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A class of elliptic equations boundary value problems probabilistic algorithms

Author: YangWenSheng
Tutor: LiuZaiMing
School: Central South University
Course: Probability Theory and Mathematical Statistics
Keywords: Second order elliptic equations boundary value problem Probabilistic algorithms Brownian motion Brownian Motion with Drift Strong Markov property
CLC: O175.2
Type: Master's thesis
Year: 2006
Downloads: 54
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Abstract


Second-order elliptic partial differential equation boundary value problem is widespread in many areas of physics, engineering and technology, so many of the exact solution of the boundary value problems very difficult, due to the complexity of the conditions on the definite solution of regional or regional boundaries, and even can not be obtained. With the rapid development of computer technology, the numerical solution more and more attention has been paid. Explore the establishment of a new method to improve the development of traditional methods, numerical study is of great significance and prospects for the development of practical problems. Proposed an efficient probabilistic numerical methods, harmonic equations and a class of Poisson problems had literature of this thesis by further in-depth study, this method is applied to a class of elliptic boundary value problems, this problem probabilistic algorithm in bounded and unbounded regions. Based on the close ties between the diffusion process elliptic equations, probability theory, the method from the solution of the boundary value problem can be expressed as the expectation of a diffusion process or random process functional discretization of the problem, and the use of Brown certain unique nature of the diffusion process of the movement, the drift Brownian motion. The study showed that: Compared with the traditional numerical method, this method in lower dimension, point-by-point solution to solve unbounded region have certain advantages. The introduction part of elliptic partial differential equations background, research status quo of the boundary value problem solving boundary value method, and introduced the idea of ??probabilistic algorithms. The second chapter outlines the boundary value problem of elliptic equations, Brownian motion, Markov property prior knowledge. Chapter III on the basis of existing research, the establishment of a class of bounded probability algorithm for the numerical solution of high-dimensional elliptic boundary value problems in the region. Chapter IV probabilistic algorithms extended to a more complex unbounded region boundary value problem, as an auxiliary ball through the external structure in unbounded domains, unbounded region transformed into definite solution region boundary, then the border on the split, the discrete problem, and then find the numerical solution of the discrete problem. Including not only the use of the drift of Brownian motion and its strong Markov property, but also the use of any point of departure from the ball outside the joint distribution of the location and time of the first spherical. Chapter 5 gives a numerical example to show that the method is effective and simple. Finally, probabilistic algorithms summarizes.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations
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