Explicit solutions to the time-fractional generalized dissipative Kawahara equation

Marwan Alquran; Mohammed Ali; Omar Alshboul

doi:10.1016/j.joes.2022.02.013

Journal of Ocean Engineering and Science >

2024 , Vol. 9 >Issue 4: 348 - 352

DOI: https://doi.org/10.1016/j.joes.2022.02.013

Original article

Explicit solutions to the time-fractional generalized dissipative Kawahara equation

Marwan Alquran ,
Mohammed Ali ,
Omar Alshboul

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Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan

Received 31 August 2021, Revised 28 January 2022, Accepted 16 February 2022, Available online 19 February 2022, Version of Record 22 July 2024.E-mail:marwan04@just.edu.jo;myali@just.edu.jo;oaalshboul19@sci.just.edu.jo

Received date: 2021-08-31

Revised date: 2022-01-28

Accepted date: 2022-02-16

Online published: 2024-08-13

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Abstract

In this paper, the generalized dissipative Kawahara equation in the sense of conformable fractional derivative is presented and solved by applying the tanh-coth-expansion and sine-cosine function techniques. The quadratic-case and cubic-case are investigated for the proposed model. Expected solutions are obtained with highlighting to the effect of the presence of the alternative fractional-derivative and the effect of the added dissipation term to the generalized Kawahara equation. Some graphical analysis is presented to support the findings of the paper. Finally, we believe that the obtained results in this work will be important and valuable in nonlinear sciences and ocean engineering.

Highlights

● The conformable-time-fractional generalized dissipative Kawahara equation is presented and solved by using the tanh-coth-expansion and sine-cosine-function methods.

● The quadratic-case and cubic-case are investigated for the proposed model.

● The new solutions highlighted the effect of the presence of the alternative fractional-derivative and the effect of the added dissipation term to the generalized Kawahara equation.

● Some graphical analysis is presented to support the findings of the current work.

Key words： Time-fractional Kawahara equation; Conformable-derivative; Tanh-coth-expansion method; Sine-cosine function method

Cite this article

Marwan Alquran , Mohammed Ali , Omar Alshboul . Explicit solutions to the time-fractional generalized dissipative Kawahara equation[J]. Journal of Ocean Engineering and Science, 2024 , 9(4) : 348 -352 . DOI: 10.1016/j.joes.2022.02.013

1. Introduction

Finding exact solutions to nonlinear models is very important part in applied mathematical sciences due to their significant role in investigating and explaining the mechanisms of propagation nonlinear waves in different complex media such as in oceans, water-waves, physics, optics, and much more [1], [2], [3], [4]. In order to obtain a clear view of the dynamic predictions of many natural phenomena and link them to their historical records, the mathematical representations of these nonlinear models were revisited and the time-varying rates were considered as fractional derivatives [5], [6], [7], [8], [9], [10]. The fractional-derivative is a generalization to the integer-derivative, and it possesses inherited memory. As a motivation by this trend, we study a time-fractional version to a model that arising in the field of ocean engineering and science. The suggested model is the generalized Kawahara equation (KE).

The KE is the modified version of the fifth-order KdV, and it describes the dynamics of shallow water waves and capillary gravity water waves. This equation takes the form Kawahara [11]

(1)

ψ_{t} (x, t) + \frac{b}{m} (ψ^{m} {(x, t)}_{x}) + A ψ_{x x x} (x, t) - B ψ_{x x x x x} (x, t) = 0 .

