In this paper an analysis will be presented for the ADI (alternating direction implicit) method of Peaceman and Rachford applied to initial-boundary value problems for partial differential equations in two space dimensions. We shall use the method of lines approach. Motivated by developments in the field of stiff nonlinear ordinary differential equations, our analysis will focus on problems whe...

in this article we implement an operational matrix of fractional integration for legendre polynomials. we proposed an algorithm to obtain an approximation solution for fractional differential equations, described in riemann-liouville sense, based on shifted legendre polynomials. this method was applied to solve linear multi-order fractional differential equation with initial conditions, and the...

The aim of this paper is by using the fractional complex transform and the optimal homotopy analysis by method (OHAM) to find the analytical approximate solutions for nonlinear partial fractional differential Zakharov-Kuznetsov equation. Fractional complex transformation is proposed to convert nonlinear partial fractional differential Zakharov-Kuznetsov equation to nonlinear partial differentia...

The existence of random attractors for a large class of stochastic partial differential equations (SPDE) driven by general additive noise is established. The main results are applied to various types of SPDE, as e.g. stochastic reaction-diffusion equations, the stochastic p-Laplace equation and stochastic porous media equations. Besides classical Brownian motion, we also include space-time frac...

n this paper, at first the  concept of Caputo fractionalderivative is generalized on time scales. Then the fractional orderdifferential equations are introduced on time scales. Finally,sufficient and necessary conditions are presented for the existenceand uniqueness of solution of initial valueproblem including fractional order differential equations.

We develop a unified Petrov-Galerkin spectral method for a class of fractional partial differential equations with two-sided derivatives and constant coefficients of the form 0 D t u+ ∑d i=1 [cli aiD 2μi xi u+cri xiD 2μi bi u]+γ u = ∑d j=1 [κl j a jD 2ν j x j u+κr j x jD 2ν j b j u]+ f , where 2τ ∈ (0, 2), 2μi ∈ (0, 1) and 2ν j ∈ (1, 2), in a (1+d)-dimensional space-time hypercube, d = 1, 2, 3,...

We develop a unified Petrov-Galerkin spectral method for a class of fractional partial differential equations with two-sided derivatives and constant coefficients of the form 0D t u+ ∑d i=1[cli aiD 2μi xi u+ cri xiD 2μi bi u] +γ u = ∑d j=1[ κlj ajD 2νj xj u+κrj xjD 2νj bj u] +f , where 2τ ∈ (0, 2), 2μj ∈ (0, 1) and 2νj ∈ (1, 2), in a (1 +d)-dimensional space-time hypercube, d = 1, 2, 3, · · · ,...

In recent years, fractional differential equations are widely used in the many academic disciplines--viscoelastic mechanics, Fractal theory and so on. Furthermore, fractional differential equations can be used to describe some abnormal phenomenon. For instance, fractional convection-diffusion equation can be used to describe the fluid of abnormal infiltration phenomenon in the medium. In this p...