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Reaction diffusion equation

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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

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

where each component of the vector q(x,t)q(x,t) represents the concentration of one substance, DD is a diagonal matrix of diffusion coefficients, and RR 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.

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