This paper introduce a numerical solution of time fractional stochastic advection-diffusion equa- tion (FSA-DE) wherein time fractional derivative is described in Caputo sence of order α (0 < α < 1). First, a L1 approximation is employed to estimate the Caputo derivative. Then, the spatial derivative is discretized by a second-order finite difference scheme. Moreover, we combine the implicit finite difference (IFD) scheme with the proper orthogonal decomposition (POD) method to reduce the used cpu time. In other words, we obtain POD based reduced-order IFD scheme. As a result, the new scheme can be viewed as the modification of the exiting job (Mirzaee et al., 2020 [23]). The numerical results provide to verify the feasibility and efficiency of the new method.

In this paper, for the first time, the distributed order time-fractional forced Korteweg-de Vries equation is studied. We use a numerical method based on the shifted Legendre operational matrix of distributed order fractional derivative with Tau method to find approximate solution of distributed order forced Korteweg-de Vries equation. This shifted Legendre operational matrix of distributed order fractional derivative with Tau method are used to reduce the solution of the distributed order time-fractional forced Korteweg-de Vries equations to a system of algebraic equations. An error analysis and convergence are obtained. Finally, to display the applicability and validity of the numerical method some examples are implemented.