ANALYSIS OF FRACTIONAL KAWAHARA AND MODIFIED KAWAHARA EQUATIONS BASED ON CAPUTO-FABRIZIO DERIVATIVE OPERATOR

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, nonlinear time fractional Kawahara and modified Kawahara equations based on CaputoFabrizio derivative operator is analysed using iterative Laplace transform method to obtain approximate solutions. The substantive features of the manuscript is to offer the stability conditions of solution for proposed technique. The acquired approximate solutions are in comparison with the precise solutions to confirm the applicability, performance and accuracy of the method. Moreover, the 3D plots of obtained numerical solution of the concerned equations for various specific cases are presented.


INTRODUCTION
From the past three decades, the most captivating rise in scientific and engineering applications have been found within the framework of Fractional calculus. It has fascinated the attention of many scholars due to its usefulness 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]. In recent times, many scholars have tried to model various physical or biological processes using fractional differential equations. Moreover, obtaining numerical solutions of these equations is turn out to be wide area of research and interest for researchers. Some of the most used and efficient analytical or numerical methods for solving these fractional differential equations are given as the Finite difference method [8], Adomian decomposition method ADM [6,7], Homotopyperturbation method HPM [9], Homotopy analysis method [10], Adams-Bashforth-Moulton method [11], Variational iteration method VIM [12,13], monotone iterative method [14,15], etc. Recently, Daftardar-Gejji and Jafari [16] suggested an iterative method which is known as new iterative method (NIM). Furthermore, applying Laplace transform utilizing NIM [17] is turned out to be most efficiect and reliable method in fractional calculus for solving linear and nonlinear fractional partial differential equations.
These FDEs involves several fractional differential operators like Riemann-Liouville operator [18], Caputo operator [19], Hilfer operator [20], Katugampola operator [21], etc. However these operators possesses a power law kernel and has singularity which leads to some limitations in modelling physical problems. To overcome this difficulty, in recent times Caputo and Fabrizio have proposed a reliable operator having nonlocal and nonsingular kernel in the form of exponential function known as Caputo-Fabrizio operator [22,23].
Nonlinear wave phenomena has significant importance in various parts of mathematical physics and engineering such as dispersion, reaction, diffusion and convection. Moreover, one of the well-known nonlinear evolution equation is the fifth order Kawahara equation which appeares in the study of shallow water waves having magneto-acoustic waves in a plasma, surface tension and capillary-gravity waves. This equation has attarcted several authors in recent times [24,25]. To describe solitary-wave propagation in media, in 1972, Kawahara [26] suggested the kawahara equation. Moreover, the modified Kawahara equation has some useful applications in physics such as,capillary-gravity water waves, plasma waves, water waves with surface tension, etc. [27,28,29].
Inspired by above literature, in this paper we have applied iterative Laplace transform with to find approximate solutions of time fractional Kawahara and modified Kawahara equations having Caputo-Fabrizio operator. These equations are given below as follows: where h, l are nonzero real constants and initial condition is Equations (1.1) and (1.3) becomes the original Kawahara and modified Kawahara equations for β = 1 [26] The remaining of this manuscript is arranged as below. Section 2 is presentation of some basic definitions and lemmas of fractional calculus. Preliminary idea of iterative Laplace transform method is illustrated in section 3. In Section 4, stability criteria for obtained approximate solutions of considered equations are displayed. The numerical simulations, plots and tables for the obtained solutions are demonstrated in section 5. In section 6, we give our conclusions.

