Reaction–diffusion system

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Published on May 12, 2023 3:50:36 PM ● Video Link: https://www.youtube.com/watch?v=9Y7gHcT-pik

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Reaction–diffusion systems are mathematical models which correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: 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.
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 (neutron diffusion theory) and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form
$$\partial_t\boldsymbol{q} =\underline{\underline{\boldsymbol{D}}}\,\nabla^2\boldsymbol{q} +\boldsymbol{R}(\boldsymbol{q}),$$
where represents the unknown vector function, is a diagonal matrix of diffusion coefficients, and 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. Such patterns have been dubbed "Turing patterns". Each function, for which a reaction diffusion differential equation holds, represents in fact a concentration variable.

Source: https://en.wikipedia.org/wiki/Reaction–diffusion_system
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