In the literature, the cosh-function method is suggested in [12] as a solution to (1) for the case

m = 1, 2

. In [13] the tanh-method and exp-method are used to solve KE for the case of

m = 1, 2

and

b = 1

. Our aim is to study the solution of KE with dissipative term and conformable-time derivative. In fact, the proposed model is

(2)

\begin{matrix} D_{t}^{α} ψ (x, t) + a ψ_{x} (x, t) + \frac{b}{m} {(ψ^{m} (x, t))}_{x} + A ψ_{x x x} (x, t) \\ - B ψ_{x x x x x} (x, t) = 0, 0 < α \leq 1, \end{matrix}

where

A, B

are the dispersion parameters,

a

is the dissipative coefficient,

b

is the nonlinearity-strength, and

D_{t}^{α}

is the conformable-derivative which is defined by Khalil et al. [14] as

(3)

D_{t}^{α} ψ (x, t) = \lim_{ϵ \to 0} \frac{ψ (x, t + ϵ t^{1 - α}) - ψ (x, t)}{ϵ} .

We refer to (3) as the conformable-time dissipative KE.

Since the fractional derivative plays the role of inheriting some historical properties of a natural phenomenon or a physical system, researchers revisited many well-known nonlinear models with the presence of conformable-derivative. The dynamics of these fractional problems have been investigated through extracting their explicit solutions. For this purposes, many effective techniques are used, such as, different versions of the

(G^{'} / G)

-expansion scheme [15], [16], [17], modified exp-expansion scheme [18], extended Kudryashov method [19], [20], [21], Polynomial function scheme [22], [23], [24], Jacobi elliptic function method [25], [26], Sine-Gordon equation [27], [28], F-expansion and simple equation algorithms [29], and many others.

The contribution of this research is that the model given in (2) is studied for the first time. In fact, we get explicit solutions to it by using two effective methods valid for this type of equations. Moreover, we study the propagation of the obtained solitary solutions to the fractional dissipative Kawahara upon varying the values of the conformable-derivative.

The paper is organised as follows: In Section 2, we suggested the tanh-coth expansion technique [30], [31] and the sine-cosine function method [32], [33] to study the solution of (2) when the nonlinearity term is of order

m = 2

. In Section 3, the same format is given to solve (2) for

m = 3

. Finally, we summarized our findings in Section 4.

2. Exact solutions to the fractional dissipative KE: $m = 2$

The conformable-time dissipative KE with

m = 2

takes the following form

(4)

\begin{matrix} D_{t}^{α} ψ (x, t) + a ψ_{x} (x, t) + b ψ (x, t) ψ_{x} (x, t) \\ + A ψ_{x x x} (x, t) - B ψ_{x x x x x} (x, t) = 0 . \end{matrix}

By using the following relation [14]

(5)

D_{t}^{α} (ψ (ϕ (t))) = t^{1 - α} D_{t} ϕ (t) D_{t} ψ (ϕ (t)),

the new wave-variable

ζ = x - c \frac{t^{α}}{α}

transform (4) into the following differential equation

(6)

(c - a) ψ (ζ) + \frac{b}{2} ψ^{2} (ζ) + A ψ^{″} (ζ) - B ψ^{(4)} = 0 .

We aim to implement two methods to retrieve explicit solutions to (6).

2.1. Tanh-coth expansion method

The tanh-coth expansion method suggests that the solution of (6) is

(7)

ψ = ψ (ζ) = \sum_{i = 0}^{n} λ_{i} Z^{i},

where

Z = Z (ζ)

is the solution of the auxiliary differential equation

(8)

Z^{'} = μ (1 - Z^{2}),

whose solution is

(9)

Z = \tanh (μ ζ),

(10)

Z = \coth (μ ζ) .

We apply the balance procedure on the terms

ψ^{2}

against

ψ^{(4)}

to get the relation

2 n = n + 4

. Thus, the index

n

given in (7) has the value of 4. Accordingly, the solution of (6) is

(11)

w (ζ) = λ_{0} + λ_{1} Z + λ_{2} Z^{2} + λ_{3} Z^{3} + + λ_{4} Z^{4} .

