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ReactionPDETerm

ReactionPDETerm

ReactionPDETerm[vars,a]

represents a reaction term with reaction coefficient and with model variables vars.

ReactionPDETerm[{u,{x₁,…,x_n}},a,pars]

uses model parameters pars.

Details

Reaction terms are used to model absorption or emission in a number of domains, such as biology, chemistry and physics.
Reaction with a reaction coefficient is the process of absorbing of the dependent variable :

ReactionPDETerm returns differential operators term to be used as a part of partial differential equations:

ReactionPDETerm can be used to model reaction equations with dependent variable , independent variables and time variable .
Stationary model variables vars are vars={u[x₁,…,x_n],{x₁,…,x_n}}.
Time-dependent model variables vars are vars={u[t,x₁,…,x_n],{x₁,…,x_n}} or vars={u[t,x₁,…,x_n],t,{x₁,…,x_n}}.
The reaction term in context with other PDE terms is given by:

The reaction coefficient has the following form:
scalar a
For a system of PDEs with dependent variables {u₁,…,u_m}, the reaction represents:

The reaction term in context systems of PDE terms:

The reaction coefficient is a tensor of rank 2 of the form where each submatrix is a scalar that can specified in the same way as for a single dependent variable.
The reaction coefficient can depend on time, space, parameters and the dependent variables.
The coefficient does not affect the meaning of NeumannValue.
All quantities that do not explicitly depend on the independent variables given are taken to have zero partial derivative.

Examples

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Basic Examples (4)

Define a stationary reaction term:

Define a stationary reaction term with a parameter:

Solve a reaction diffusion equation build with basic terms:

Visualize the result:

Solve for the eigenvalues of a reaction diffusion equation:

Scope (7)

Define a time-dependent reaction term:

Define a symbolic reaction term:

Define a 3D axisymmetric time-independent reaction term:

Define a 2D stationary reaction term:

Define a reaction term with multiple dependent variables:

Define a Helmholtz model:

Solve for the eigenvalues of the Helmholtz equation:

Solve the Helmholtz equation with a source term:

Visualize the solution:

Solve a nonlinear reaction diffusion equation build with basic terms:

Visualize the result:

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