On the exponential solutions to three extracts from extended fifth-order KdV equation

Aly R. Seadawy; R.I. Nuruddeen; K.S. Aboodh; Y.F. Zakariya

doi:10.1016/j.jksus.2019.01.007

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32 (

); 765-769

doi:

10.1016/j.jksus.2019.01.007

On the exponential solutions to three extracts from extended fifth-order KdV equation

Aly R. Seadawy^{^a,⁎}, R.I. Nuruddeen^{^b}, K.S. Aboodh^{^c,^d}, Y.F. Zakariya^{^e}

a Department of Mathematics, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

b Department of Mathematics, Federal University Dutse, Jigawa State, Nigeria

c Department of Mathematics, Faculty of Science & Technology, Omdurman Islamic University, Khartoum, Sudan

d Department of Mathematics, Faculty of Science & Arts, University of Bisha, Bisha, Saudi Arabia

e Department of Science Education, Ahmadu Bello University, Zaria, Nigeria

⁎Corresponding author. aly742001@yahoo.com (Aly R. Seadawy),

Received: 2018-12-9, Accepted: 2019-1-17, Published: 2019-01-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

An extended fifth order Korteweg-de-Vries (efKdV) equation is an important equation in fluids dynamics for the description of nonlinear wave processes, and contains quite a number of KdV-type equations including the Sawada-Kotera equation, the Caudrey-Dodd-Gibbon equation, the Lax equation, the Kaup-Kuperschmidt equation and the Ito equation among others. However, in this paper, we examine the efKdV by extracting three different fifth order Korteweg-de-Vries (fKdV) equations. Solitary wave solutions to the extracts are found by means of an exponential function ansatz specifically constructed for the study. The obtained solutions can be used in description of shallow-water waves.

Keywords

Fifth order KdV equation

KdV equations

Exponential function ansatz

Solitary wave

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1 Introduction

Nonlinear partial differential equations are known for their roles in many nonlinear science applications including the hydrodynamics, fluid dynamics and mechanics, plasma physics and nonlinear dispersive wave among others. An important model in this regard is the Korteweg-de-Vries (KdV) equation (Olver, 1984) that has many applications such as in description of shallow water waves and ion-acoustic waves in plasmas. The KdV equation has undergone several modifications and extensions leading to various forms of KdV equations appearing in three, five, seven or more order forms and different dimensions; read (Hereman, 2000; Hereman, 2009; Helal and Seadawy, 2009; Bakodah, 2013; Adomian, 1996; Nuruddeen, 2018; Djidjeli et al., 1995; Kaya, 2005; Khater et al., 2000; Khater et al., 2006; Lei et al., 2002; Sawada and Kotera, 1974; El-Wakil et al., 2011; Pandir et al., 2013; Islam et al., 2015 ) for various forms and methods.

Further, of recent, Yun-hu and Yong (Yun-hu and Yong, 2013) analyzed the extended fifth order Korteweg-de-Vries (efKdV) equation of the form

(1)

u_{t} + p_{1} u^{2} u_{x} + p_{2} u_{x} u_{xx} + p_{3} {uu}_{xxx} + p_{4} u_{xxxxx} + p_{5} {uu}_{x} + p_{6} u_{xxx} + p_{7} u_{x} = 0,

where

p_{j} (j = 1, 2, \dots, 7)

are constants. They (Yun-hu and Yong, 2013) established all the conservation laws through Lax pair and talked of many special cases of efKdV equations. One can also see the recent paper by Nikolay (2016) for the analysis of the generalized modified fKdV with dissipation.

However in this study, we aim to study three different fKdV equations to be extracted from the efKdV equation given in Eq. (1) as follows:

1). The first one is obtained by setting $P_{5} = P_{6} = P_{7} = 0$ in Eq. (1), to yields

(2)

u_{t} + p_{1} u^{2} u_{x} + p_{2} u_{x} u_{xx} + p_{3} {uu}_{xxx} + p_{4} u_{xxxxx} = 0 .