Next, we substitute (11) and its necessary derivatives in (6) to get a polynomial of eighth-order in the variable

Z

with the following coefficients:

(12)

\begin{matrix} Z^{0} & : & λ_{0} (a - c) + 2 λ_{2} μ^{2} (A + 8 B μ^{2}) + \frac{b λ_{0}^{2}}{2} - 24 B λ_{4} μ^{4}, \\ Z^{1} & : & λ_{1} (a - 2 A μ^{2} + b λ_{0} - 16 B μ^{4} - c) + 6 λ_{3} μ^{2} (A + 20 B μ^{2}), \\ Z^{2} & : & λ_{2} (a - 8 A μ^{2} + b λ_{0} - 136 B μ^{4} - c) + 12 λ_{4} μ^{2} (A + 40 B μ^{2}) + \frac{b λ_{1}^{2}}{2}, \\ ↓ \\ Z^{7} & : & λ_{3} (b λ_{4} - 360 B μ^{4}), \\ Z^{8} & : & \frac{b λ_{4}^{2}}{2} - 840 B λ_{4} μ^{4} . \end{matrix}

Equating the coefficient of

Z^{i} : i = 0, 1, \dots, 8

to zero, results in a non algebraic system with the unknowns

λ_{0}, λ_{1}, λ_{2}, λ_{3}, λ_{4}, μ

and

c

. Solving the obtained system gives

(13)

\begin{matrix} λ_{0} & = & \frac{105 A^{2}}{169 b B}, λ_{1} = λ_{3} = 0, λ_{2} = - \frac{210 A^{2}}{169 b B}, λ_{4} = \frac{1680 B μ^{4}}{b}, \\ μ & = & \mp \frac{\sqrt{A}}{2 \sqrt{13} \sqrt{B}}, c = \frac{169 a B + 36 A^{2}}{169 B} . \end{matrix}

Accordingly, two solutions to (4) are obtained and given by the following:

(14)

ψ_{1} (x, t) = \frac{105 A^{2}}{169 b B} - \frac{210 A^{2} \tanh^{2} (\frac{\sqrt{A} (x - \frac{t^{α} (169 a B + 36 A^{2})}{169 α B})}{2 \sqrt{13} \sqrt{B}})}{169 b B} + \frac{105 A^{2} \tanh^{4} (\frac{\sqrt{A} (x - \frac{t^{α} (169 a B + 36 A^{2})}{169 α B})}{2 \sqrt{13} \sqrt{B}})}{169 b B},

and

(15)

ψ_{2} (x, t) = \frac{105 A^{2}}{169 b B} - \frac{210 A^{2} \coth^{2} (\frac{\sqrt{A} (x - \frac{t^{α} (169 a B + 36 A^{2})}{169 α B})}{2 \sqrt{13} \sqrt{B}})}{169 b B} + \frac{105 A^{2} \coth^{4} (\frac{\sqrt{A} (x - \frac{t^{α} (169 a B + 36 A^{2})}{169 α B})}{2 \sqrt{13} \sqrt{B}})}{169 b B} .

Figure 1, presents the 3D solutions of (14) and (15) for the integer case

α = 1

. In Figs. 2 and 3, we study the effect of the conformable-derivative acting on the propagation of

ψ_{1} (x, t)

and

ψ_{2} (x, t)

. It can be observed that for

α > 0.2

, the solutions are mapping continuously to each other with the same physical shape. However, for

α < 0.2

, we report a chaotic behaviors of the solutions where we can tell nothing [34]. Based on these observations, we may say that the fractional derivative can be considered as a memory index in a way that after a specific value, correct heritage-information can be retrieved regarding the physical structures of the given model.

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Fig.1 3D soliton and singular-soliton solutions of (14) and (15), respectively, for the integer case $α = 1$ and the assigned parameters $a = b = A = B = 1$ .

View original graphic|Download|PPT slide

Fig.2 The effect of the conformable-derivative acting on the propagation of $ψ_{1} (x, t)$ (14) when $t = 0.5$ and $a = b = A = B = 1$ .

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Fig.3 The effect of the conformable-derivative acting on the propagation of $ψ_{2} (x, t)$ (15) when $t = 0.5$ and $a = b = A = B = 1$ .