BASICS OF FRACTIONAL CALCULUS
In this section, we present some useful definitions and lemmas of fractional calculus.
Similar to Caputo derivative operator, the CF operator gives CF D The benefit of Caputo-Fabrizio operator is that there is no singularity for t = s in the new kernel as compared to Caputo operator In particular, we have

ITERATIVE LAPLACE TRANSFORM METHOD
In this section, a general nonhomogeneous Caputo-Fabrizio fractional differential equation is considered which is given as below with initial condition Where g(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 (3.5) then we get where ψ(x,t) is the term derived from source term.
Further, we use new iterative method to obtain infinite series solution. This method is introduced by Daftardar-Gejji and Jafari [16].
since R is linear, The decomposion of nonlinear operator N is given as In view of (3.7), (3.8) and (3.9), the equation (3.6) is equivalent to further, consider the recurrence relation as follows The approximate solution with p−term is given as The convergence condition of the above approximate solution is obtained in [32] 4. STABILITY ANALYSIS 4.1. Stability analysis of the fractional Kawahara equation. Let (B, · ) as a Banach space. Further, define Γ as self-map of B. and ν m+1 = f (Γ, ν m ) shows exact recurring process.
The fixed-point set on Γ is denoted by F(Γ). Moreover, Γ has atleast one element such that Comparably, we think about that, this sequence {ω m } has an upper bound. This iteration is called as Picard's iteration and it is Γ− stable, if all these criterias are fulfilled for ν m+1 = Γν m .
Theorem 4.1. Consider a Banach space (B, · ) and define Γ as self-map on B fulfilling Proof. Here, we will show that Γ consists a fixed point. Hence, for all (m, r) ∈ N × N, we consider the following.
By applying norm on both sides of (4.3) and without loss of generality, we obtain Next, utilizing triangular inequality and simplifying further (4.4) we get, Since, u m and u r are bounded functions, we have u m ≤ δ 1 and u r ≤ δ 2 . Therefore, simplifying (4.8), we obtain Hence, the self-mapping Γ has a fixed point. This completes the proof.
Further, we prove that Γ satisfies all the criterias in Theorem 4.1. Let (4.9) holds then using Theorem 4.4. Consider a self-map Θ defined as Proof. Here, we will show that Θ consists a fixed point. Hence, for all (m, r) ∈ N × N, we consider the following.
By applying norm on both sides of (4.13) and without loss of generality, we obtain Next, utilizing triangular inequality and simplifying further (4.14) we get, Since, u m and u r are bounded functions, we have u m ≤ δ 3 and u r ≤ δ 4 . Therefore, simplifying (4.18), we obtain where F 4 , F 5 and F 6 are functions of Hence, the self-mapping Θ has a fixed point. This completes the proof.
Further, we prove that Θ satisfies all the criterias in Theorem 4.3. Let (4.19) holds then using

NUMERICAL SIMULATIONS
This section deals with the interpretation of the analytical results for the time fractional Kawahara and modified Kawahara equations with the graphical illustrations. We have used Mathematica software to compute approximate solutions. The exact solution to (1.1) is given in [33] as

Approximate solution for time fractional Kawahara equation
The initial condition (1.2) is rewritten as Applying laplace transform on both side of (1.1) we get Rearranging terms we obtain The series solution is given as follows, The nonlinear term v ∂ v ∂ x is written as v n ∂ v n ∂ x = ∑ ∞ n=0 P n ; whereas P n is further decomposed as follows The n−term approximate solution is given by Therefore, using (5.6) the first three terms of approximate solution of (1.1) are obtained as Continuing in the same way, remaining terms of the iteration formula (5.6) are obtained. The exact solution for the classical modified Kawahara equation is given by [33] Applying laplace transform on both side of (1.3) we get, Rearranging terms we obtain, Next, the inverse Laplace transform on (5.10), gives The series solution is given as, The nonlinear term v 2 ∂ v ∂ x is written as v 2 n ∂ v n ∂ x = ∑ ∞ n=0 J n ; whereas J n is further decomposed as follows The n−term approximate solution is given by

CONCLUSIONS
In this work, iterative Laplace transform method is applied lucratively to obtain the approximate solutions of time fractional Kawahara and modified Kawahara equations based on Caputo-Fabrizio fractional derivative. We have also obtained the stability conditions of approximate solution. The present investigation illuminates the effetiveness of the considered derivative operator. It is seen that the results obtained byiterative Laplace transform method are more stimulating as compared to results available in the literature. We can conclude from the numerical results that this is very simple, reliable and powerful technique for finding approximate solutions of many fractional physical models arise in applied sciences.