2). The second by setting $P_{1} = P_{2} = P_{6} = P_{7} = 0$ in Eq. (1), which gives

(3)

u_{t} + p_{3} {uu}_{xxx} + p_{4} u_{xxxxx} + p_{5} {uu}_{x} = 0 .

3). The third by setting $P_{1} = P_{2} = P_{5} = P_{7} = 0$ in Eq. (1), that reads

(4)

u_{t} + p_{3} {uu}_{xxx} + p_{4} u_{xxxxx} + p_{6} u_{xxx} = 0 .

Further, different analytical methods have been proposed in the literature to determining exact solutions to various evolutions equations modelled in partial and or fractional differential equations, see Ali and Kalisch (2014), Grujic and Kalisch (2009), Aly Seadawy (2017), Seadawy (132 2017,), Seadawy (2017), Nuruddeen and Nass (2017), Seadawy and Lu (2017), Helal and Seadawy (2011), Alquran et al. (2017), Khalid and Nuruddeen (2017), Khalid et al. (2018), Bakodah et al. (2016), Seadawy (2015), Seadawy (2017), Aly (2018), Khalid et al. (2018), Khalid et al. (2018) for details. The paper is organized as follows: Section 2 presents the concept of the constructed method; Section 3 is for the application; Section 4 discusses the obtained results and Section 5 is for conclusion.

2 The methodology

In constructing exact solutions to the extracrted equations in Eqs. (2)–(4), we devise an exponential function ansatz and presented here by considering the general form of differential equation below:

(5)

P (u, D_{t} u, D_{x} u, D_{t} D_{x}, D_{tt} u, D_{xx} u, \dots) = 0 .

We then seek for an exponential function solution of Eq. (5) of the form:

(6)

u (x, t) = A_{1} + A_{2} \frac{e^{k (x - ct)}}{{(1 + e^{k (x - ct)})}^{2}},

where k and c are nonzero constants, and

A_{1}

, and

A_{2}

, are determined with

A_{2} \neq 0

in any case.

Thus, substituting Eq. (6) and its possible derivatives into Eq. (5) gives a polynomial in $e^{jk (x - ct)}$ for $j = 0, 1, 2, \dots, n$ . Equating the coefficient of each power of $e^{jk (x - ct)}, (j = 0, 1, 2, \dots, n)$ to zero, we obtain a system of algebraic equations which will then be solved for the unknowns (we use Mathematica software) to obtain the solution of Eq. (5).

3 Application

In this section, we present exact exponential function traveling wave solutions to the three extracts given in Eqs. (2)–(4) which we extracted from the extended fifth-order KdV (efKdV) equation given in (1) as follows:

3.1

3.1 The first extract

We consider the first extract of the extended fifth-order KdV (efKdV) equation given in Eq. (2);

(7)

u_{t} + p_{1} u^{2} u_{x} + p_{2} u_{x} u_{xx} + p_{3} {uu}_{xxx} + p_{4} u_{xxxxx} = 0,

Substituting the exponential solution ansatz given in Eq. (6) into Eq. (7) and equating the coefficient of each power of

e^{jk (x - ct)}, (j = 0, 1, 2, \dots, 6)

to zero we get the following system of algebraic equations:

{ckA}_{2} - {kA}_{1}^{2} A_{2} p_{1} - k^{3} A_{1} A_{2} p_{3} - k^{5} A_{2} p_{4} = 0,

- {ckA}_{2} + {kA}_{1}^{2} A_{2} p_{1} + k^{3} A_{1} A_{2} p_{3} + k^{5} A_{2} p_{4} = 0,

3 {ckA}_{2} - 3 {kA}_{1}^{2} A_{2} p_{1} - 2 {kA}_{1} A_{2}^{2} p_{1} - k^{3} A_{2}^{2} p_{2} + 9 k^{3} A_{1} A_{2} p_{3} - k^{3} A_{2}^{2} p_{3} + 57 k^{5} A_{2} p_{4} = 0,