2.2. Sine-cosine function method

The sine-cosine function method suggests that the solution of (6) is

(16)

ψ (ζ) = λ \sin^{n} (μ ζ),

(17)

ψ (ζ) = λ \cos^{n} (μ ζ) .

Then, we substitute (16) in (6) to obtain

(18)

\begin{matrix} 0 & = & - 2 B μ^{4} n (n^{3} - 6 n^{2} + 11 n - 6) + 2 μ^{2} (n - 1) n \sin^{2} (μ z) \\ (A + 2 B μ^{2} (n^{2} - 2 n + 2)) \\ + 2 \sin^{4} (μ z) (a + μ^{2} (- n^{2}) (A + B μ^{2} n^{2}) - c) \\ + b λ \sin^{n + 4} (μ z) . \end{matrix}

Now, we equate the exponent

n + 4

to zero, and we set each coefficient of

\sin^{i} (μ z) : i = 0, 2, 4

to zero. Then, we solving the resulting system to have the following:

(19)

λ = \frac{105 A^{2}}{169 b B}, μ = \mp \frac{i \sqrt{A}}{2 \sqrt{13} \sqrt{B}}, c = \frac{169 a B + 36 A^{2}}{169 B} .

Accordingly, the solution of (4) is

(20)

ψ_{3} (x, t) = \frac{105 A^{2} csch^{4} (\frac{\sqrt{A} (x - \frac{t^{α} (169 a B + 36 A^{2})}{169 α B})}{2 \sqrt{13} \sqrt{B}})}{169 b B} .

On the other side, if (17) is used, one more solution will be obtained and given by

(21)

ψ_{4} (x, t) = \frac{105 A^{2} sech^{4} (\frac{\sqrt{A} (x - \frac{t^{α} (169 a B + 36 A^{2})}{169 α B})}{2 \sqrt{13} \sqrt{B}})}{169 b B} .

Remark: By using the identities

csch^{2} (μ z) = \coth^{2} (μ z) - 1

and

sech^{2} (μ z) = 1 - \tanh^{2} (μ z)

, the solutions of (4) given in (20) and (21) are identical, respectively, to (15) and (14).

3. Exact solutions to the fractional dissipative KE: m=3

The conformable-time dissipative KE with

m = 3

takes the following form:

(22)

\begin{matrix} D_{t}^{α} ψ (x, t) + a ψ_{x} (x, t) + b ψ {(x, t)}^{2} ψ_{x} (x, t) \\ + A ψ_{x x x} (x, t) - B ψ_{x x x x x} (x, t) = 0 . \end{matrix}

The wave-variable

ζ = x - c \frac{t^{α}}{α}

transforms (22) into the following differential equation

(23)

(c - a) ψ (ζ) + \frac{b}{3} ψ^{3} (ζ) + A ψ^{″} (ζ) - B ψ^{(4)} = 0 .

3.1. Tanh-coth expansion method

Balancing the order of the term

ψ^{3} (ζ)

against

ψ^{(4)} (ζ)

gives

n = 2

. Thus, the tanh-coth solution of (23) is

(24)

ψ (ζ) = a_{0} + λ_{1} Z + λ_{2} Z^{2} .

Implicit differentiations of (24) leads to

(25)

\begin{matrix} ψ^{″} (ζ) = 2 μ^{2} (Z^{2} - 1) (λ_{2} (3 Z^{2} - 1) + λ_{1} Z), \\ ψ^{(4)} (ζ) = 8 μ^{4} (Z^{2} - 1) \\ (λ_{1} Z (3 Z^{2} - 2) + λ_{2} (15 Z^{4} - 15 Z^{2} + 2)) . \end{matrix}

By substitution of (24)-(25) in (23), we get a polynomial of sixth-order in the variable

Z

with the following coefficients:

