On the exact and numerical solutions to a new (2 + 1)-dimensional Korteweg-de Vries equation with conformable derivative

The aim of this paper is to introduce a novel study of obtaining exact solutions to the (2+1) dimensional conformable KdV equationmodeling the amplitude of the shallow-waterwaves in uids or electrostatic wave potential in plasmas. The reduction of the governing equation to a simpler ordinary di erential equation by wave transformation is the rst step of the procedure. By using the improved tan(φ/2)expansion method (ITEM) and Jacobi elliptic function expansion method, exact solutions including the hyperbolic function solution, rational function solution, soliton solution, traveling wave solution, and periodic wave solution of the considered equation have been obtained. We achieve also a numerical solution corresponding to the initial value problem by conformable variational iteration method (C-VIM) and give comparative results in tables. Moreover, by using Maple, some graphical simulations are done to see the behavior of these solutions with choosing the suitable parameters.


Introduction
Nonlinear evolution equations (NLEEs) have been used for many years to express the modern world phenomena we encounter in nonlinear sciences such as mathematical biology, plasma physics, elastic media, nance, uid mechanics, control theory, chemistry, optics, engineering sciences, etc. The concept of fractional derivative operator dates back to the work of L'hospital in the 17th century. We need fractional partial di erential equations (FPDEs) to be able to and interpret physical models that occur in most applied sciences. Recently, we have been observing these equations, especially in physical models that contain space and time variables. In fact, fractional derivatives better explain the various physical phenomena encountered. It is quite crucial to examine fractional spacetime di erential equations which are nonlocal operator instead of integer ordered di erential operator which is a local operator. For, as is known, this means that the next state of a system depends not only on its present state but also on all its former states. This is the main advantage of fractional di erential equations over integer-order di erential equations and this makes the model created more realistic [1]. There are several de nitions of fractional derivative, such as Riemann-Liouville, Caputo, Caputo-Fabrizio, Atangana-Baleanu [2][3][4][5][6].
The classical ( + )-dimensional KdV equation is a well known equation that is utilized to characterize the waves on shallow water surfaces. In addition, this equation has a wide range of applications in various branches such as bubble liquid mixtures, waves in enharmonic crystals, ion acoustic wave and magneto-hydrodynamic waves in a warm plasma, cold lossless (collisionless) plasma as well as shallow water waves [32][33][34][35][36][37]. One of the main features of the KdV equation is that the speed of solitary wave is related to the magnitude of the solitary wave, and the other is that the solutions can represent solitary wave solutions noted as solitons which have quantum mechani-cal e ects which occur in particle physics and quantum eld theory [33,37]. Many researchers have investigated numerous versions of this famous equation with di erent procedures and techniques [38][39][40][41][42][43][44][45][46][47][48][49]. A few of the various interesting features of this well-known equation are an in nite number of conservation laws (higher order), bi-Hamilton structures, symmetries and the Lax pair. The ( + )-dimensional KdV equation was derived by Boiti et al. using the idea of the weak Lax pair [50]. For v = u and y = x, Eq. (2) reads the ( + )-dimensional KdV equation. Compared to ( + )dimensional case, the ( + )-dimensional case explains a more involved nonlinear phenomenon. As highlighted in the lines above that physical systems can be better expressed with fractional order derivatives used for global modeling.
The main objective of this paper is to use ITEM to nd the new exact solutions of the time fractional ( + )dimensional KdV equation given as folllows where D α t (.) is conformable derivative of order α. When α = , Eq. (3) changes to the ( + ) -dimensional KdV equation which has been proved based on the extended Lax pair and has been announced in [46].
The organization of this paper is as follows: In Section 2, some basic de nitions of the conformable derivative are recalled. In Sections 3,4 and 5, the key idea of ITEM, Jacobi elliptic function expansion method and conformable variational iteration method is described. In section 6, the acquisition of the considered physical model is brie y presented. In Sections 7,8 and 9, applications of the methods are given. Finally, some conclusions are presented in the last section.

Conformable derivative
R. Khalil et al. proposed a derivative that coincides with the classical derivative when α = and that can rectify the shortcomings of the previous de nitions. Here, de nition and some properties of the conformable derivative are presented [7,8].

