Nonlinear Fractional Jaulent-Miodek and Whitham-Broer-Kaup Equations within Sumudu Transform

and Applied Analysis 3 by considering a general fractional nonlinear nonhomogeneous partial differential equation with the initial condition of the following form: D α t U (x, t) = L (U (x, t)) + N (U (x, t)) + f (x, t) , α > 0, (13) subject to the initial condition D k 0 U (x, 0) = gk, (k = 0, . . . , n − 1) , D n 0 U (x, 0) = 0, n = [α] , (14) where D t denotes without loss of generality the Caputo fraction derivative operator, f is a known function, N is the general nonlinear fractional differential operator, and L represents a linear fractional differential operator. Applying the Sumudu transform on both sides of (10), we obtain S [D α t U (x, t]) = S [L (U (x, t))] + S [N (U (x, t))] + S [f (x, t)] . (15) Using the property of the Sumudu transform, we have S [U (x, t)] = u α S [L (U (x, t))] + u α S [N (U (x, t))] + u α S [f (x, t)] + g (x, t) . (16) Now applying the Sumudu inverse on both sides of (12) we obtain U (x, t) = S −1 [u α S [L (U (x, t))] + u α S [N (U (x, t))]] + G (x, t) , (17) where G(x, t) represents the term arising from the known function f(x, t) and the initial conditions. Now we apply the following HPM: U (x, t) = ∞ ∑ n=0 p n Un (x, t) . (18) The nonlinear term can be decomposed to NU(x, t) = ∞ ∑ n=0 p n Hn (U) , (19) using the He’s polynomialHn(U) given as Hn (U0, . . . , Un) = 1 n! ∂ n ∂p [ [ N( ∞ ∑ j=0 p j Uj (x, t)) ] ] , n = 0, 1, 2 . . . . (20) Substituting (15) and (16) gives ∞ ∑ n=0 p n Un (x, t) = G (x, t) + p [S −1 [u α S [L( ∞ ∑ n=0 p n Un (x, t))] + u α S [N( ∞ ∑ n=0 p n Un (x, t))]]] , (21) which is the coupling of the Sumudu transform and the HPM using He’s polynomials. Comparing the coefficients of like powers of p, the following approximations are obtained [29, 30]: p 0:U0 (x, t) = G (x, t) , p 1:U1 (x, t) = S −1 [u α S [L (U0 (x, t)) + H0 (U)]] , p 2:U2 (x, t) = S −1 [u α S [L (U1 (x, t)) + H1 (U)]] , p 3:U3 (x, t) = S −1 [u α S [L (U2 (x, t)) + H2 (U)]] , p n:Un (x, t) = S −1 [u α S [L (Un−1 (x, t)) + Hn−1 (U)]] . (22) Finally, we approximate the analytical solution U(x, t) by truncated series: U (x, t) = lim N→∞ N ∑ n=0 Un (x, t) . (23) The above series solutions generally converge very rapidly [29, 30]. 4. Applications In this section, we apply this method for solving the system of the fractional differential equation. We will start with (1). 4.1. Approximate Solution of (1). Following carefully the steps involved in the STHPM, after comparing the terms of the same power of p and choosing the appropriate initials conditions, we arrive at the following series solutions: u0 (x, t) = G (x, t) = − c1 c2 + 2c1√−α − β 2 sech (c1x) , V0 (x, t) = G1 (x, t) = − c 2 1 (α + β 2 ) + 2c 2 1 (α + β 2 ) sech(c1x) 2 + 2c 2 1 β√−α − β2sech (c1x) tanh (c1x) , u1 (x, t) = S −1 [u α S [L (u0 (x, t)) + H0 (u)]] = c 2 1 t ηsech(c1x) 3


