in this paper, a high-order and conditionally stable stochastic difference scheme is proposed for the numerical solution of $rm ithat{o}$ stochastic advection diffusion equation with one dimensional white noise process. we applied a finite difference approximation of fourth-order for discretizing space spatial derivative of this equation. the main properties of deterministic difference schemes,...

1. Elliptic Differential Operators 1 1.1. Partial differential operators 1 1.2. Sobolev spaces and Hölder spaces 10 1.3. Apriori estimates and elliptic regularity 20 1.4. Elliptic operators on compact manifolds 28 2. Non-linear Elliptic Equations 35 2.1. Banach manifolds and Fredholm operators 35 2.2. Moduli space of nonlinear elliptic equations 39 2.3. The Sard-Smale theorem and transversality...

In the case ω = 1, this space is denoted W (Ω). Sobolev spaces without weights occur as spaces of solutions for elliptic and parabolic partial differential equations. In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is “disturbed” in the sense that some degeneration or singularity appears. This “bad” behaviour can be caused by the coefficient...

This paper presents some of the results of the simulation in the radiation section of an industrial boiler using an advanced mathematical model. Calculations are described for the flow, heat transfer, and chemical reaction processes occurring within a gas-fired cylindrical furnace. The calculation procedure is a two dimensional one in which the main hydrodynamic variables are the velocity and s...

3 4 CONTENTS Preface These lecture notes are intented as an introduction to linear second order elliptic partial differential equations. It can be considered as a continuation of a chapter on elliptic equations of the lecture notes [17] on partial differential equations. In [17] we focused our attention mainly on explicit solutions for standard problems for elliptic, parabolic and hyperbolic eq...