Term information
The diffusion equation is an equation for diffusion which states that the rate of change of the density of the diffusing substance, at a fixed point in space, equals the sum of the diffusion coefficient times the Laplacian of the density, the amount of the quantity generated per unit volume per unit time, and the negative of the quantity absorbed per unit volume per unit time. More generally, any equation which states that the rate of change of some quantity, at a fixed point in space, equals a positive constant times the Laplacian of that quantity. source: http://www.answers.com/topic/diffusion-equation The diffusion equation is a partial differential equation which describes density fluctuations 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. The equation is usually written as: \frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla \cdot \big[ D(\phi,\mathbf{r}) \, \nabla\phi(\mathbf{r},t) \big], where \, \phi(\mathbf{r},t) is the density of the diffusing material at location \mathbf{r} and time t and \, D(\phi,\mathbf{r}) is the collective diffusion coefficient for density \,\phi at location \mathbf{r}; the nabla symbol \, \nabla represents the vector differential operator del acting on the space coordinates. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear. If \, D is constant, then the equation reduces to the following linear equation: \frac{\partial\phi(\mathbf{r},t)}{\partial t} = D\nabla^2\phi(\mathbf{r},t), also called the heat equation. More generally, when D is a symmetric positive definite matrix, the equation describes anisotropic diffusion, which is written (for three dimensional diffusion) as: \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] source: http://en.wikipedia.org/wiki/Diffusion_equation
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mfrac> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>Φ</mi> <mo>⁢</mo> <mfenced close=")" open="("> <mstyle mathvariant="bold"> <mi>r</mi> </mstyle> <mi>t</mi> </mfenced> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <mo>∇</mo> <mo>⁢</mo> <mo>⋅</mo> <mo>⁢</mo> <mfenced close="]" open="["> <mrow> <mrow> <mi>D</mi> <mo>⁢</mo> <mfenced close=")" open="("> <mi>Φ</mi> <mstyle mathvariant="bold"> <mi>r</mi> </mstyle> </mfenced> </mrow> <mo>⁢</mo> <mrow> <mi> </mi> <mo>⁢</mo> <mo>∇</mo> <mo>⁢</mo> <mi>Φ</mi> <mo>⁢</mo> <mfenced close=")" open="("> <mstyle mathvariant="bold"> <mi>r</mi> </mstyle> <mi>t</mi> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math>
\frac{\partial\Phi(\mathbf{r},t)}{\partial t} = \nabla \cdot [ D(\Phi,\mathbf{r}) \ \nabla\Phi(\mathbf{r},t)]
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>Φ</mi> <mfenced close=")" open="("> <mstyle mathvariant="bold"> <mi>r</mi> </mstyle> <mi>t</mi> </mfenced> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mo>∇</mo> <mo>⋅</mo> <mfenced close="]" open="["> <mrow> <mi>D</mi> <mfenced close=")" open="("> <mi>Φ</mi> <mstyle mathvariant="bold"> <mi>r</mi> </mstyle> </mfenced> <mi> </mi> <mo>∇</mo> <mi>Φ</mi> <mfenced close=")" open="("> <mstyle mathvariant="bold"> <mi>r</mi> </mstyle> <mi>t</mi> </mfenced> </mrow> </mfenced> </mrow> <annotation encoding="SnuggleTeX">\[ \frac{\partial\Phi(\mathbf{r},t)}{\partial t} = \nabla \cdot [ D(\Phi,\mathbf{r}) \ \nabla\Phi(\mathbf{r},t)] \]</annotation> </semantics> </math>
Term relations
- equation
- partial differential equation
- has_variable some (
variable and
is_about some time) - has_variable some (
variable and
is_about some dense) - has_variable some (
variable and
is_about some (quality_of some diffusion)) - has_variable some (
variable and
is_about some position)