Time averaging for nonautonomous/random linear parabolic equations

Time Averaging for Random Nonlinear Abstract Parabolic Equations

It is well known that the averaging principle is a powerful tool of investigation of ordinary differential equations, containing high frequency time oscillations, and a vast work was done in this direction (cf. [1]). This principle was extended to many other problems, like ordinary differential equations in Banach spaces, delayed differential equations, and so forth (for the simplest result of ...

Averaging for Fundamental Solutions of Parabolic Equations

Herein, an averaging theory for the solutions to Cauchy initial value problems of arbitrary order, "-dependent parabolic partial di erential equations is developed. Indeed, by directly developing bounds between the derivatives of the fundamental solution to such an equation and derivatives of the fundamental solution of an \averaged" parabolic equation, we bring forth a novel approach to compar...

Solving parabolic stochastic partial differential equations via averaging over characteristics

The method of characteristics (the averaging over the characteristic formula) and the weak-sense numerical integration of ordinary stochastic differential equations together with the Monte Carlo technique are used to propose numerical methods for linear stochastic partial differential equations (SPDEs). Their orders of convergence in the mean-square sense and in the sense of almost sure converg...

Self-averaging in time reversal for the parabolic wave equation

We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. This allows us to construct an approximate martingale for the phase space Wigner transform of two wave fields. Using a priori L-bounds available in the time-reversal setting, we prove that the Wigner transform ...

Time-averaging under fast periodic forcing of parabolic partial differential equations: Exponential estimates