Approximate Analytic Solutions of Time-Fractional Hirota-Satsuma Coupled KdV Equation and Coupled MKdV Equation

and Applied Analysis 3 Theorem 5. If u(x, t) = f(x)g(t), function f(x) = xh(x), where λ > −1 and h(x) has the generalized Taylor series expansion h(x) = ∑∞ n=0 a n (x − x 0 ) αn, (i) β < λ + 1 and α arbitrary, or (ii) β ≥ λ+1, α arbitrary, and a n = 0 for n = 0, 1, . . . , m− 1, wherem − 1 < β ≤ m, then the generalized differential transform (8) becomes U α,β (k, h) = 1 Γ (αk + 1) Γ (βh + 1) [D αk x0 (D β t0 ) h u (x, t)] (x0 ,t0) . (9) If u(x, t) = Dγ x0 V(x, t), m − 1 < γ ≤ m, and V(x, t) = f(x) g(t), then U α,β (k, h) = Γ (αk + γ + 1) Γ (αk + 1) V α,β (k + γ α , h) . (10) Some details of the aformentioned theorems can be found in [30]. 3. Applications of GDTM 3.1. Fractional Hirota-Satsuma Coupled KdV Equation. Consider the following time-fractional Hirota-Satsuma coupled KdV equation: D α t u = 1 2 u xxx − 3uu x + 3(Vw) x , D α t V = −V xxx + 3uV x , D α t w = −w xxx + 3uw x , t > 0, 0 < α ≤ 1, (11) subject to the initial conditions u (x, 0) = 1 3 (β − 8γ 2 ) + 4γ 2tanh2 (γx) , V (x, 0) = −4 (3γ 4 c 0 − 2βγ 2 c 2 + 4γ 4 c 2 ) 3c 2 2 + 4γ 2