zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Random walk method for the two- and three-dimensional Laplace, Poisson and Helmholtz’s equations. (English) Zbl 0988.65004
Summary: The random walk method (RWM) is developed here for solving the Laplace, Poisson, and Helmholtz equations in two and three dimensions. The RWM is a local method, i.e. the solution at an arbitrary point can be determined without having to obtain the complete field solution. The method is based on the properties of diffusion processes, the Itô formula, the Dynkin formula, the Feynman-Kac functional, and Monte Carlo simulation. Simplicity, stability, accuracy, and generality are the main features of the proposed method. The RWK is inherently parallel and this fact has been fully exploited in this paper. Extensive numerical results have been presented in order to understand the various parameters involved in the method.

65C50Other computational problems in probability
65C05Monte Carlo methods
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
60G50Sums of independent random variables; random walks
60J60Diffusion processes
60J65Brownian motion
Full Text: DOI