Numerical simulations of stochastic conformable space–time fractional Korteweg-de Vries and Benjamin–Bona–Mahony equations


 In this paper, we investigate the effect of white noise on conformable time and space fractional KdV and BBM equations. For this purpose, we convert these equations with external noise to homogeneous conformable time and space fractional KdV and BBM equations with defined transformation and then we solve them by modified Kudryashov method. We bring our numerical results in some figures in the last section.


Introduction
L'Hospital expressed the idea of the fractional derivative in his letter to Leibniz [23], since that time it has attracted the attention of many researchers who have tried to propose a de nition on a fractional derivative like Liouville, Caputo, Hadamard, Riemann [10,18,31]. Mathematical models established by using fractional derivatives have better overlapping with experimental data rather than the models with integer order derivatives. The most use of these fractional derivatives is their applications as the modelling term in many elds of sciences such as chemistry, physics, biology, nancial modelling, control theory and other elds [24,27,30]. A new idea of fractional derivative involving two singular kernels has been suggested in [6] and presented its properties and applications to fractional di erential equations. In [14], it has been investigated the Riemann-Liouville and the Caputo fractional Leila Pedram, Department of Applied Mathematics, Imam Khomeini International Davoud Rostamy, Department of Applied Mathematics, Imam Khomeini International, E-mail: rostamyd@yahoo.com derivative of the Dirac delta function and its Laplace transform to explore the solution for fractional-order system. In [44], short memory fractional derivatives and a short memory fractional modelling approach are introduced. In [37], the q-homotopy analysis transform method is used to the mathematical model of the cancer chemotherapy effect in the sense of Caputo fractional. P. Veeresha et al. have used the q-homotopy analysis transform method for nding the solution for a fractional Richards equation describing the water transport in unsaturated porous media and time-fractional coupled Burgers equations [35,39]. In [36], the approximated analytical solutions for nonlinear dispersive fractional Zakharov-Kuznetsov equations are obtained with the help of two novel techniques, called fractional natural decomposition method and q-homotopy analysis transform method.
Khalil et al. [19] introduced a de nition for the fractional derivative which is satis ed most properties of classical derivative despite the other de nition of the fractional derivative. Its applications can be referred to the quantum mechanics and the uid dynamics [4,5,8,20,32,46]. In [45] B. Xin et al. developed Bertrand duopoly game to that based on conformable fractional derivative. S. He et al. solved the conformable fractional memcapacitor system by using conformable di erential transform method [16]. A new class of smooth solutions for the Newton's law of cooling with conformable fractional derivative was gained [28].
When one or more terms of a di erential equation are a stochastic process, it is called stochastic di erential equation (SDE). SDEs have many applications in physics, biology, chemistry, mechanics and economics [9,25,34]. The Korteweg-de Vries (KdV) equation was concluded by Korteweg and de Vries as a model nonlinear equation for the propagation of shallow water waves along a canal and wave motion in plasmas [21], so the KdV equation is useful for the modelling biological and physical phenomenon. The third order nonlinear partial di erential KdV equation happens in many elds of physics such as in water waves, plasmas and ber optics [13,17]. When weakly dispersive long waves would be with moving sur-face pressure or the oor is bumpy, we add a forcing term to the unperturbed KdV equation which will then be called forced KdV equation [3,22]. In [29] Pelinovsky et al. developed the nonlinear-dispersive model of the resonant mechanism of the tsunami wave generation by the forced Korteweg-de Vries equation. In this paper, we centralize on the forced conformable derivative KdV and BBM equations that forcing term is white noise, which is normal if the external stress is produced by a turbulence velocity eld.
Many authors solved KdV equation with di erent ways such as inverse scattering transform [2,15] and tanh-coth method [43]. K. Djidjeliab et al. sketched the numerical methods to solve the third-and fth-order Korteweg-de Vries equations [11]. Also, Mirza and Haider in [26] obtained rational solution of supersymmetric KdV equation. In [41] Wadati considered stochastic KdV equation.
The generalized Benjamin-Bona-Mahony (BBM) equation is ∂u ∂t + u n ∂u ∂x + ∂ u ∂x + u ∂u ∂x = [42]. We consider it when n = which describes shallow water waves in the water and is an alternative to the KdV equation [7].
In this work, we look at the e ects of the external noise on the perturbed solution of the KdV and BBM equations which are the nonlinear partial di erential equations with the conformable derivative (CD) of u of order α for time and space modelled by the stochastic CFDKdV equation and CFDBBM equation for any nite time interval < t ≤ t and the nonhomogeneous term ζ (t) demonstrates Gaussian white noise. The notations T t α u(x, t) := ∂ α u ∂t α for < α ≤ is the conformable derivative of u of order α with respect to t that we will de ne T t α in the next section.
The structure of this paper is organized as follows. We bring the de nition of the conformable fractional derivative and some properties of it in section 2. In section 3, we describe the modi ed Kudryashov method. In section 4 and 6, we convert the stochastic conformable fractional derivative KdV and BBM equations to the unperturbed conformable fractional derivative equations and we solve them with the modi ed Kudryashov method in section 5 and 7. Finally, we show our results in the gures in section 8.

Conformable fractional derivative and its some properties
In this section, we express the de nition and some properties of the conformable fractional derivative.
Someone who wants to know more properties of the conformable fractional derivative like Laplace transform, exponential functions, the Gronwall's inequality, etc. can nd them in [1].

