In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moveover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Som...

An algorithm based on operator splitting has been successfully implemented for solving unsteady, advectiondominated transport problems in 3-D. Specifically, the general operator-integration-factor splitting method of Maday et al. is applied to the unsteady advection–diffusion equation with source/sink terms. The algorithm incorporates a 3-D characteristic Galerkin scheme to treat advection, and...

In the last three decades, interest in fractional calculus has experienced rapid growth and at present we can find many papers devoted its theoretical and application aspects (see the work of Machado et al. (2011) and the references therein). Fractional order models of real systems are often more adequate than the usually used integer order models in electrochemistry (Ichise et al., 1971), adve...

We consider the problem of numerically approximating the solution of the coupling of the flow equation in a porous medium, with the advection-diffusion equation in the presence of uncertainty on the permeability of the medium. Random coefficients are classically used in the flow equation to modelize uncertainty. More precisely, we propose the numerical analysis of a method developed to compute ...

According to Godunov theorem for numerical calculations of advection equations, there exist no higher-order schemes with constant positive difference coefficients in a family of polynomial schemes with an accuracy exceeding the first-order. In case of advection-diffusion equations, so far there have been not found stable schemes with positive difference coefficients in a family of numerical sch...

In this paper, the piecewise parabolic method is presented for solving one-dimensional advection-diffusion type equation and its application to burger equation. First, given solution domain discretized by using a uniform Discretization grid point. Next applying integration in terms of spatial variable, we converting it into system first-order ordinary differential temporarily variable. Next, Ta...

Systems in which reaction terms are coupled to diffusion and advection transports arise in a wide range of chemical engineering applications, physics, biology and environmental. In these cases, the components of the unknown can denote concentrations or population sizes which represent quantities and they need to remain positive. Classical finite difference schemes may produce numerical drawback...

A new direct solution method for the advection-diffusion equation is presented. By employing a semi-implicit time discretisation, the equation is rewritten as a heat equation with source terms. The solution is obtained by discretely approximating the integral convolution of the associated Green’s function with advective source terms. The heat equation has an exponentially decaying Green’s funct...

This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D(0) = 0) and advection-degenerate (at h'(0) = 0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection term h(u): (1)  h'(u...