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Hybrid computer solutions of partial differential equations by Monte Carlo methods Little, Warren David

Abstract

A continuous Markov process is examined for the purpose of developing Monte Carlo methods for solving partial differential equations. Backward Kolmogorov equations for conditional probability density functions and more general equations satisfied by auxiliary probability density functions are derived. From these equations and the initial and boundary conditions that the density functions satisfy, it is shown that solutions of partial differential equations at an interior point of a region can be written as the expected value of randomly-selected initial and boundary values. From these results, Monte Carlo methods for solving homogeneous and nonhomogeneous elliptic, and homogeneous parabolic partial differential equations are proposed. Hybrid computer techniques for mechanizing the Monte Carlo methods are given. The Markov process is simulated on the analog computer and the digital computer is used to control the analog computer and to form the required averages. Methods for detecting the boundaries of regions using analog function generators and electronic comparators are proposed. Monte Carlo solutions are obtained on a hybrid system consisting of a PACE 231 R-V analog computer and an ALWAC III-E digital computer. The interface for the two computers and a multichannel discrete-interval binary-noise source are described. With this equipment, solutions having a small variance are obtained at a rate of approximately five minutes per solution. Example solutions are given for Laplace's equation in two and three dimensions, Poisson's equation in two dimensions and the heat equation in one, two and three dimensions.

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