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Sets both a <Dirichlet boundary condition> and a <Neumann boundary condition> for a single part of the boundary.

Can be used for <hyperbolic partial differential equations>.

We understand intuitively that this imposes stricter requirements on solutions, which makes it easier to guarantee uniqueness, but also harder to have existence. TODO intuitively why hyperbolic need this extra level of restriction.


Cauchy boundary condition

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Linear combination of a <Dirichlet boundary condition> and <Neumann boundary condition> at each point of the boundary.

Examples:
* <heat equation> when metal plaque is immersed in a large external environment of fixed temperature.

  In this case, the normal derivative at the boundary is proportional to the difference between the temperature of the boundary and the fixed temperature of the external environment.

  The result as time tends to infinity is that the temperature of the plaque tends to that of the environment.

  Shown a solved example in the <FreeFem> tutorial: https://doc.freefem.org/tutorials/thermalConduction.html (https://github.com/FreeFem/FreeFem-doc/blob/1d5996d8b891fd553fd318321249c2c30f693fc3/source/tutorials/thermalConduction.rst)


Robin boundary condition

http://www.cns.gatech.edu/~predrag/courses/PHYS-6124-12/StGoChap6.pdf 6.1 "Classification of PDE's" clarifies which boundary conditions are needed for existence and uniqueness of each <classification of second order partial differential equations into elliptic, parabolic and hyperbolic>[type of second order of PDE]:
* <elliptic partial differential equation> and <parabolic partial differential equation>: <Dirichlet boundary condition> or <Neumann boundary condition>
* <hyperbolic partial differential equation>: <Cauchy boundary condition>


Which boundary conditions lead to existence and uniqueness of a second order PDE

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Specifies the derivative in a direction normal to the boundary.

Can be used for <elliptic partial differential equations> and <parabolic partial differential equations>.


Ciro Santilli @cirosantilli 37

 Incoming links: Neumann boundary condition