The Time-Fractional Coupled-Korteweg-de-Vries Equations

and Applied Analysis 3 Subject to the initial condition D α−k 0 U (x, 0) = f k (x) , (k = 0, . . . , n − 1) , D α−n 0 U (x, 0) = 0, n = [α] , D k 0 U (x, 0) = g k (x) , (k = 0, . . . , n − 1) , D n 0 U (x, 0) = 0, n = [α] , (11) where ∂α/∂tα denotes the Caputo or Riemann-Liouville fraction derivative operator, f is a known function, N is the general nonlinear fractional differential operator, and L represents a linear fractional differential operator. The method first step here is to transform the fractional partial differential equation to the fractional partial integral equation by applying the inverse operator ∂α/∂tα of both sides of (10) to obtain the following. In the case of Riemann-Liouville fractional derivative