Solving boundary value problems with complex parameters by the Monte Carlo method
type: Item
instance of: scholarly article
title: Solving boundary value problems with complex parameters by the Monte Carlo method
zbMATH Open document ID: 0878.65116
DOI: 10.1007/BF02104668
author: B. V. Men'shchikov
author: G. A. Mikhaĭlov
published in: Siberian Mathematical Journal
publication date: 1997-07-27T00:00:00Z
review text: The authors consider the three-dimensional Dirichlet problem for the Helmholtz equation \[ \Delta u + cu = -g,\qquad u|_{\Gamma} = \psi \] in a domain \(D\subset\mathbb{R}^3\) with boundary \(\Gamma\), where \(c=a+bi\) is a complex parameter. The estimates under consideration are constructed on the base of the process of a ``random walk by spheres'' in the domain \(D\). New in this paper is the fact, that it extends the estimators of the Monte Carlo type, constructed earlier by the same authors for a real parameter, to the case of a complex parameter. The obtained results enable in the second part of the paper to implement the Fourier transform while solving the mixed problem for a parabolic equation \[ u_t-\Delta u = f(r,t),\qquad u|_{t=0} = \phi(r),\qquad u|_{\Gamma}=v(r,t)\qquad\text{with}\qquad v(r,0)|_{\Gamma}=\psi(r) \] by the Monte Carlo method.
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 1023929
zbMATH Keywords: Monte Carlo method
zbMATH Keywords: Helmholtz equation
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Fourier transform
zbMATH Keywords: random walks by spheres
zbMATH Keywords: complex parameters
zbMATH Keywords: parabolic equation
reviewed by: Jaromír Antoch
MaRDI profile type: Publication
cites work: Some Continuous Monte Carlo Methods for the Dirichlet Problem
cites work: Q3740770
cites work: New Monte Carlo methods with estimating derivatives
cites work: Q3734960
cites work: To the theory of the estimators of the Monte Carlo method which are connected with a ``random walk by spheres''
cites work: Q5731975
description: scientific article; zbMATH DE number 1023929