A NOVEL APPROACH FOR FRACTIONAL KAWAHARA AND MODIFIED KAWAHARA EQUATIONS USING ATANGANA-BALEANU DERIVATIVE OPERATOR

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this manuscript, iterative laplace transform method is applied to obtain approximate solutions of the nonlinear time fractional Kawahara and modified Kawahara equations based on Atangana-Baleanu derivative operator. The noticeable features of the manuscript is to providing the existence and uniqueness conditions of solution for proposed technique and the graphical presentations of numerical solution of the concerned equations for various specific cases. The obtained approximate solutions are compared with the exact solutions to verify the applicability, efficiency and accuracy of the method.


INTRODUCTION
Nonlinear wave phenomena play an essential role in various parts of mathematical physics and engineering such as dispersion, diffusion, reaction and convection. One such well-known nonlinear evolution equations is the fifth order Kawahara equation. These equation appeares in the study of shallow water waves having magneto-acoustic waves in a plasma, surface tension and capillary-gravity waves. Furthermore, these equations becomes the area of active research in recent times [28,29,30]. In 1972, Kawahara [32] proposed the kawahara equation first to describe solitary-wave propagation in media. Furthermore, the modified Kawahara equation has some useful applications in physics such as,capillary-gravity water waves, plasma waves, water waves with surface tension, etc. [31,33,34,35,36].
Fractional calculus allows differentiation and integration of arbitrary order and hence, becomes more popular in the past few decades in various fields of science and engineering, such as fluid mechanics, diffusive transport, electrical networks, electromagnetic theory, different branches of physics, biological sciences and groundwater problems , [1,2,3,4,5]. There are several useful applications of fractional calculus such as dissipation [19], modelling of processes such as anomalous diffusion [20,21], control theory [22], relaxation [23], etc. Many mathematicians and researchers have tried to model several physical or biological processes using fractional differential equations. Solving these equations is turn out to be wide area of research and interest for researchers from various fields. Some of the recent analytical and numerical methods for solving linear and nonlinear fractional differential equations are the Adomian decomposition method ADM [6,7,8], Variational iteration method VIM [9,10], Homotopyperturbation method HPM [11], Homotopy analysis method [12], Finite difference method [13], monotone iterative method [15] and so on. In recent times, an iterative method was proposed by Daftardar-Gejji and Jafari [17,18] which is known as new iterative method (NIM). This method is very useful and simple in fractional calculus for solving linear and nonlinear fractional partial differential equations.
However these operators possesses a power law kernel, exponential kernel and has singularity.
Hence these operators posseses some limitations in modelling physical problems. To overcome this difficulty, in recent times Atangana and Baleanu have proposed a reliable operator having nonlocal and nonsingular kernel in the form of Mittag-Leffler function known as Atanagana-Baleanu operator [27].
Motivated by above, in this paper we have applied iterative Laplace transform with to find approximate solutions of time fractional Kawahara and modified Kawahara equations having Atanagana-Baleanu operator. These equations are given below as follows: where m, n are nonzero real constants and initial condition is.
Equations (1.1) and (1.3) becomes the original Kawahara and modified Kawahara equations for The rest of this paper is organized as follows. In Section 2, preliminaries of fractional calculus is presented. In Section 3 basic idea of iterative Laplace transform method is given. Section 4, presents existence and uniqueness criteria for obtained solutions of time fractional Kawahara and modified Kawahara equations. The numerical results and plots for the obtained solutions are presented in Section 5. Finally, we give our conclusions in Section 6.

PRELIMINARIES
Definition 2.1. The left and right sided Caputo fractional derivative for order κ > 0 is defined is a normalization function and E κ (·) is the Mittag-Leffler function.
is a constant, integral will be resulted with zero.

FUNDAMENTAL IDEA OF ITERATIVE LAPLACE TRANSFORM METHOD
In this section, we consider the arbitrary fractional order differential equation with Atangana Baleanu operator to demonstrate the fundamental solution procedure of the proposed algorithm by Daftardar-Gejji and Jafari [17] with initial condition Where f (x,t) denotes source term, R and N are given linear and non-linear operator respectively. Applying Laplace transform on (3.1) we get Rearranging terms we get Next, we apply inverse laplace transform on (3.4) then we get Further, we apply iterative method [17]. We consider series solution as below given by, since R is linear, The nonlinear operator N is decomposed as further, consider the recurrence relation as given below . . .
The n−term approximate solution is given by

