A Partial Differential Equation Framework for Modeling Economic Growth

Abstract

This work develops a novel mathematical framework for modeling energy dynamics in economic systems, combining production theory with spatial diffusion and transport phenomena. The foundation is a reaction-diffusion partial differential equation that extends classical growth models through the incorporation of endogenous spatial coupling effects. The model captures three fundamental equilibrium states whose stability properties are governed by the interplay between production efficiency and depreciation rates. Numerical simulations demonstrate dynamical behavior ranging from uniform convergence to spatially modulated transitions and cascading collapse sequences, depending on the structure of spatial coupling parameters.

The analysis reveals how prosperous regions can elevate entire networks through energy redistribution mechanisms. A second model formulation incorporates explicit transport mechanisms through higher-order spatial derivatives, producing nonlinear coupling terms that capture network effects. Traveling wave solutions demonstrate characteristic propagation speeds that scale with system discretization, while numerical implementation via backward Euler schemes provides stability for simulating these nonlinear dynamics.