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| In [[mathematics]], the '''Robin boundary condition''' ({{IPAc-en|ˈ|r|ɔː|b|ɪ|n}}; properly {{IPA-fr|ʁoˈbɛ̃|lang}}), or '''third type boundary condition''', is a type of [[boundary condition]], named after [[Victor Gustave Robin]] (1855–1897).<ref>Gustafson, K., (1998). Domain Decomposition, Operator Trigonometry, Robin Condition, ''Contemporary Mathematics'', '''218'''. 432–437.</ref> When imposed on an [[ordinary differential equation|ordinary]] or a [[partial differential equation]], it is a specification of a [[linear combination]] of the values of a [[function (mathematics)|function]] ''and'' the values of its derivative on the [[boundary (topology)|boundary]] of the domain.
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| Robin boundary conditions are a weighted combination of [[Dirichlet boundary condition]]s and [[Neumann boundary condition]]s. This contrasts to [[mixed boundary condition]]s, which are boundary conditions of different types specified on different subsets of the boundary. Robin boundary conditions are also called '''impedance boundary conditions''', from their application in [[Electromagnetism|electromagnetic]] problems.
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| If Ω is the domain on which the given equation is to be solved and <math>\partial\Omega</math> denotes its [[boundary (topology)|boundary]], the Robin boundary condition is:
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| :<math>a u + b \frac{\partial u}{\partial n} =g \qquad \text{on} ~ \partial \Omega\,</math>
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| for some non-zero constants ''a'' and ''b'' and a given function ''g'' defined on <math>\partial\Omega</math>. Here, ''u'' is the unknown solution defined on <math>\Omega</math> and <math>{\partial u}/{\partial n}</math> denotes the [[normal derivative]] at the boundary. More generally, ''a'' and ''b'' are allowed to be (given) functions, rather than constants.
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| In one dimension, if, for example, <math>\Omega = [0,1]</math>, the Robin boundary condition becomes the conditions:
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| :<math>a u(0) - bu'(0) =g(0)\,</math>
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| :<math>a u(1) + bu'(1) =g(1).\,</math> | |
| notice the change of sign in front of the term involving a derivative: that is because the normal to <math>[0,1]</math> at 0 points in the negative direction, while at 1 it points in the positive direction.
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| Robin boundary conditions are commonly used in solving [[Sturm–Liouville problems]] which appear in many contexts in science and engineering.
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| In addition, the Robin boundary condition is a general form of the '''insulating boundary condition''' for [[convection–diffusion equation]]s. Here, the convective and diffusive fluxes at the boundary sum to zero:
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| :<math>u_x(0)\,c(0) -D \frac{\partial c(0)}{\partial x}=0\,</math>
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| where ''D'' is the diffusive constant, ''u'' is the convective velocity at the boundary and ''c'' is the concentration. The second term is a result of [[Fick's law of diffusion]].
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| ==See also==
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| *[[Dirichlet boundary condition]]
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| *[[Neumann boundary condition]]
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| *[[Mixed boundary condition]]
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| *[[Cauchy boundary condition]]
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| ==References==
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| <references />
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| *Gustafson, K. and T. Abe, (1998a). (Victor) Gustave Robin: 1855–1897, ''The Mathematical Intelligencer'', 20, 47–53.
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| *Gustafson, K. and T. Abe, (1998b). The third boundary condition – was it Robin's?, ''The Mathematical Intelligencer'', '''20''', 63–71.
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| *{{cite book
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| | last = Eriksson
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| | first = K.
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| | coauthors = Estep, D.; Johnson, C.
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| | title = Applied mathematics, body and soul
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| | publisher = Berlin; New York: Springer
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| | year = 2004
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| | pages =
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| | isbn = 3-540-00889-6
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| }}
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| *{{cite book
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| | last = Atkinson
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| | first = Kendall E.
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| | coauthors = Han, Weimin
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| | title = Theoretical numerical analysis: a functional analysis framework
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| | publisher = New York: Springer
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| | year = 2001
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| | pages =
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| | isbn = 0-387-95142-3
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| }}
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| *{{cite book
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| | last = Eriksson
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| | first = K.
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| | coauthors = Estep, D.; Hansbo, P.; Johnson, C.
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| | title = Computational differential equations
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| | publisher = Cambridge; New York: Cambridge University Press
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| | year = 1996
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| | pages =
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| | isbn = 0-521-56738-6
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| }}
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| *{{cite book
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| | last = Mei
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| | first = Zhen
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| | title = Numerical bifurcation analysis for reaction-diffusion equations
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| | publisher = Berlin; New York: Springer
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| | year = 2000
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| | isbn = 3-540-67296-6
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| }}
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| [[Category:Boundary conditions]]
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