Editing Diffusion equation (section)

== Statement ==

The equation is usually written as:
<math display="block">\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla \cdot \big[ D(\phi,\mathbf{r}) \ \nabla\phi(\mathbf{r},t) \big],</math>
where {{math|''ϕ''('''r''', ''t'')}} is the [[density]] of the diffusing material at location {{math|'''r'''}} and time {{mvar|t}} and {{math|''D''(''ϕ'', '''r''')}} is the collective [[diffusion coefficient]] for density {{mvar|ϕ}} at location {{math|'''r'''}}; and {{math|∇}} represents the vector [[differential operator]] [[del]]. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.

The equation above applies when the diffusion coefficient is [[Isotropy|isotropic]]; in the case of anisotropic diffusion, {{mvar|D}} is a symmetric [[positive definite matrix]], and the equation is written (for three dimensional diffusion) as:
<math display="block">\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>
The diffusion equation has numerous analytic solutions.<ref>{{Cite journal |last1=Barna |first1=I.F. |last2=Mátyás |first2=L. |year=2022 |title=Advanced Analytic Self-Similar Solutions of Regular and Irregular Diffusion Equations |journal=Mathematics |volume=10 |issue=18 |pages=3281 |doi=10.3390/math10183281|doi-access=free |arxiv=2204.04895 }}</ref> 

If {{mvar|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>
which is identical to the [[heat equation]].