The particle diffusion equation was originally derived by Adolf Fick in The diffusion equation can be trivially derived from the continuity equation , which states that a change in density in any part of the system is due to inflow and outflow of material into and out of that part of the system. Effectively, no material is created or destroyed:.
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The diffusion equation can be obtained easily from this when combined with the phenomenological Fick's first law , which states that the flux of the diffusing material in any part of the system is proportional to the local density gradient:. If drift must be taken into account, the Smoluchowski equation provides an appropriate generalization.
Analytical solutions to a nonlinear diffusion–advection equation | SpringerLink
The diffusion equation is continuous in both space and time. One may discretize space, time, or both space and time, which arise in application. Discretizing time alone just corresponds to taking time slices of the continuous system, and no new phenomena arise. In discretizing space alone, the Green's function becomes the discrete Gaussian kernel , rather than the continuous Gaussian kernel.
In discretizing both time and space, one obtains the random walk. The product rule is used to rewrite the anisotropic tensor diffusion equation, in standard discretization schemes, because direct discretization of the diffusion equation with only first order spatial central differences leads to checkerboard artifacts.
Applications of Nonlinear Diffusion Equations
The rewritten diffusion equation used in image filtering:. The spatial derivatives can then be approximated by two first order and a second order central finite differences. From Wikipedia, the free encyclopedia.
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See also: Discrete Gaussian kernel. We show that the solutions of the nonlinear diffusion problems can be approximated by those of semilinear reaction-diffusion systems. This indicates that the mechanism of nonlinear diffusion might be captured by reaction-diffusion interaction.
go The reaction-diffusion systems include only simple reactions and linear diffusions. Resolving semilinear problems is typically easier than dealing with nonlinear problems. Therefore, our ideas are expected to reveal effective approaches to the study of nonlinear problems.
Applying the similinear approixmation to numerical analysis, we constructed and analyzed a linear numerical scheme for the nonlinear diffusion systems.