

μ, (7)

whereφ, β andμare parameters to be determined. We use

f(ξ)=φsinβ(μξ),

f (ξ)=φβμsinβ−1(μξ)cos(μξ), f (ξ)

= −(sin(μ ξ))βφ β μ2

β (sin(μ ξ))2β+1 (sin(μ ξ))2 , . . .

(8) or

f(ξ)=φcosβ(μξ),

f (ξ)= −φβμcosβ−1(μξ)sin(μξ), f (ξ)=φβ(β−12cosβ−2(μξ)

−φβ2μ2cosβ(μξ), . . . . (9) We replace eq. (8) or eq. (9) into eq. (5) and balance the terms of the sine function if eqs (8) are used or balance

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Figure 3. 2D graphs of eq. (18) for (a)t =1 and (b)t =2 witha = −1,b=1,c=1,−5x5 and various values ofα.

the terms of the cosine function if eqs (9) are used. We then equate all terms with the same power in sinm(μξ) or cosm(μξ) and set their coefficients to zero to get a system of algebraic equations containing the unknown parametersφ, βandμ. We then solve the system to get all possible values ofφ, β andμ.

4. Solutions to the non-linear fifth-order time-fractional KdV equation

In this section, the solution of conformal time-fractional KdV equation have been presented systematically by using the sine–cosine method.

By using the fractional travelling wave transforma- tions, one may have

u(x,t)= f(ξ), whereξ =xωtα

α . (10)

Substituting the above expression and its related derivative terms in eq. (1) and after that, integrating once

and equating the constant of integration to zero gives the ordinary differential equation (ODE)

ωf + a

3 f3+b f +c f(4)=0. (11) One can now obtain the solution of eq. (11) by sub- stituting eq. (8) in eq. (11). Note that here

f = −(sin(μ ξ))βφ β μ2(β (sin(μ ξ))2β+1) (sin(μ ξ))2

(12) and

f(4)= 1 (sin(μξ))4

×

⎧⎨

(φ(sin(μξ))β3(sin(μξ)))4)

+((−2β3+6β2−8β+4)(sin(μξ))2) +((β3−6β2+11β−64β)

⎫⎬

. (13)

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64 Page 6 of 8 Pramana – J. Phys. (2022) 96:64

Figure 4. 3D graphs (a,c,e,g) of eq. (18) and their corresponding contour plots (b,d,f,h) fora = −1,b = 1,c =1,

50x50, 0t 50 and various values ofα.

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Figure 5. 2D graphs of eq. (18) for (a)x=1, (b)x=2, (c) x =3 witha = −1,b =1,c=1, 0 t 5 and various values ofα.

This yields,

−ωsinβ(μξ)+a

3φsin3β(μξ)

b(sin(μξ))βφβμ2(β(sin(μξ))2β+1) (sin(μξ))2

+c 1 (sin(μξ))4

×

⎧⎨

(φ(sin(μξ))β3(sin(μξ)))4) +

(−2β3+6β2−8β+4)(sin(μξ))2 +((β3−6β2+11β−6)μ4β)

⎫⎬

⎭=0. (14) Equating powers we haveβ = −2. This simplifies to 1

3

(484−122−3ω)φ

(sin(μ ξ))2 +6μ2(−202+b)φ (sin(μ ξ))4

+1 3

(3604+2

(sin(μ ξ))6 =0. (15)

We now equate the coefficients to zero and solve the resulting system of equations using Maple. This gives φ =3b

−1

10ac, μ=

√5√ b/c

10 , ω= −4b2

25c. (16) This gives the solution of eq. (11) to be

f(ξ)=3b −1

10acsin2

5√ b/c

10 ξ

(17) withω= −4b2/25c.

Substituting the values as defined in eq. (10) into eq.

(17) along with the value ofω the new form of exact travelling wave solution of eq. (1) can be obtained as u(x,t)=3b

−1 10acsin2

× √

5√ b/c 10

x −−(4b2/25c)tα α

.(18)

5. Results and discussions

In this section, various case studies related to the above solution have been made with respect to different param- eters. Figure1shows the three-dimensional graph of the solution and its contour plot fora = −1,b=1,c =1,

−3≤ x ≤3, 0≤t ≤5 and various values ofα. Figures 2and3give two-dimensional plots for the solution by varyingx and fixing the value oft along with various values of α. For figures 2a and 2b, we have consid- eredt =1 and 2 respectively whereas other parameters are considered the same as in figure1. Similar to these plots, figures3a and3b have been plotted by extending the range ofx, that is for−5≤x ≤5. It can be clearly seen from figures 2 and3 that if we keep on increas- ing the fractional-order α, the corresponding wave of the solution is also moving from left to right. Moreover, figure4represents the same types of graphs as figure1 for−50≤ x ≤ 50 and 0≤ t ≤ 50. For figure4other parameters are considered the same as in figure1. From these two- and three-dimensional plots, it can be clearly seen that if we extend the range ofx, the heights of the solution waves gradually decrease. Figure5represents the two-dimensional plots of the solution by varying t and keeping the value of x as constant along with different values ofα. Other parameters considered are self-explanatory. From these plots, it can also be seen that for a particular time interval, increasing the partic- ular value of x reduces the height of the waves. From figures1and4it can be observed that exact solutions

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64 Page 8 of 8 Pramana – J. Phys. (2022) 96:64 obtained are solitary waves for various fractional-order

derivatives. Also from figures2,3and5it can be seen that different fractional parameters give different shapes of the solitary waves. It is worth mentioning that for α = 1 it gives the exact travelling wave solution of integer-order KdV equation.

6. Conclusions

It is well known that KdV-type equations are very impor- tant to study various nonlinear phenomena of plasma physics, quantum mechanics, optics etc. Accordingly, a new exact travelling wave solution of the time-fractional fifth-order non-linear KdV equation has been computed successfully using sine–cosine method. Here, conformal fractional derivative has been considered for the analy- sis. Also it has been verified by the Maple software that the solution obtained by the present method exactly sat- isfies the original equation. Moreover, the sine–cosine method used is found to be effective and straightfor- ward over the existing method such as the trial equation method [22] in the sense that less numbers of steps are required here to obtain the solution. As such, the method can be applied to many other nonlinear fractional par- tial differential equations. It can be observed that for a and c both positive or both negative, one may get imaginary solution which has large potential for further studies. However, the aim is also to study same types of problem by considering the other definitions of frac- tional derivative and new methods to obtain new form of solutions.

Acknowledgements

The authors would like to thank the anonymous reviewer and editor for their valuable suggestions and comments to improve the quality and clarity of the manuscript.

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