EXISTENCE
where ABC D κ represents the fractional operator of type Atangana-Baleanue-Caputo (ABC) hav- The equation (4.1) can be converted to the Volterra-type integral equation by using the ABC fractional integral which is written by referring definition 2.3 as follows: Consider u 1 and u 2 be two functions, then we obtain the following where x 5 are the differential operators. Since, u 1 and u 2 are bounded functions, we have u 1 ≤ ρ 1 u 2 ≤ ρ 2 .
The recursive form of equation (4.2) defined as follows Next, we get the difference between the successive iterative terms in the form of following By using equation (4.4) and then using norm on equation (4.5), we get This completes the proof of the theorem.
Proof: First, we consider bounded function u(x,t) satisfying the Lipschitz condition. From equations (4.5) and (4.7) we get the following equation Hence, the solution is smooth, moreover existence is proved for the obtained solution. Next, we show that the equation (4.8) is the solution for the equation (4.1). For this, we consider where δ n (x,t) are reminder terms of series solution. Then, we must show that these terms approach to zero at infinity, that is, δ ∞ (x,t) → 0.
Therefore, Continuing this way recursively at t 0 we get: where M = u−u n−1 . After taking limit of both sides as n tends to infinity, we get δ n (x,t) →

0.
Next, it is necessity to demonstrate uniqueness for the solution of the proposed problem.
Suppose, u * (x,t) be the another solution, then we get On applying norms on both side of above equation we get After simplification we get λ ≤ 0. (4.14) From the above inequility, we get that if Therefore, (4.15) is required condition for uniqueness.
Also λ 1 = τ 4 3 (ρ 2 3 + ρ 3 ρ 4 + ρ 2 4 ) + τ 5 + τ 6 . Therefore, we have This shows that the Lipschitz condition is obtained for H . Moreover, we see that if 0 ≤ τ 4 3 (ρ 2 3 + ρ 3 ρ 4 + ρ 2 4 ) + τ 5 + τ 6 < 1, then it implies the contraction. The recursive form of equation (4.17) defined as follows Next, we get the difference between the successive iterative terms in the form of following expression ϑ n (x,t) = u n (x,t) − u n−1 (x,t) By using equation (4.19) and then using norm on equation (4.20), we get Proof: First, we consider bounded function u(x,t) satisfying the Lipschitz condition. From equations (4.20) and (4.22) we get the following equation Hence, the solution is smooth, moreover existence is proved for the obtained solution. Next, we show that the equation (4.23) is the solution for the equation (4.16). For this, we consider where ψ n (x,t) are reminder terms of series solution. Then, we must show that these terms approach to zero at infinity, that is, ψ ∞ (x,t) → 0.
Therefore, continuing this way recursively at t 0 we get: where M = u−u n−1 . After taking limit of both sides as n tends to infinity, we get ψ n (x,t) → 0.
Next, it is necessity to demonstrate uniqueness for the solution of the proposed problem. Suppose, u * (x,t) be the another solution, then we get On applying norms on both side of above equation we get After simplification we get From the above inequility, we get that if Therefore, (4.30) is required condition for uniqueness.

NUMERICAL APPLICATION
In this section, we demonstrate the efficiency of iterative Laplace transform method by ap-  The exact solution to (1.1) is given in [33] as The initial condition (1.2) is rewritten as Applying laplace transform on both side of (1.1) we get Rearranging terms we get Further, the inverse Laplace transform on (5.3), yields The series solution obtained by the method is given by, The nonlinear term u ∂ u ∂ x is written as u n ∂ u n ∂ x = ∑ ∞ n=0 P n ; whereas P n is further decomposed as follows we get the recursive formula as follows The n−term approximate solution is given by (5.7) u(x,t) = u 0 (x,t) + u 1 (x,t) + u 2 (x,t) + · · · + u n−1 (x,t).

CONCLUSIONS
In this study, we have obtained the approximate solutions of time fractional Kawahara and modified Kawahara equations based on Atangana-Baleanue fractional derivative operator by using Laplace transform with iterative method. It is seen that the solutions obtained converges very rapidly to the exact solutions in only second order approximations i.e. approximate solutions are very near to the exact solutions. We can conclude from the numerical results that the present technique is straightforward, efficient and provides very high accuracy. This is very simple, reliable and powerful technique for finding approximate solutions of many fractional physical models arising in science and engineering.