On the relation between elementary partial difference equations and partial differential equations
type: Item
instance of: scholarly article
title: On the relation between elementary partial difference equations and partial differential equations
zbMATH Open document ID: 0928.03071
DOI: 10.1016/S0168-0072(98)00006-2
published in: Annals of Pure and Applied Logic
publication date: 2000-01-09T00:00:00Z
review text: The author continues the nonstandard investigation of the formal translations of ``finite'' or ``discrete'' partial difference equations into partial differential equations of the first and, now, of the second order with the aim of applying these processes to equations such as the most basic heat transfer equation. The author uses internal set theory concepts and one must take care when the terms ``finite'' or ``discrete'' are used. The author usually means (infinite but) hyperfinite in terms of a Robinson-styled nonstandard analysis. This approach leads immediately to S-continuous solutions to such difference equations, which are then extended in the usual manner to continuous solutions. Numerous other results are investigated with the aim of leading to this basic application. The paper is not a paper in infinitesimal modeling, however. No formal methods are given to obtain specific difference equations. The facts are that it is well-known that there are nonstandard methods that can be used to translate an observed physical scenario directly into a particular partial differential equation by use of a physical *-transfer ``infinitesimalizing'' process. This yields derivations for specific partial differential equations that do not rely upon the approach presented within this paper.
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 03H10
Mathematics Subject Classification ID: 00A79
zbMATH DE Number: 1308063
zbMATH Keywords: partial difference equations
zbMATH Keywords: shadow
zbMATH Keywords: stroboscopy
zbMATH Keywords: heat transfer equation
zbMATH Keywords: internal set theory
zbMATH Keywords: nonstandard analysis
zbMATH Keywords: \(S\)-continuous solutions to difference equations
zbMATH Keywords: infinitesimalization of an observed physical scenario
author: I. P. Van den Berg
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308063