(26)

\begin{matrix} Z^{0} & : & λ_{0} (a - c) + 2 λ_{2} μ^{2} (A + 8 B μ^{2}) + \frac{b λ_{0}^{3}}{3}, \\ Z^{1} & : & λ_{1} (a - 2 A μ^{2} + b λ_{0}^{2} - 16 B μ^{4} - c), \\ Z^{2} & : & λ_{2} (a - 8 μ^{2} (A + 17 B μ^{2}) - c) + b λ_{2} λ_{0}^{2} + b λ_{1}^{2} λ_{0}, \\ ↓ \\ Z^{5} & : & λ_{1} (b λ_{2}^{2} - 24 B μ^{4}), \\ Z^{6} & : & \frac{b λ_{2}^{3}}{3} - 120 B λ_{2} μ^{4} . \end{matrix}

Equating the coefficient of

Z^{i} : i = 0, 1, \dots, 6

to zero, results in a non algebraic system with the unknowns

λ_{0}, λ_{1}, λ_{2}, μ

, and

c

. Solving the obtained system gives

(27)

\begin{matrix} λ_{0} & = & \pm \frac{3 A}{\sqrt{10} \sqrt{b} \sqrt{B}}, λ_{1} = 0, λ_{2} = \mp \frac{6 \sqrt{10} \sqrt{B} μ^{2}}{\sqrt{b}}, \\ μ & = & \mp \frac{\sqrt{A}}{2 \sqrt{5} \sqrt{B}}, c = \frac{25 a B + 4 A^{2}}{25 B} . \end{matrix}

Accordingly, two solutions to the quadratic Kawahara equation are obtained and given by the following:

(28)

ψ_{5} (x, t) = \mp (\frac{3 A \tanh^{2} (\frac{\sqrt{A} (x - \frac{t^{α} (25 a B + 4 A^{2})}{25 α B})}{2 \sqrt{5} \sqrt{B}})}{\sqrt{10} \sqrt{b} \sqrt{B}} - \frac{3 A}{\sqrt{10} \sqrt{b} \sqrt{B}}),

and

(29)

ψ_{6} (x, t) = \mp (\frac{3 A \coth^{2} (\frac{\sqrt{A} (x - \frac{t^{α} (25 a B + 4 A^{2})}{25 α B})}{2 \sqrt{5} \sqrt{B}})}{\sqrt{10} \sqrt{b} \sqrt{B}} - \frac{3 A}{\sqrt{10} \sqrt{b} \sqrt{B}}) .

Figure 4, presents the 3D solutions of (28) and (29) for the integer case

α = 1

. In Figs. 5 and 6, we study the effect of the conformable-derivative acting on the propagation of

ψ_{5} (x, t)

and

ψ_{6} (x, t)

. We observe the same findings as depicted in Figs. 2 and 3.

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Fig.4 3D soliton and singular-soliton solutions of (28) and (29), respectively, for the integer case $α = 1$ and the assigned parameters $a = b = A = B = 1$ .

View original graphic|Download|PPT slide

Fig.5 The effect of the conformable-derivative acting on the propagation of $ψ_{5} (x, t)$ (28) when $t = 0.5$ and $a = b = A = B = 1$ .

View original graphic|Download|PPT slide

Fig.6 The effect of the conformable-derivative acting on the propagation of $ψ_{6} (x, t)$ (29) when $t = 0.5$ and $a = b = A = B = 1$ .

3.2. Sine-cosine function method

In this section, we skip the details of implementing the sine-cosine function method in solving (22), and we only present its findings:

(30)

n = - 2, λ = \mp \frac{3 A}{\sqrt{10} \sqrt{b} \sqrt{B}}, μ = \pm \frac{i \sqrt{A}}{2 \sqrt{5} \sqrt{B}}, c = \frac{25 a B + 4 A^{2}}{25 B} .

