Comparative study of two techniques on some nonlinear problems based ussing conformable derivative

Abstract In this paper, three eminent types of time-fractional nonlinear partial differential equations are considered, which are the fractional foam drainage equation, fractional Gardner equation, and fractional Fornberg–Whitham equation in the sense of conformable fractional derivative. The approximate solutions of these considered problems are constructed and discussed using the conformable fractional variational iteration method and conformable fractional reduced differential transform method. The conformable derivative is one of the admirable choices to handle nonlinear physical problems of different fields of interest. Comparisons of approximate solution obtained by two techniques, to each other and with the exact solutions are also presented and affirm that the considered methods are efficient and reliable techniques to study other nonlinear fractional equations and models in the sense of conformable derivative. To explain the effects of several parameters and variables on the movement, the approximate results are shown in tables and two-and three-dimensional surface graphs.


Introduction
Fractional partial di erential equations are used to model most of the natural problems, such as nance, physical, chemical, uid dynamics, biological and other elds of engineering. For diagnosing these physical models as well as in addition follow these physical models in realistic med-ical studies, it is essential to discover their approximate and exact solutions. In calculus a lot of numerical, analytic and approximate methods are developed for nonlinear models [1][2][3][4][5][6][7].
There are miscellaneous researches correlated to this derivative. Atangana et al. [8] give a few de nitions of the conformable derivative and time-space fractional heat differential equations being resolved with the aid of the conformable derivative delivered by Cenesiz and Kurt [9]. New exact solutions of conformable Burgers' type equations obtained by Çenesiz et al. [10]. Conformable di erential method was introduced by Ünal and Gökdoğan [11]. In addition, the conformable fractional method has been carried out to many models with the aid of many authors [12][13][14][15]. By using conformable operator Yavuz explains the dynamical behaviors of models [16,17]. The conformable fractional derivative is taken into consideration for this examine because it is simple for the calculation. A Chinese professor He [18,19] introduced variational iteration method. Such as linear and nonlinear wave equation, systems of fourth-order di erential equations are solved by the variational iteration method [20]. Conformable variational iteration method is built on the conformable derivative for fractional ordinary di erential equations and fractional partial di erential equations [21].
Keskin and Oturanc [22,23] introduced reduced di erential transform method. The technique gives a dependable and pro cient process for extensive kind of chemical engineering, medical and uid model problems, inclusive of fractional order or non-fractional partial di erential equations [24]. By the de nition of conformable derivative, Acan et al. [25] o ered a new form of reduced di erential transform technique that is known as conformable reduced di erential transform, implemented to a few fractional di erential equations.
Take a look, foam drainage equation is very momentous because many technological and commercial programs were developed, which consist cleansing, water puri cation, mineral extraction. Numerical solution of fractional di erential equations is supplied in [26]. Traveling Wave solution of fractional di erential equations with the aid of using rational (G / G)-expansion technique is presented in [27].
We analyzed the time fractional Gardner equation. The Gardner equation is an advantageous example for the de nition of interior solitary waves in shallow water, while Buckmaster's equation is applied in thin viscous uid sheet ows. Gardner equation is extensively used in several physics and engineering problems consisting of uid physics, chemical engineering, and quantum discipline principle. The equation presents outstanding function in ocean waves. In shallow seas Gardner equation de nes internal solitary waves. This equation scrutinized within the literature because it's far used to model a di usion of nonlinear phenomena. Gardner equation is solved by way of using F-expansion in [28] and has been generally examined by several other methods [29][30][31].
The Fornberg-Whitham equation has been found to require peakon results as a simulation for limiting wave heights as well as the frequency of wave breaks. In fractional calculus has gained considerable signi cance and popularity, primarily because of its well-shown applications in a wide range of apparently disparate areas of engineering and science. Fractional Fornberg-Whitham equation is solved by using fractional variational iteration method in [32]. Kumar [33] to resolve the fractional Fornberg-Whitham equation via Laplace transform. Besides, the Laplace Adomian decomposition approach is likewise used in [34]. Many scholars have therefore researched the fractional extensions of the Fornberg-Whitham model for the Caputo fractional-order derivative [35][36][37].

