we focus on the use of two stable and accurate explicit finite difference schemes in order to approximate the solution of stochastic partial differential equations of it¨o type, in particular, parabolic equations. the main properties of these deterministic difference methods, i.e., convergence, consistency, and stability, are separately developed for the stochastic cases.

The goal of this lecture is to discuss more sophisticated time-marching schemes, and in particular, the major distinction between explicit and implicit methods.

• Instructions to approach or avoid fictitious social groups cause changes in implicit evaluations. • These changes are not fully mediated by changes in explicit evaluations. • These changes occur even in the absence of changes in explicit evaluations. • These changes occur even if highly diagnostic incompatible valence information is provided. • These findings constrain current and future mode...

The paper is concerned with stabilization of a scalar delay differention equation ẋ(t)− m ∑ k=1 Ak(t)x[hk(t)] = 0, t ≥ 0, x(ξ) = φ(ξ), ξ < 0, by introducing impulses in certain moments of time x(τj) = Bjx(τj − 0), j = 1, 2, . . . . Explicit stability results are presented both for the equation with positive coefficients and for the equation with Ak being of arbitrary sign. Supported by Israel M...

This paper presents a series expansion for the evolution of nonlinear systems which are analytic in the state and linear in the controls. An explicit recursive expression is obtained assuming that the input vector fields are constant. Additional simplifications take place in the analysis of systems described by second order polynomial vector fields. Sufficient conditions are derived to guarante...

Fluid-structure interaction (FSI) occurs when the dynamic water hammer forces; cause vibrations in the pipe wall. FSI in pipe systems due to Poisson and junction coupling has been the center of attention in recent years. It causes fluctuations in pressure heads and vibrations in the pipe wall. The governing equations of this phenomenon include a system of first order hyperbolic partial differen...

Four explicit finite difference schemes, including Lax-Friedrichs, Nessyahu-Tadmor, Lax-Wendroff and Lax-Wendroff with a nonlinear filter are applied to solve water hammer equations. The schemes solve the equations in a reservoir-pipe-valve with an instantaneous and gradual closure of the valve boundary. The computational results are compared with those of the method of characteristics (MOC), a...

In this paper, we develop a new stability and convergence theory for highly indefinite elliptic partial differential equations by considering the Helmholtz equation at high wave number as our model problem. The key element in this theory is a novel k-explicit regularity theory for Helmholtz boundary value problems that is based on decomposing the solution into in two parts: the first part has t...

Convection-diffusion equations provide the basis for describing heat and mass transfer phenomena as well as processes of continuum mechanics. To handle flows in porous media, the fundamental issue is to model correctly the convective transport of individual phases. Moreover, for compressible media, the pressure equation itself is just a time-dependent convection-diffusion equation. For differen...