As a result, the following two solutions are reported to the conformable-time cKE:

(31)

ψ_{7} (x, t) = \mp (\frac{3 A csch^{2} (\frac{\sqrt{A} (x - \frac{t^{α} (25 a B + 4 A^{2})}{25 α B})}{2 \sqrt{5} \sqrt{B}})}{\sqrt{10} \sqrt{b} \sqrt{B}}),

and

(32)

ψ_{8} (x, t) = \mp (\frac{3 A sech^{2} (\frac{\sqrt{A} (x - \frac{t^{α} (25 a B + 4 A^{2})}{25 α B})}{2 \sqrt{5} \sqrt{B}})}{\sqrt{10} \sqrt{b} \sqrt{B}}) .

We should point here that

ψ_{7} (x, t)

and

ψ_{8} (x, t)

are equivalent, respectively, to

ψ_{6} (x, t)

and

ψ_{5} (x, t)

4. Conclusions

The conformable-fractional dissipative generalized Kawahara equation is introduced for the first time in the paper. We applied two numerical schemes to extract explicit solitary solutions to the proposed model for two states; the quadratic and the cubic cases. Moreover, we provided a graphical analysis that visualizes the effect of the value of the fractional derivative on the propagation of the obtained solutions. In addition, we observed that the solutions admit the bifurcation or the chaotic behaviours around a specific value of the fractional order. The findings of the current work confirm the fact that the fractional derivative acts as a tool for retrieving some properties inherited in a particular scientific model.

As a future work, we recommend the following actions: (1) Include other types of fractional derivatives and compare it with the obtained results of this work. (2) Apply analytical schemes and recover approximate solutions. (3) Study other fractional nonlinear models arise in ocean engineering and science.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

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Abstract

Cite this article

1. Introduction

2. Exact solutions to the fractional dissipative KE: m=2

2.1. Tanh-coth expansion method

Fig.1 3D soliton and singular-soliton solutions of (14) and (15), respectively, for the integer case α=1 and the assigned parameters a=b=A=B=1 .

Fig.2 The effect of the conformable-derivative acting on the propagation of ψ1(x,t) (14) when t=0.5 and a=b=A=B=1 .

Fig.3 The effect of the conformable-derivative acting on the propagation of ψ2(x,t) (15) when t=0.5 and a=b=A=B=1 .

2.2. Sine-cosine function method

3. Exact solutions to the fractional dissipative KE: m=3

3.1. Tanh-coth expansion method

Fig.4 3D soliton and singular-soliton solutions of (28) and (29), respectively, for the integer case α=1 and the assigned parameters a=b=A=B=1 .

Fig.5 The effect of the conformable-derivative acting on the propagation of ψ5(x,t) (28) when t=0.5 and a=b=A=B=1 .

Fig.6 The effect of the conformable-derivative acting on the propagation of ψ6(x,t) (29) when t=0.5 and a=b=A=B=1 .

3.2. Sine-cosine function method

4. Conclusions

References

Links

2. Exact solutions to the fractional dissipative KE: $m = 2$

Fig.1 3D soliton and singular-soliton solutions of (14) and (15), respectively, for the integer case $α = 1$ and the assigned parameters $a = b = A = B = 1$ .

Fig.2 The effect of the conformable-derivative acting on the propagation of $ψ_{1} (x, t)$ (14) when $t = 0.5$ and $a = b = A = B = 1$ .

Fig.3 The effect of the conformable-derivative acting on the propagation of $ψ_{2} (x, t)$ (15) when $t = 0.5$ and $a = b = A = B = 1$ .

Fig.4 3D soliton and singular-soliton solutions of (28) and (29), respectively, for the integer case $α = 1$ and the assigned parameters $a = b = A = B = 1$ .

Fig.5 The effect of the conformable-derivative acting on the propagation of $ψ_{5} (x, t)$ (28) when $t = 0.5$ and $a = b = A = B = 1$ .

Fig.6 The effect of the conformable-derivative acting on the propagation of $ψ_{6} (x, t)$ (29) when $t = 0.5$ and $a = b = A = B = 1$ .