Introduction
Nonlinear partial differential equations arise in various areas of physics, mathematics, and engineering [1][2][3][4].We notice that in fluid dynamics, the nonlinear evolution equations show up in the context of shallow water waves.Some of the commonly studied equations are the Korteweg-de Vries (KdV) equation, modified KdV equation, Boussinesq equation [5], Green-Naghdi equation, Gardeners equation, and Whitham-Broer-Kaup and Jaulent-Miodek (JM) equations.Analytical solutions of these equations are usually not available.Since only limited classes of equations are solved by analytical means, numerical solution of these nonlinear partial differential equations is of practical importance.Therefore, finding new methods and techniques to deal with these type of equations is still an open problem in this area.The purpose of this paper is to find an approximated solution for the system of fractional Jaulent-Miodek and Whitham-Broer-Kaup equations (FWBK) via the Sumudu transform method.The fractional systems of partial differential equations under investigation here are given below.
The nonlinear FWBK equation which will be considered in this paper has the following form: FWBK equation (1) describes the dispersive long wave in shallow water, where (, ) is the field of horizontal velocity, V(, ) is the height which deviates from the equilibrium position of liquid, and  and  are constants that represent different powers.If  = 0 and  = 1, (1) reduces to the classical long-wave equations which describe the shallow water wave with diffusion [6].If  = 1 and  = 0, (1) becomes the modified Boussinesq equations [7,8].FJM equation ( 2) appears in several areas of science such as condense matter physics [9], fluid mechanics [10], plasma physics [11], and optics [12] and associates with energydependent Schrödinger potential [13,14].
The paper is organized as follows.In Section 2, we introduce briefly some of the basic tools of fractional order and of the Sumudu transform method.We show the numerical results in Section 4. The conclusions can be seen in Section 5.

Background of Sumudu Transform
Definition 8 (see [25]).The Sumudu transform of a function (), defined for all real numbers  ≥ 0, is the function   (), defined by Theorem 9 (see [26]).Let () be the Sumudu transform of () such that (ii) there exist a circular region Γ with radius  and positive constants  and  with |(1/)/| <  − , then the function () is given by For the proof see [26].

Basics of the Sumudu Transform Homotopy Perturbation
Method.We illustrate the basic idea of this method [27][28][29][30][31][32] by considering a general fractional nonlinear nonhomogeneous partial differential equation with the initial condition of the following form: subject to the initial condition where    denotes without loss of generality the Caputo fraction derivative operator,  is a known function,  is the general nonlinear fractional differential operator, and  represents a linear fractional differential operator.
Applying the Sumudu transform on both sides of (10), we obtain Using the property of the Sumudu transform, we have Now applying the Sumudu inverse on both sides of ( 12) we obtain where (, ) represents the term arising from the known function (, ) and the initial conditions.Now we apply the following HPM: The nonlinear term can be decomposed to using the He's polynomial H  () given as Substituting ( 15) and ( 16) gives which is the coupling of the Sumudu transform and the HPM using He's polynomials.Comparing the coefficients of like powers of , the following approximations are obtained [29,30]: Finally, we approximate the analytical solution (, ) by truncated series: The above series solutions generally converge very rapidly [29,30].

Applications
In this section, we apply this method for solving the system of the fractional differential equation.We will start with (1).

Approximate Solution of (1)
. Following carefully the steps involved in the STHPM, after comparing the terms of the same power of  and choosing the appropriate initials conditions, we arrive at the following series solutions: And so on in the same manner one can obtain the rest of the components.However, here, few terms were computed and the asymptotic solution is given by  (, ) =  0 (, ) +  1 (, ) +  2 (, ) +  3 (, ) + ⋅ ⋅ ⋅ ,   (2).For (2), in the view of the Sumudu transform method, by choosing the appropriate initials conditions we are at the following series solutions:

Conclusion
We derived approximated solutions of nonlinear fractional Jaulent-Miodek and Whitham-Broer-Kaup equations using the relatively new analytical technique the STHPM.We presented the brief history and some properties of fractional derivative concept.It is demonstrated that STHPM is a powerful and efficient tool for the system of FPDEs.In addition, the calculations involved in STHPM are very simple and straightforward.The STHPM is chosen to solve this nonlinear problem because of the following advantages that the method has over the existing methods.This method does not require the linearization or assumptions of weak nonlinearity.The solutions are not generated in the form of general solution as in the Adomian decomposition method (ADM) [33,34].No correction functional or Lagrange multiplier is required in the case of the variational iteration method [35,36].It is more realistic compared to the method of simplifying the physical problems.If the exact solution of the partial differential equation exists, the approximated solution via the method converges to the exact solution.STHPM provides us with a convenient way to control the convergence of approximation series without adapting ℎ, as in the case of [37] which is a fundamental qualitative difference in the analysis between STHPM and other methods.And also there is nothing like solving a partial differential equation after comparing the terms of same power of  like in the case of homotopy perturbation method (HPM) [38].