- 3 {ckA}_{2} + 3 {kA}_{1}^{2} A_{2} p_{1} + 2 {kA}_{1} A_{2}^{2} p_{1} + k^{3} A_{2}^{2} p_{2} - 9 k^{3} A_{1} A_{2} p_{3} + k^{3} A_{2}^{2} p_{3} - 57 k^{5} A_{2} p_{4} = 0,

⋮

Solving the above system, we get the following sets of solutions:

$A_{1} = \frac{- k^{2} p_{2} - 2 k^{2} p_{3} - \sqrt{k^{4} (p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4})}}{4 p_{1}},$

$A_{2} = 3 (\frac{k^{2} p_{2} + 2 k^{2} p_{3} + \sqrt{k^{4} (p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4})}}{p_{1}}),$

$c = \frac{1}{8} (- 12 k^{4} p_{4} + \frac{k^{4} p_{2}^{2} + 2 k^{4} p_{2} p_{3} + k^{2} p_{2} \sqrt{k^{4} (p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4})}}{p_{1}})$ , and

$A_{1} = \frac{- k^{2} p_{2} - 2 k^{2} p_{3} + \sqrt{k^{4} (p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4})}}{4 p_{1}},$

$A_{2} = 3 (\frac{k^{2} p_{2} + 2 k^{2} p_{3} - \sqrt{k^{4} (p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4})}}{p_{1}}),$

$c = \frac{1}{8} (- 12 k^{4} p_{4} + \frac{k^{4} p_{2}^{2} + 2 k^{4} p_{2} p_{3} - k^{2} p_{2} \sqrt{k^{4} (p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4})}}{p_{1}})$ , where, $p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4} > 0$ .

The first set gives the exact solution

(8)

\begin{matrix} u (x, t) = & \frac{- k^{2} p_{2} - 2 k^{2} p_{3} - \sqrt{k^{4} (p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4})}}{4 p_{1}} + \\ 3 \frac{k^{2} p_{2} + 2 k^{2} p_{3} + \sqrt{k^{4} (p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4})}}{p_{1} {(1 + e^{k (x - c_{1} t)})}^{2}} e^{k (x - c_{1} t)} . \end{matrix}

(see Fig. 1.)

The 3D and the contour plots for Eq. (8).

Further, the second set gives the solution:

(9)

\begin{matrix} u (x, t) = & \frac{- k^{2} p_{2} - 2 k^{2} p_{3} + \sqrt{k^{4} (p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4})}}{4 p_{1}} + \\ 3 \frac{k^{2} p_{2} + 2 k^{2} p_{3} - \sqrt{k^{4} (p_{2}^{2} + 4 p_{2} p_{3} + 4 p_{3}^{2} - 40 p_{1} p_{4})}}{p_{1} {(1 + e^{k (x - c_{2} t)})}^{2}} e^{k (x - c_{2} t)} . \end{matrix}

where k is left as a free parameter in Eqs. (8) and (9) and

c_{1}

is the c in the first set and

c_{2}

is the c in the second set. (see Fig. 2.)

The 3D and the contour plots for Eq. (9).

3.2

3.2 The second extract

We consider the second extract of the extended fifth-order KdV (efKdV) equation given in Eq. (3);

(10)

u_{t} + p_{3} {uu}_{xxx} + p_{4} u_{xxxxx} + p_{5} {uu}_{x} = 0,

Substituting the exponential solution ansatz given in Eq. (6) into Eq. (10) and equating the coefficient of each power of

e^{jk (x - ct)}, (j = 0, 1, 2, \dots, 6)

to zero we get the following system of algebraic equations:

- {ckA}_{2} + k^{3} A_{1} A_{2} p_{3} + k^{5} A_{2} p_{4} + {kA}_{1} A_{2} p_{5} = 0,

- 3 {ckA}_{2} - 9 k^{3} A_{1} A_{2} p_{3} + k^{3} A_{2}^{2} p_{3} - 57 k^{5} A_{2} p_{4} + 3 {kA}_{1} A_{2} p_{5} + {kA}_{2}^{2} p_{5} = 0,

