Escrever uma avaliação PREreview

This work develops a theoretical framework for Partial Difference Equations (P∆E) as a natural mathematical language for modeling discrete- time, discrete-space systems. Motivated by the limitations of continuous partial differential equations (PDE) in representing inherently discrete phenomena, we begin by defining P∆E in terms of discrete function spaces and shift operators, contrasting them with ordinary difference equations (O∆E) and PDE, and clarifying the scope of our study. We then examine linear P∆E, outlining their main types, providing formal definitions, and presenting selected analytic solutions in simple cases. Building on this, we introduce the discrete functional analytic setting: discrete function spaces, Hilbert space structure, and discrete operators, including difference and shift operators, and study their alge- braic and adjoint properties. The discrete Green’s function is also defined within this framework. As a demonstration of the framework’s unifying power, we reformulate a wide range of well-known discrete models, including elementary cellular automata, coupled map lattices, Conway’s Game of Life, the Abelian sandpile model, the Olami–Feder–Christensen earthquake model, forest- fire models, the Ising model, the Kuramoto Firefly Model, the Greenberg–Hastings Model, and the Langton’s ant as explicit P∆E. For each case, we focus on obtaining a concise and mathematically elegant formulation rather than detailed dynamical analysis. Finally, we compare the “discrete universe” of P∆E with the contin- uous universe of PDE, highlighting their structural parallels and their respective connections to discrete mathematics and continuous analysis. This reveals P∆E and PDE as mathematical “twins”, analogous in form yet rooted in fundamentally different underlying mathematics. The generality of the P∆E formalism suggests broad applicability, from modeling biological and ecological processes to analyzing complex networks, emergent computation, and other spatiotemporally extended systems.