The Exact Solutions of Certain Linear Partial Difference Equations

TL;DR

This work develops an exact-solution framework for linear partial difference equations by translating discrete recurrences into algebraic relations via multivariable power-series generating functions. It provides explicit closed-form solutions for simple 1D cases and extends them to the general case and to multidimensional settings, including high-order partial difference equations (HOPDE), with solutions expressed through multinomial coefficients and shifted initial data. The approach highlights the efficiency of generating-function methods in solving PDDEs and yields scalable formulas for higher dimensions, including 3D and beyond, under a unified combinatorial structure. The abstract also notes a speculative connection to black-hole information theory, indicating a potential interpretive framework (via Theorem 4.2) for encoding information on event horizons, alongside practical applicability to discrete heat diffusion and random-walk problems.

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.

Theorems & Definitions (8)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Theorem 4.1
• Theorem 4.2
• Lemma 4.3
• Theorem 4.4