标题： 基于空间索洛模型的物质资本数学建模 显示英文标题

标题： Mathematical modeling of physical capital using the spatial Solow model

摘要： 这项研究涉及通过国家边界的物理资本扩散的数学建模。 物理资本被认为是国家经济增长的重要变量。 在这里，我们使用了经济索洛模型的一个扩展来描述走私如何影响国家的经济增长。 在本研究中，我们依赖于一个非凹生产函数，而不是经典的科布-道格拉斯生产函数。 为了模拟物理资本通过国家边界的扩散，我们开发了一个基于抛物型偏微分方程的模型，该方程描述了物理资本的动力学以及诺伊曼型边界条件。 走私存在于许多国家之间的边界，并可能包括燃料、机械和食品。 通过边界的走私是一个对国家经济具有问题性的问题。 走私问题通常主要与非官方汇率有关，该汇率不同于官方汇率或补贴。 数值模拟是通过显式有限差分方案获得的，该方案展示了物理资本如何通过国家边界扩散。 对物理资本的研究是未来几年许多国家经济增长的一个关键问题。 结果表明，当诺伊曼型边界条件不为零时，物理资本的动力学与在没有通过国家边界进行物理资本流动的经典空间索洛模型中观察到的经典经济行为不同。 最后，可以得出结论，避免通过边境的走私是影响国家经济增长的重要因素。 显示英文摘要

摘要： This research deals with the mathematical modeling of the physical capital diffusion through the borders of the countries. The physical capital is considered an important variable for the economic growth of a country. Here we use an extension of the economic Solow model to describe how the smuggling affects the economic growth of the countries. In this study we rely on a production function that is non-concave instead of the classical Cobb-Douglas production function. In order to model the physical capital diffusion through the borders of the country, we developed a model based on a parabolic partial differential equation that describes the dynamics of physical capital and boundary conditions of Neumann type. Smuggling is present in many borders between countries and may include fuel, machinery and food. This smuggling through the borders is a problematic issue for the country's economies. The smuggling problem usually is related mainly to a non-official exchange rate that is different than the official rate or subsides. Numerical simulations are obtained using an explicit finite difference scheme that shows how the physical capital diffusion through the border of the countries. The study of physical capital is a paramount issue for the economic growth of many countries for the next years. The results show that the dynamics of the physical capital when boundary conditions of Neumann type are different than zero differ from the classical economic behavior observed in the classical spatial Solow model without physical capital flux through the borders of countries. Finally, it can be concluded that avoiding the smuggling through the frontiers is an important factor that affects the economic growth of the countries.