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This research deals with the mathematical modeling of the physical capitaldiffusion through the borders of the countries. The physical capital isconsidered an important variable for the economic growth of a country. Here weuse an extension of the economic Solow model to describe how the smugglingaffects the economic growth of the countries. In this study we rely on aproduction function that is non-concave instead of the classical Cobb-Douglasproduction function. In order to model the physical capital diffusion throughthe borders of the country, we developed a model based on a parabolic partialdifferential equation that describes the dynamics of physical capital andboundary conditions of Neumann type. Smuggling is present in many bordersbetween countries and may include fuel, machinery and food. This smugglingthrough the borders is a problematic issue for the country's economies. Thesmuggling problem usually is related mainly to a non-official exchange ratethat is different than the official rate or subsides. Numerical simulations areobtained using an explicit finite difference scheme that shows how the physicalcapital diffusion through the border of the countries. The study of physicalcapital is a paramount issue for the economic growth of many countries for thenext years. The results show that the dynamics of the physical capital whenboundary conditions of Neumann type are different than zero differ from theclassical economic behavior observed in the classical spatial Solow modelwithout physical capital flux through the borders of countries. Finally, it canbe concluded that avoiding the smuggling through the frontiers is an importantfactor that affects the economic growth of the countries.

Mathematical modeling of physical capital using the spatial Solow model