Diffusion: Absorption of Water, Atmolysis, Atomic Diffusion, Bohm Diffusion, Boltzmann-Matano Analysis, Convection-Diffusion Equa  (English, Paperback, LLC Books, Source Wikipedia)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 54. Chapters: Absorption of water, Atmolysis, Atomic diffusion, Bohm diffusion, Boltzmann-Matano analysis, Convection-diffusion equation, Dialysis tubing, Diffusion layer, Diffusion of innovations, Diffuson, Eddy diffusion, Effective diffusion coefficient, Ehrenfest model, Facilitated diffusion, Fick's laws of diffusion, File dynamics, Fink effect, Forward osmosis, Fractional anisotropy, Gas exchange, Grain boundary diffusion coefficient, Heat equation, Knudsen diffusion, Lattice diffusion coefficient, Liesegang rings, Liquid junction potential, Mass diffusivity, Maxwell-Stefan diffusion, Molecular diffusion, Momentum diffusion, Narrow escape problem, Nernst-Planck equation, Normalization process model, Osmolyte, Oxygen equivalent, Photon diffusion, Relativistic heat conduction, Reverse diffusion, Self-diffusion, Semipermeable membrane, Streamline diffusion. Excerpt: The heat equation is a parabolic partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. For a function u(x, y, z, t) of three spatial variables (x, y, z) (see cartesian coordinates) and the time variable t, the heat equation is More generally in any coordinate system: where is a positive constant, and or denotes the Laplace operator. In the physical problem of temperature variation, u(x, y, z, t) is the temperature and is the thermal diffusivity. For the mathematical treatment it is sufficient to consider the case = 1. The heat equation is of fundamental importance in diverse scientific fields. In mathematics, it is the prototypical parabolic partial differential equation. In probability theory, the heat equation is connected with the study of Brownian motion via the Fokker-Planck equation. In financial mathematics it is used to solve the Black-Scholes partial differential equation. The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes. Suppose one has a function u which describes the temperature at a given location (x, y, z). This function will change over time as heat spreads throughout space. The heat equation is used to determine the change in the function u over time. The image to the right is animated and describes the way heat changes in time along a metal bar. One of the interesting properties of the heat equation is the maximum principle which says that the maximum value of u is either earlier in time than the region of concern or on the edge of the region of concern. This is essentially saying that temperature comes either from some source or from earlier in time because heat permeates but is not created from nothingness. This is a property of parabolic partial differential equations and is not difficult to prove mathematically (see below). Another interesting property is that even