Description of the modi ed Kudryashov method
We use the modi ed Kudryashov method to obtain the exact solutions of the nonlinear conformable derivative partial di erential equations. In this section, we bring a brief review of the modi ed Kudryashov method [12]. Let F be a nonlinear equation u = u(x, t) and it has partial derivatives in the following form where α ∈ ( , ] be the derivative order and T x α (u) and T t α (u) are the conformable derivatives of u with respect to x and t.
Here, we bring the fundamental three steps of the modi ed Kudryashov method: Step 1. We convert (3) from PDE to ODE. For that, we use the transformation (4) Now, we have this nonlinear ODE for new variable η: where R is a function of u(η) and the ordinary derivatives are respect to η.
Step 2. We suppose that the solution of (5) be shown as here, a i , ≤ i ≤ N are constants such that a N ≠ . The N is determined by considering equation (5) such that we investigate homogeneous balancing between the highest order of derivatives and the nonlinear terms of this equation. Furthermore, P(η) = +da η is a solution of the auxiliary equation (7) P where d ≠ , a > , a ≠ .
Step 3. We substitute (6) into (5) along with (7). Finally, a , a , . . . , a N are gained by equating all the coe cients of the powers of P i (η), (i = , , , . . .) to zero. We solve this algebraic equations system by Maple and we can nd amount of a , a , . . . , a N and constants and coe cients which are used in the (4).

The stochastic CFDKdV
In this section, we consider the stochastic CFDKdV (1). First of all, we describe some of the concepts of the stochastic calculus.

De nition 4. [33] For t ∈ T, X t , a collection of random variables, is called a Stochastic process.
Brownian motion was discovered by Robert Brown in 1827 [33].

De nition 5. [33]
A Brownian motion is a Stochastic process X t if it has three following conditions for t ≥ 1. X = .
2. X t increases independently and steadily.
3. X t has Gaussian increments i.e. it is normal with mean and variance σ t.
De nition 6. [33] Let for t ≥ , X t de nes a standard Brownian motion process, the white noise is {dX(t), ≤ t < ∞}.
Wadati in [41] studied the stochastic KdV equation with classic derivative where ζ (t) is the external noise which is dependent on time as follows We use his scheme for solving the stochastic CFDKdV equation (1). The stochastic CFDKdV equation (1) can be transformed into an unperturbed CFDKdV equation by the transformation We have for a function of X and T: We apply the transformations in equations (8) and (9) to equation (1) De ning ζ (t) = ∂ α W ∂T α , for α ∈ ( , ]. Equation (10) becomes the unperturbed CFDKdV equation.

Unperturbed CFDKdV
We solve unperturbed CFDKdV equation (14) by the modied Kudryashov method, The transformation U(x, t) = U(η) that η = k x α α − k t α α , converts the CDKdV equation (14) to where derivative is respect to η. We integrate equation (15), then we have where A is a constant. We obtain N = , due to the homogeneous balance between U and U , the highest order of nonlinear terms and the highest order of derivatives, in (16). Then we assume the solution of equation (5) is Substitution of equation (17) and its second derivative into equation (16) gives We make all the coe cients of the powers of each P(η) equal to zero. Hence We have the above nonlinear system that contains ve algebraic equations for a , a , a and k . By solving this system, So, the solution of equation (16) is . (19) In the original variables, the solution for the equation (14) is

Unperturbed CFDBBM
We solve unperturbed CFDBBM equation (24) by the modi ed Kudryashov method, The transformation U(x, t) = U(η) that η = k x α α − k t α α , converts the CFDBBM equation (24) to where derivative is respect to η. We integrate equation (25), then we have where A is a constant. We get N = , due to the homogeneous balance between U and U , the highest order of nonlinear terms and the highest order of derivatives, in (26). Then we assume the solution of equation (5) is Substitution of equation (27) and its second derivative into equation (26) gives P k a (ln a) + k a + P k a (ln a) − k a (ln a) + k a a + P − k a (ln a) + k a (ln a) + k a − k a + k a + k a a + P k a (ln a) + k a − k a + k a a We make all the coe cients of the powers of each P(η) equal to zero. Hence k a (ln a) − k a (ln a) + k a a = , − k a (ln a) + k a (ln a) + k a − k a k a (ln a) + k a − k a + k a a = , We have the above nonlinear system that contains ve algebraic equations for a , a , a and k . By solving this system, So, the solution of equation (26) is In the original variables, the solution for the equation (24) is

Numerical experimentals the stochastic CFDKdV and CFDBBM
In this section, we illustrate the e ect of noise on the solutions of CFDKdV and CFDBBM equations with di erent values of α. Figures 1 to 10 are related to CFDKdV equation (1) and Figures 11 to 20 are related to CFDBBM equation (2). We show both the solutions of the stochastic and unperturbed forms of equations. Numerical results show that the e ect of noise is greater in the small order of conformable fractional derivative, α, and in α close to one, we see less e ect of the random noise in the solution of CFDKdV and CFDBBM equations.

Conclusion
In this paper, stochastic CFDKdV and CFDBBM equations are considered which are the equations that an external noise inter into them. First, we have converted stochastic CFDKdV and CFDBBM equations to unperturbed equations by de ning a transformation. Then the conformable homogeneous equations are solved with modi ed Kudryashov method analytically. We have shown the solutions of the stochastic CFDKdV and CFDBBM equations in gures. It can be concluded that as α, the order of conformable fractional derivative, gets smaller, the noise e ect will be greater on the solution of CFDKdV and CFDBBM equations. Let the di erential equation (1) or (2) be in form of for α, β ∈ ( , ], in the case of α = β the solution is presented in this paper, but for the case of α ≠ β the de ned transformations in this paper could not work out and we should nd a new way to solve it. Also, applying the numerical methods which are used in [38,40] for solving the equations discussed in this paper can be an interesting topic for future work.

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