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A SHARP ERROR ESTIMATE FOR THE NUMERICAL SOLUTION OF MULTIVARIATE DIRICHLET PROBLEM FOR THE HEAT EQUATION

For the multidimensional Dirichlet problem of the heat equation on a cylinder, this study examines convergence properties with rates of approximate solutions, obtained by a naturally arising difference scheme over inscribed uniform grids. Sharp quantitative estimates are given by the use of first and second moduli of continuity of some first and second order partial derivatives of the exact solution. This is accomplished by using the probabilistic method an appropriate random walk.