Numerical solution to a Parabolic-ODE Solow model with spatial diffusion and technology-induced motility

TL;DR

The paper extends the Solow growth framework by formulating a coupled parabolic PDE system with spatial diffusion of capital $k(x,t)$ and technology $A(x,t)$ on a bounded domain, incorporating technology-induced mobility via $-\\nabla\\cdot(\\chi k \\nabla A)$ and a nonconvex production function $f(k)=\\frac{\\alpha_1 k^{p}}{1+\\alpha_2 k^q}$. It develops 1D and 2D explicit Generalized Finite Difference Method (GFDM) discretizations based on moving least squares and proves conditional convergence under a time-step restriction. Numerical experiments on irregular meshes illustrate the method’s accuracy and reveal dynamics such as capital concentration following technology diffusion and potential poverty-trap regimes, highlighting implications for regional growth patterns. The work provides a meshless, scalable framework for simulating spatial Solow-type systems and sets the stage for future coupling enhancements and empirical validation.

Abstract

This work studies a parabolic-ODE PDE's system which describes the evolution of the physical capital "$k$" and technological progress "$A$", using a meshless in one and two dimensional bounded domain with regular boundary. The well-known Solow model is extended by considering the spatial diffusion of both capital anf technology. Moreover, we study the case in which no spatial diffusion of the technology progress occurs. For such models, we propound schemes based on the Generalized Finite Difference method and proof the convergence of the numerical solution to the continuous one. Several examples show the dynamics of the model for a wide range of parameters. These examples illustrate the accuary of the numerical method.

Figures (10)

• Figure 1: 1D irregular clouds of points
• Figure 2: Solution $k$ for $\delta=0.05$.
• Figure 3: Solution $k$ for $\delta=0.02$.
• Figure 4: Solution $k$ for $\delta=0.02$.
• Figure 5: 2D irregular cloud of points.

Theorems & Definitions (3)

• Remark 4.1
• Theorem 4.1
• Remark 4.2