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