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Brownian Motion in Parabolic Space

A new mathematical system applicable to whatever Brownian problems where the Fickian diffusion equation (F-equation) is applicable was established. The F-equation, which is a parabolic type partial differential equation in the evolution equation, has ever been used for linear diffusion problems in the time-space (t, x, y, z). In the parabolic space (xt–0.5, yt–0.5, zt–0.5), the present study reveals that the F-equation becomes an ellipse type Poisson equation and furthermore the elegant analytical solutions are possible. Applying the new system to one-dimension nonlinear interdiffusion problems, the solutions were previously obtained as the analytical expressions. The obtained solutions were also elegant in accordance with the experimental results. In the present study, nonlinear diffusion problems are discussed in the two and three dimensional cases. The Brownian problem is widely relevant not only to material science but also to other various science fields. Hereafter, the new mathematical system will be thus extremely useful for the analysis of the Brownian problem in various science fields.