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The '''diffusion equation''' is a [[partial differential equation]] which describes density dynamics in a material undergoing [[diffusion]].  It is also used to describe processes exhibiting diffusive-like behaviour, for instance the 'diffusion' of [[alleles]] in a population in [[population genetics]].
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==Statement==
 
The equation is usually written as:
 
{{Equation box 1
|equation=<math>\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla \cdot \big[ D(\phi,\mathbf{r}) \ \nabla\phi(\mathbf{r},t) \big], </math>
|indent=:
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
 
where ''ϕ''('''r''', ''t'') is the [[density]] of the diffusing material at location '''r''' and time ''t'' and ''D''(''ϕ'', '''r''') is the collective [[diffusion coefficient]] for density ''ϕ'' at location '''r'''; and ∇ represents the vector [[differential operator]] [[del]]. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.  
 
More generally, when ''D'' is a symmetric [[positive definite matrix]], the equation describes [[Anisotropy|anisotropic]] diffusion, which is written (for three dimensional diffusion) as:
 
{{Equation box 1
|equation=<math>\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \sum_{i=1}^3\sum_{j=1}^3 \frac{\partial}{\partial x_i}\left[D_{ij}(\phi,\mathbf{r})\frac{\partial \phi(\mathbf{r},t)}{\partial x_j}\right]</math>
|indent=:
|cellpadding
|border
|border colour = #50C878
|background colour=#ECFCF4}}
 
If ''D'' is constant, then the equation reduces to the following [[linear differential equation]]:
 
:<math>\frac{\partial\phi(\mathbf{r},t)}{\partial t} = D\nabla^2\phi(\mathbf{r},t), </math>
 
also called the [[heat equation]].
 
== Historical origin ==
 
The [[Fick's law of diffusion|particle diffusion equation]] was originally derived by [[Adolf Fick]] in 1855.<ref>A. Fick, ''Ueber Diffusion'', Pogg. Ann. Phys. Chem. 170 (4. Reihe 94), 59-86 (1855).</ref>
 
== Derivation ==
 
The diffusion equation can be derived in a straightforward way 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:
 
:<math>\frac{\partial\phi}{\partial t}+\nabla\cdot\mathbf{j}=0</math>,
 
where '''j''' is the flux of the diffusing material. The diffusion equation can be obtained easily from this when combined with the phenomenological [[Fick's law|Fick's first law]], which assumes that the flux of the diffusing material in any part of the system is proportional to the local density gradient:
 
:<math>\mathbf{j}=-D(\phi)\,\nabla\phi(\mathbf{r},t)</math>.
 
If [[drift (telecommunication)|drift]] must be taken into account, the [[Smoluchowski equation]] provides an appropriate generalization.
 
==Discretization==
{{see also|Discrete Gaussian kernel}}
 
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]].
 
==Discretization (Image)==
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. The rewritten diffusion equation used in image filtering:
 
<math> \frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla\cdot \left[D(\phi,\mathbf{r})\right] \nabla \phi(\mathbf{r},t) + {\rm tr} \Big[ D(\phi,\mathbf{r})\big(\nabla\nabla^T \phi(\mathbf{r},t)\big)\Big] </math>
 
where "tr" denotes the [[Trace (linear algebra)|trace]] of the 2nd rank [[tensor]], and superscript "''T''" denotes [[transpose]], in which in image filtering ''D''(''ϕ'', '''r''') are symmetric matrices constructed from the [[eigenvectors]] of the image [[structure tensor]]s . The spatial derivatives can then be approximated by two first order and a second order central [[finite difference]]s. The resulting diffusion algorithm can be written as an image [[convolution]] with a varying kernel (stencil) of size 3 × 3 in 2D and 3 × 3 × 3 in 3D.
 
== See also ==
* [[Heat equation]]
* [[Fokker-Planck equation]]
* [[Fick's law of diffusion]]: [[Fick's law of diffusion#Second law|Fick's Second Law]]
* [[Maxwell-Stefan equation]]
* [[Radiative transfer equation and diffusion theory for photon transport in biological tissue]]
 
==References==
<references/>
2. Carslaw, H. S. and Jager, J. C. (1959). Conduction of Heat in Solids. Oxford: Clarendon Press
 
3. Crank, J. (1956). The Mathematics of Diffusion. Oxford: Clarendon Press
 
4. Thambynayagam, R. K. M (2011). The Diffusion Handbook: Applied Solutions for Engineers: McGraw-Hill
 
== External links ==
* [http://www.ee.byu.edu/cleanroom/DopConCalc.phtml Diffusion Calculator for Impurities & Dopants in Silicon]
* [http://dl.dropbox.com/u/46147408/tutorials/diffusion.pdf A tutorial on the theory behind and solution of the Diffusion Equation.]
* [http://dragon.unideb.hu/~zerdelyi/Diffusion-on-the-nanoscale/index.html Classical and nanoscale diffusion (with figures and animations)]
 
{{DEFAULTSORT:Diffusion Equation}}
[[Category:Diffusion]]
[[Category:Partial differential equations]]
[[Category:Parabolic partial differential equations]]
 
[[it:Leggi di Fick]]

Revision as of 09:59, 1 March 2014

Let's look an actual registry scan and a few of what we will see whenever we do 1 on your computer. This test was done on a computer that was not functioning because it must, operating at slow speed plus having certain issues with freezing up.

Before really obtaining the software it's best to check on the firms that make the software. If you could find details found on the form of reputation every organization has, maybe the risk of malicious programs could be reduced. Software from reputed firms have helped me, plus other consumers, to create my PC run quicker.. If the product description does not look good to you, refuses to include details about the software, does not include the scan functions, we should go for another one that ensures you're paying for what you desire.

It doesn't matter whether you're not extremely clear about what rundll32.exe is. However remember that it plays an significant role inside maintaining the stability of our computers plus the integrity of the system. When certain software or hardware can not answer normally to the system procedure, comes the rundll32 exe error, that might be caused by corrupted files or missing information inside registry. Usually, error message usually shows up at booting or the beginning of running a system.

In order to remove the programs on a computer, Windows Installer need to be inside a healthy state. If its installation is corrupted you could obtain error 1721 inside Windows 7, Vista plus XP throughout the program removal task. Simply re-registering its component files would solve a problem.

Another prevalent cause of PC slow down is a corrupt registry. The registry is a important component of computers running on Windows platform. When this gets corrupted the PC will slowdown, or worse, not begin at all. Fixing the registry is easy with all the employ of the program and tuneup utilities.

We should moreover see to it that it is easy to download plus install. We should avoid those treatments that might require we a surprisingly complicated set of instructions. Furthermore, you need to no longer need any different system needs.

Reboot PC - Simply reboot a PC to see if the error is gone. Frequently, rebooting the PC readjusts the internal settings and software plus therefore fixes the issue. If it doesn't then move on to follow the instructions under.

You are able to click here to find out how to speed up Windows and heighten PC perfomance. And you can click here to download a registry cleaner to help we clean up registry.

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