Algorithm of the improved tan(ϕ/ )-expansion method for NLEEs
This method was summarized and improved for achieving the analytic solutions of NLEEs by Mana an et al in 2015 [51]. Assume a nonlinear partial di erential equation is given in general form as follows After simple algebraic operations, this equation is transformed into an ordinary di erential equation (ODE) with ξ = x − µt transformation Q(u, u , −µu , u , µ u , ...) = .
Then, assume that the searched wave solutions of Eq. (5) have the following representation and B k ( ≤ k ≤ m) are constants to be determined and p is arbitrary constant, such that Am ≠ , Bm ≠ and ϕ = ϕ(ξ ) is the solution of the following rst order di erential equation: If we try to nd the solution of the (7), then we obtain special solutions that vary according to the state of the coe cients:  As usual, for determining m, the highest order derivative should be balanced with the highest order nonlinear terms in Eq. (5). In the case of m = q/p (where m = q/p be a fraction in the lowest term), we need to do a conversion on the unknown function u as follows: Then substitute Eq. (8) into Eq. (5). By using of new Eq. (5), the value of m can be determined. If m be a negative integer, similar process can be followed with the transformation u(ξ ) = (v(ξ )) m .
Following these operations, according to the m value obtained above, let substitute (6) into Eq. (5). Therefore we obtain a set of algebraic equations that contains tan(ϕ/ ) k , cot(ϕ/ ) k , (k = , , , ...). Then setting each coe cients of tan(ϕ/ ) k , cot(ϕ/ ) k to zero, we can get a set of over-determined equations for A , A k , B k (k = , , ..., m), a, b, c and p . Since obtained algebraic equations system will be di cult to solve manually, symbolic computation as Maple can be used at this stage. Finally, A , A , B , ..., Am , Bm , µ, p are replaced in the Eq. (6).

Algorithm of the Jacobi elliptic function expansion method
In this section, we recall the Jacobi elliptic function expansion method [52]. Consider a nonlinear partial di erential equation is (9) where u = u(ξ ), u = du dξ , ....In order to construct more general periodic and solitary wave solution of Eq. (3) by employing the Jacobi elliptic function expansion method, it is assumed that u (ξ ) can be formulated as a nite series of Jacobi elliptic sine and cosine functions. The ansatz are given below and where n, a j and b j (j = , , , , ...) are constants. sn (ξ ) = sn (ξ |m) and cn (ξ ) = cn (ξ |m) where m ( < m < ) is called a modulus of the elliptic function, are double periodic and satisfy the following properties: The value of n is determined againby balancing the nonlinear term and the highest derivative. Therefore, the highest degree of d p u dξ p is taken as O d p u dξ p = n + p, p = , , , ... (13) and the nonlinear term as O u q d p u dξ p = (q + ) n + p, q = , , , , .... (14) Then substituting the ansatz (11) and (12) into Eq. (10) and equating the coe cients of all powers of elliptic functions to zero, we get a system of algebraic equations for a j and b j (j = , , , , ...).

Succint overview of the conformable variational iteration method
In this section, we will present how the conformable variational iteration method (C-VIM) works for conformable nonlinear evolution type equations [53,54]. Let us assume that the following conformable nonlinear evolution equations in operator form where L is a linear operator, N is a non-linear operator, g is an non-homogeneous term and t Tα is conformable derivative of order α. To solve di erential equation (15) via C-VIM write the di erential equation (15) in the form by Theorem 1 (property (vi)), As in classical variational iteration method, the trial functional for (16) can be constructed as where λ is a general Lagrangian multiplier and it can be optimally determined by the aid of variational theory [55][56][57]. Hereũn is a restricted variation [55][56][57] where δũn = 0. As the rst step of this approach, λ multiplier should bederived by the help of variational theory and integration by parts. Using the determined Lagrangian multiplier and any selected function u , the u n+ iteration, which is the successive approximations of u (x) for n ≥ , will be obtained readily. Hence, we get the solution as

Governing equation
Recently in 2019, a new (2+1)-dimensional KdV equation has been proved based on the extended Lax pair [46]. To derive of the ( + )-dimensional KdV equation, ( + )dimensional zero curvature equation [42,58,59] considered where with the compatibility conditions and ζ t = . Plugging (18) into Eq. (17), a system of algebraic equations is obtained. Based on the work done by Ablowitz [42], Relations between coe cients is found by using algebraic equation system and series expansions of functions [46]. If n = , the new ( + )-dimensional Schrödinger equation [59] is obtained If n = , with the appropriate selection of coe cients [46], it immediately generates the following new ( + )dimensional KdV equation If (20) will be discussed .

