The Exact Solutions of Certain Linear Partial Difference Equations

TL;DR

This paper develops methods to find exact solutions for linear partial difference equations using generating functions, extending to higher dimensions and high-order cases, with applications in various scientific fields.

Contribution

It introduces a systematic approach using generating functions to solve linear partial difference equations, including high-dimensional and high-order cases, advancing analytical solution techniques.

Findings

Exact solutions for simple linear partial difference equations are obtained.

The method extends to n-dimensional and high-order partial difference equations.

Generating functions are shown to be an efficient tool for solving these equations.

Abstract

Difference equations have many applications and play an important role in numerical analysis, probability, statistics, combinatorics, computer science, quantum consciousness, etc. We first prove that the partial differential equation is equivalent to partial difference equation with an example of heat equation. Additionally, we use generating functions to find the exact solutions of some simple linear partial difference equations. Then we extend it to more general partial difference equations of higher dimensions and obtain their solutions. Notice that Theorem 4.2 could provide a mathematical framework for understanding how information within a black hole is encoded on its event horizon, a key concept in the black hole information paradox. In addition, we extend it to n-dimensional case, Theorem 4.4, the high-order partial difference equations (HOPDE). We conclude that using multivariable power series as generating function is a very efficient method to solve partial difference equations.