Analytical Solutions of the Boltzmann Transformation Equation

Abstract

The so-called continuity equation derived by Fick is one of the most fundamental and extremely important equations in physics and/or in materials science. As is well known, this partial differential equation is also called the diffusion equation or the heat conduction equation and is applicable to physical phenomena of the conservation system. Incorporating the parabolic law relevant to a random movement into it, Boltzmann obtained the ordinary differential equation (B-equation). Matano then applied the B-equation to the analysis of the nonlinear problem for the interdiffusion experiment. The empirical Boltzmann-Matano (B-M) method has been successful in the metallurgical field. However, the nonlinear B-equation was not mathematically solved for a long time. Recently, the analytical solutions of the B-equation were obtained in accordance with the results of the B-M method. In the present study, an applicable limitation of the B-equation to the interdiffusion problems is investigated from a mathematical point of view.