- 2 {ckA}_{2} - 10 k^{3} A_{1} A_{2} p_{3} - 11 k^{3} A_{2}^{2} p_{3} + 302 k^{5} A_{2} p_{4} + 2 {kA}_{1} A_{2} p_{5} + {kA}_{2}^{2} p_{5} = 0,

2 {ckA}_{2} + 10 k^{3} A_{1} A_{2} p_{3} + 11 k^{3} A_{2}^{2} p_{3} - 302 k^{5} A_{2} p_{4} - 2 {kA}_{1} A_{2} p_{5} - {kA}_{2}^{2} p_{5} = 0,

3 {ckA}_{2} + 9 k^{3} A_{1} A_{2} p_{3} - k^{3} A_{2}^{2} p_{3} + 57 k^{5} A_{2} p_{4} - 3 {kA}_{1} A_{2} p_{5} - {kA}_{2}^{2} p_{5} = 0,

{ckA}_{2} - k^{3} A_{1} A_{2} p_{3} - k^{5} A_{2} p_{4} - {kA}_{1} A_{2} p_{5} = 0 .

Solving the above system, we get the following set of solution:

$A_{1} = - \frac{5 (k^{2} p_{3} p_{4} - p_{4} p_{5})}{2 p_{3}^{2}},$

$A_{2} = \frac{30 k^{2} p_{4}}{p_{3}},$

$c = - \frac{3 k^{4} p_{3}^{2} p_{4} - 5 p_{4} p_{5}^{2}}{2 p_{3}^{2}}$ ; which produces a solution:

(11)

u (x, t) = - \frac{5 (k^{2} p_{3} p_{4} - p_{4} p_{5})}{2 p_{3}^{2}} + \frac{30 k^{2} p_{4} e^{k (x + \frac{3 k^{4} p_{3}^{2} p_{4} - 5 p_{4} p_{5}^{2}}{2 p_{3}^{2}} t)}}{p_{3} {(1 + e^{k (x + \frac{3 k^{4} p_{3}^{2} p_{4} - 5 p_{4} p_{5}^{2}}{2 p_{3}^{2}} t)})}^{2}},

where k is left as a free parameter. (see Fig. 3.)

The 3D and the contour plots for Eq. (11).

3.3

3.3 The third extract

We consider the third extract of the extended fifth-order KdV (efKdV) equation given in Eq. (4);

(12)

u_{t} + p_{3} {uu}_{xxx} + p_{4} u_{xxxxx} + p_{6} u_{xxx} = 0 .

Proceeding as above, we get the following system of algebraic equations:

- {ckA}_{2} + k^{3} A_{1} A_{2} p_{3} + k^{5} A_{2} p_{4} + k^{3} A_{2} p_{6} = 0,

- 3 {ckA}_{2} - 9 k^{3} A_{1} A_{2} p_{3} + k^{3} A_{2}^{2} p_{3} - 57 k^{5} A_{2} p_{4} - 9 k^{3} A_{2} p_{6} = 0,

- 2 {ckA}_{2} - 10 k^{3} A_{1} A_{2} p_{3} - 11 k^{3} A_{2}^{2} p_{3} + 302 k^{5} A_{2} p_{4} - 10 k^{3} A_{2} p_{6} = 0,

2 {ckA}_{2} + 10 k^{3} A_{1} A_{2} p_{3} + 11 k^{3} A_{2}^{2} p_{3} - 302 k^{5} A_{2} p_{4} + 10 k^{3} A_{2} p_{6} = 0,

3 {ckA}_{2} + 9 k^{3} A_{1} A_{2} p_{3} - k^{3} A_{2}^{2} p_{3} + 57 k^{5} A_{2} p_{4} + 9 k^{3} A_{2} p_{6} = 0,

{ckA}_{2} - k^{3} A_{1} A_{2} p_{3} - k^{5} A_{2} p_{4} - k^{3} A_{2} p_{6} = 0 .

