Application of Neumann-Ulam scheme to solving the first boundary value problem for a parabolic equation |
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Authors: | A. S. Sipin |
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Affiliation: | 1. Vologda State Pedagogical University, ul. S. Orlova 6, Vologda, 160035, Russia
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Abstract: | Statistical estimates of the solutions of boundary value problems for parabolic equations with constant coefficients are constructed on paths of random walks. The phase space of these walks is a region in which the problem is solved or the boundary of the region. The simulation of the walks employs the explicit form of the fundamental solution; therefore, these algorithms cannot be directly applied to equations with variable coefficients. In the present work, unbiased and low-bias estimates of the solution of the boundary value problem for the heat equation with a variable coefficient multiplying the unknown function are constructed on the paths of a Markov chain of random walk on balloids. For studying the properties of the Markov chains and properties of the statistical estimates, the author extends von Neumann-Ulam scheme, known in the theory of Monte Carlo methods, to equations with a substochastic kernel. The algorithm is based on a new integral representation of the solution to the boundary value problem. |
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