On the relation between elementary partial difference equations and partial differential equations
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.
\(S\)-continuous solutions to difference equations
infinitesimalization of an observed physical scenario
scientific article; zbMATH DE number 1308063
scientific article