Green's function-stochastic approach to solving linear, nonlinear and nonhomogeneous evolution transportproblems.

摘要

A general, analytical Green's function-stochastic approach for solving linear, nonlinear and non-homogeneous evolution and transport problems is presented.;Analysis of practical and natural engineering situations reveal the basic elements from which an understanding of such systems is best put together. Partial differential equations that evolve from such analysis sometimes become difficult to solve by classical approaches. While numerical solutions are often sought in some studies, Green's function methods often offer a powerful alternative.;Green's function in the case of linear models, provide integral solutions that depend strictly on known boundary conditions, initial conditions, and when necessary, space and/or time varying system forcing functions. The approach of Green's function best reveals the responses to the problems effected by Dirac delta functions. Green's function methods are especially useful when the system is subject to random boundary and/or initial conditions, and/or random forcing. Under these conditions, Green's function based solution describes exactly the random response of the modeled system. In addition, the Green's -based solution allows explicit calculation of the space-and possibly time-dependent mean response, as well as the space and time-dependent system variance about the mean evolves in space and time is, in turn typically.;Problems of physics and engineering that reveal probabilistic characteristics or involve boundary conditions that are random in nature as well as possibly stochastic in-system forcing can be analysed by Green's function methods. For such problems, Green's function methods can be usefully combined with methods from theory of stochastic differential equations. Here, the time and space-dependent evolution of the variables describing the system response is often modeled as a random walk/stochastic process, the evolution of which is governed by advection-diffusion stochastic differential equation. The probabilstic decription of how the stochastic process, evolves in space and time is, in typically embedded in the transition probability density, p, which gives the (conditional) probability of observing the stochastic process/random system response, at some specified position and time, given its position at an earlier time. Importantly and as detailed and exploited in this dissertation, the equation governing the evolution of the transition density, p - the Chapmann-Kolmogorov equation - corresponds exactly, under fairly general conditions, to the Green's function describing the system's response. Thus, the important problem of modeling the random response of linear systems subject to random forcing can be powerfully tackled by initially determining the system's Green's function, and explicitly identifying the system variable describing system response as stochastic process /random walker. Using this recipe provides scientists and engineers with a near-complete, rigorous, physics-based, probabilistic picture of how the system evolves under random forcing and /or random boundary and initial conditions.;This dissertation first analysed the flow problem from continuum approach employing conservation principles of mass and momentum. Such analysis led to a pure diffusion problem. The diffusion problem is then solved for a semi-infinite and finite medium with moving boundary. The Green's function method is then employed to solve the diffusion equation but this time with time - dependent boundary motions. Some useful Green's function results were obtained. The Chapmann-Kolmogorov equation is then derived and simplified to the Fokker- Planck equation for application to randomly-forced incompressible flow problems. Finally, two simple example flow problems, the random response of a semi-infinite fluid layer, and the random response of a finite layer, both driven by the random boundary motion are considered.