On some classes of partial difference equations

TL;DR

This paper extends the classification and solution methods for partial difference equations, focusing on elliptic, parabolic, and hyperbolic types, with detailed analysis of discrete Laplace, Poisson, and biharmonic equations.

Contribution

It classifies partial difference equations into three subclasses and provides detailed solution methods for elliptic equations, expanding on Baker's 1969 work.

Findings

Classification into elliptic, parabolic, hyperbolic subclasses

Solution methods for elliptic partial difference equations

Analysis of discrete Laplace, Poisson, and biharmonic equations

Abstract

In one of his work, appeared in 1969, John A. Baker initiated the systematic investigation of some partial difference equations. The main purpose of this paper is to continue and to extend these investigations. Firstly, we present how such type of equations can be classified into elliptic, parabolic and hyperbolic subclasses, respectively. After that, we show solution methods in the elliptic class. Here we will deal in details with the discrete version of the following partial differential equations: Laplace's equation, Poisson equation and the (in)homogeneous biharmonic equation.