Basic concept for fractional calculus
Some primary de nitions of fractional calculus are recalled which can be practiced in the calculation. It is famous that there are precise de nitions of fractional Integral and fractional derivatives, which includes, Grünwald-Letnikov, Riesz, Riemann-Liouville, Caputo, Hadamard and Erdélyi-Kober and lots of others [38][39][40]. De nition 2.1 The fractional integral operator of RL of order α ≥ of a function f ∈ Cµ, µ ≥ − is de ned as

De nition 2.2 The Caputo's fractional derivative is [41]
De nition 2.3 Let f be an n time di erentiable at x then the conformable fractional derivative is well-de ned by [41] ( If the above limit exists, then f is called αdi erentiable. Let α ∈ ( , ]and f , g be α-di erentiable at a point x > 0, then Tα satis es the following properties: 1.

Analysis of conformable fractional variational iteration method (CFVIM)
Firstly the conformable fractional variational iteration method is discussed for the solution of the following nonlinear time fractional partial di erential equation where L and N is linear and non-linear operator respectively, h(x) is source term and T α t is conformable fractional derivative of order α. To solve di erential Equation (5) via conformable fractional variational iteration method, write the di erential Eq. (5) in the form, As in conformable fractional variational iteration method, the correction functional for Eq. (6) can be Finally, the solution is

Analysis of conformable fractional reduced di erential transform method (CFRDTM)
Let w(x, t) is continuous and analytic with respect to time t and space x. conformable fractional reduce di erential transform of w(x, t) is de ned as Where some 0 < α ≤ 1, α is a parameter labeling the order of CFD, The inverse conformable fractional reduce di erential transform of W α k (x) is de ned as (10) For integer order derivatives the initial conditions for conformable fractional reduce di erential transform are de ned as where n is the order of conformable fractional partial differential equation. andw The solution is written as Numerical applications . The foam drainage equation subject to condition According to conformable fractional variational iteration method, the correctional functional of Eq. (15) is read as Consequently, we get Now, by applying conformable fractional reduce differential transform on Eq. (15) and (16), we get The series solution isw The exact solution of Eq. (15) is Here c is the speed of wave.

. Time-fractional Gardner equation
with the initial condition w(x, ) = + tanh x .
For ε = , (30) will reduce into According to conformable fractional variational iteration method, the correctional functional of Eq. (32) is Consequently, we get w (x, t) = − tSech x ( + Cos (x) + Sin (x)) + + Tanh And so on. Now, by applying conformable fractional reduce di erential transform on Eq. (32) and (31), we get The series solution is The exact solution of Eq. (30) is  . Time-fractional Fornberg-Whitham equation   subject to initial condition w(x, ) = e x .
According to conformable fractional variational iteration method, the correctional functional of Eq. (44) is Consequently, we get And so on. Now, by applying conformable fractional reduce di erential transform on Eq. (44) and (45), we get The series solution isw (x, t) = e x + α e x t + α e x t + α e x t + ..., The closed form solution of Eq. (44) is

Results and discussions
From the above results, Figure 1 Table 1, 2 and 3 numerically explore the comparison of obtained solutions with the exact solution. Absolute errors also determine to show the worth of these techniques.

Conclusion
In this study, we successfully applied the conformable fractional di erential transform method and conformable variational iteration method on the foam drainage equation, time-fractional Gardner equation and timefractional Fornberg-Whitham equation. We showed the existence and uniqueness of the solutions. The solutions obtained by these methods are compared with the exact solution which is presented in graphs. From the results, given graphs, it is clear that for these models CFVIM gives more accurate results as compare to the CFRDTM using only a few iterates. Numerical results indicate that these techniques are very good results when looked carefully at the graphs and are very powerful, simple and compatible for solving the nonlinear FPDEs and allow us to understand the dynamics of nonlinear mathematical model of propagation. In this work Mathematica software is used for computing the results, tables and graphs. From these conclusions, we can say that the CFVIM with fractional derivative is suitable for examining many problems in the elds of science and engineering.

Con ict of interest:
The authors declare no con ict of interest regarding the publication of this paper.