Application of ITEM to conformable (2+1) dimensional KdV equation
In this section, we apply the ITEM to Eq. (3) to obtain the traveling wave solutions. In this context, let us consider u(x, t) = u(ξ ), ξ = kx + ry − w α t α and therefore Eq. (3) becomes where " " shows the derivative according to ξ . By integrating (21) once with respect to ξ , we obtain −wu − ( k − r)u − (k − r − rk + r k)u = . (22) With balancing prodecure, where u derivative is balanced by u , m + = m, then m = is obtained. Therefore, by considering p = in (6), we get the following nite series expansion for unknown function of u(ξ ) We substitute the expression of u in (23) into (22) and collect all terms with the same order of tan(ϕ(ξ )/ ), cot(ϕ(ξ )/ ) together. Then by equating the coe cient of each polynomial to zero, we obtain a set of algebraic equations . . .
Solving the above algebraic equations (24) by help of Maple, we have numerous sets of coe cients for the solutions of (22). We only choose some of them as follows:

SET 1
We have yielded the arbitrary constants as By using Family 1, (23) becomes By using Family 2, (23) reads By using Family 5 , one constructs for (23) By using Family 8, (23) can be written as By using Family 11, we can write By using Family 12, (23) becomes By using Family 13, (23) reads By using Family 17, one constructs for (23) By using Family 18, we get

SET 2
We have yielded the arbitrary constants as    By using Family 1, (23) can be written as where By using Family 2, (23) becomes where By using Family 3, we obtain By using Family 4, one constructs for (23) By using Family 5, (23) can be written as By using Family 11, we can write By using Family 12, (23) reads By using Family 13, one constructs for (23) By using Family 14, we get where By using Family 15, we can write By using Family 17, (23) becomes u (x, y, t) = − (−r + k) (kx + ry + C) . (43) By using Family 18, (23) reads

SET 3
We have yielded the arbitrary constants as By using Family 1, (23) can be written as where By using Family 2, (23) becomes where By using Family 8, (23) reads u (x, y, t) = − (k − r) a (akx + ary + aC + ) . (47) By using Family 11, one constructs for (23) By using Family 12, (23) can be written as By using Family 13, (23) can be written as
By using Family 11, (23) reads By using Family 12, we get By using Family 13, (23) becomes By using Family 15, (23) can be written as
By using Family 18, (23) reads By using Family 19, one constructs for (23) u

SET 5:
We have yielded the arbitrary constants as By using Family 1, (23) becomes By using Family 2, (23) can be written as By using Family 8, (23) reads .
By using Family 11, one constructs for (23) By using Family 12, (23) becomes where By using Family 13, (23) can be written as By using Family 16, (23) becomes .

SET 6:
We have yielded the arbitrary constants as By using Family 1, (23) reads By using Family 2, we get By using Family 4, one constructs for (23) u By using Family 5, (23) reads By using Family 8, (23) can be written as By using Family 11, (23) becomes By using Family 12, one constructs for (23) By using Family 13, we get By using Family 17, (23) becomes u (x, y, t) = − (k − r) (kx + ry + C) .
Substituting Eq. (64) into Eq. (63) and collecting various powers of sn(ξ ), we get kr a m + k a m − r a m − ra − k ra m + ka = , k a m + kr a m − r a m − k ra m + ka a − ra a = , wa + r a m + r a − k a m − kr a m − kr a − k a + ka a + ka + k ra − ra a + k ra m − ra = , − kr a m + wa + k ra m + ka a − k a m + r a + k ra − k a + r a m − ra a − kr a = k a + wa + kr a − r a − ra − k ra + ka = .
By solving the above system of equations using any package of symbolic computations, we can determine the values of the coe cients as: Substituting Eq. (68) into Eq. (63) and collecting various powers of cn(ξ ), we get an algebraic equations system by solving this system of equations using any package of symbolic computations, we can determine the values of the coe cients as: Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Con ict of interest:
The authors state no con ict of interest.