Solving the above system, we get the following set of solution:

$A_{1} = - \frac{5 k^{2} p_{4} + 2 p_{6}}{2 p_{3}},$

$A_{2} = \frac{30 k^{2} p_{4}}{p_{3}},$

$c = - \frac{3}{2} k^{4} p_{4}$ ; which produces a solution:

(13)

u (x, t) = - \frac{5 k^{2} p_{4} + 2 p_{6}}{2 p_{3}} + \frac{30 k^{2} p_{4} e^{k (x + \frac{3}{2} k^{4} p_{4} t)}}{p_{3} {(1 + e^{k (x + \frac{3}{2} k^{4} p_{4} t)})}^{2}},

where k is left as a free parameter. (see Fig. 4.)

The 3D and the contour plots for Eq. (13).

4 Results and discussion

The present study effectively constructs exponential traveling wave solutions to three different fifth order Korteweg-de-Vries equations (fKdV) extracted from the extended fifth order Korteweg-de-Vries equation (efKdV) that has recently been studied by Yun-hu and Yong in Yun-hu and Yong (2013). The first extract Eq. (1) which was once numerically considered by Bakodah (2013) was found to have two different exponential function exact solutions. The fractional version of the second extract Eq. (2) was also recently studied by Khalid et al. (2018) using generalized exp expansion method, while the third extract Eq. (3) is first extracted and studied in this study to our knowledge with one exponential function exact solutions each. Since KdV equations play vital roles in the study of shallow-water waves, thus the present obtained traveling wave solutions would not be an exception in the description of shallow-water waves and other nonlinear wave processes. Finally, Three-dimensional (3D) and contour plots are given for the obtained solution.

5 Conclusion

In conclusion, exponential functions traveling wave solutions to three different fifth order Korteweg-de-Vries equations have been constructed. The three fKdV equations considered are extracted from the extended fifth order Korteweg-de-Vries equation that has recently been studied in the literature and found to be of various applications in nonlinear sciences. Interesting and reliable exponential function ansatz was proposed and implemented on Mathematica software in the study. We provide three-dimensional and contour plots for the obtained solutions for easier interpretations of results.

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Introduction

The methodology

Application

The first extract
The second extract
The third extract

Results and discussion

Conclusion

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[1] Adomian G., . The fifth-order Korteweg-De Vries equation. Int. J. Math. Math. Sci.. 1996;19
[Google Scholar]

[2] Ali A., Kalisch H., . On the formulation of mass, momentum and energy conservation in the KdV equation. Acta Appl. Math.. 2014;133:113-131.
[Google Scholar]

[3] Alquran M., Jaradat H.M., Syam M.I., . Analytical solution of the time-fractional Phi-4 equation using modified power series method. Nonlinear Dyn. 2017
[Google Scholar]

[4] Bakodah H.O., . Modified Adomain decomposition method for the generalized fifth order KdV equations. Am. J. Comput. Math.. 2013;3:53-58.
[Google Scholar]

[5] Bakodah H.O., Al-Zaid N.A., Mirzazadeh M., Zhou Q., . Decomposition method for solving Burger’s equation with Dirichlet and Neumann boundary conditions. Int. J. Light Electron Opt.. 2016;30
[Google Scholar]

[6] Djidjeli K., Price W.G., Twizell E.H., Wang Y., . Numerical methods for the soltution of the third and fifth-order disprsive Korteweg-De Vries equations. J. Comput. Appl. Math.. 1995;58:307-336.
[Google Scholar]

[7] El-Wakil S.A., Abulwafa E.M., Zahran M.A., Mahmoud A.A., . Time-fractional KdV equation: formulation and solution using variational methods. Nonlinear Dyn.. 2011;65:55-63.
[Google Scholar]

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On the exponential solutions to three extracts from extended fifth-order KdV equation

Abstract

Keywords

Fifth order KdV equation

KdV equations

Exponential function ansatz

Solitary wave

1 Introduction

2 The methodology

3 Application

3.1 The first extract

3.2 The second extract

3.3 The third extract

4 Results and discussion

5 Conclusion

References

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