Reaction diffusion equation

**Reaction diffusion equation** is used to model how for example chemical concentrations changes over time and space. According to Wikipedia:

They are mathematical models which explain how the concentration of one or more substances distributed in space changes under the influence of two processes: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space. This description implies that reaction–diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form

$\partial_t\vec q = D\nabla^2\vec q+R(\vec q)$

where each component of the vector $q(x,t)$ represents the concentration of one substance, $D$ is a diagonal matrix of diffusion coefficients, and $R$ accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of travelling waves and wave-like phenomena as well as other self-organized patterns like stripes, hexagons or more intricate structure like dissipative solitons.

- Reaction diffusion equation, Wikipedia.

category: mathematical methods