View source for Robin boundary condition

{{Short description|Class of problems in PDEs}}
{{Differential equations}}
{{Technical|introduction|date=March 2024}}
In [[mathematics]],  the '''Robin boundary condition''' ({{IPAc-en|ˈ|r|ɒ|b|ɪ|n}} {{respell|ROB|in}}, {{IPA|fr|ʁɔbɛ̃|lang}}), or '''third type boundary condition''', is a type of [[boundary condition]], named  after [[Victor Gustave Robin]] (1855&ndash;1897).<ref>Gustafson, K., (1998). Domain Decomposition, Operator Trigonometry, Robin Condition, ''Contemporary Mathematics'', '''218'''. 432&ndash;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. Other equivalent names in use are '''Fourier-type condition''' and '''radiation condition'''.<ref>Logan, J. David, (2001). Transport Modeling in Hydrogeochemical Systems. Springer.</ref>

==Definition==
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, or '''convective boundary conditions''', from their application in [[heat transfer]] problems (Hahn, 2012).

If Ω is the domain on which the given equation is to be solved and ∂Ω denotes its [[boundary (topology)|boundary]], the Robin boundary condition is:<ref>{{cite book|title=Finite Element Analysis with Error Estimators: An Introduction to the FEM and Adaptive Error Analysis for Engineering Students|author=J. E. Akin|publisher=Butterworth-Heinemann|year=2005|isbn=9780080472751|page=69|url=https://books.google.com/books?id=uUsm99d8WBAC&pg=PA69}}</ref>
:<math>a u + b \frac{\partial u}{\partial n} =g \qquad \text{on } \partial \Omega</math>
for some non-zero constants  ''a'' and ''b'' and a given function ''g'' defined on ∂Ω. Here, ''u'' is the unknown solution defined on Ω and {{sfrac|∂''u''|∂''n''}} denotes the [[normal derivative]] at the boundary. More generally, ''a'' and ''b'' are allowed to be (given) functions, rather than constants.

In one dimension, if, for example, Ω&nbsp;= [0,1], the Robin boundary condition becomes the conditions:
:<math>\begin{align} a u(0) - bu'(0) &=g(0) \\ a u(1) + bu'(1) &=g(1) \end{align}</math>
Notice the change of sign in front of the term involving a derivative: that is because the normal to [0,1] at 0 points in the negative direction, while at 1 it points in the positive direction.

==Application==
Robin boundary conditions are commonly used in solving [[Sturm&ndash;Liouville problems]] which appear in many contexts in science and engineering.

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:

:<math>u_x(0)\,c(0) -D \frac{\partial c(0)}{\partial x}=0</math>

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]].

==References==
{{Reflist}}

==Bibliography==
*Gustafson, K. and T. Abe, (1998a). [[doi:10.1007/BF03024402|The third boundary condition &ndash; was it Robin's?]], ''The Mathematical Intelligencer'', '''20''', #1, 63&ndash;71.
*Gustafson, K. and T. Abe, (1998b). (Victor) Gustave Robin: 1855–1897, ''The Mathematical Intelligencer'', '''20''', #2, 47&ndash;53.
*{{cite book
 | last       = Eriksson
 | first      = K. |author2=Estep, D. |author3=Johnson, C.
 | title      = Applied mathematics, body and soul
 | publisher  = Berlin; New York: Springer
 | year       = 2004
 | pages      = 
 | isbn       = 3-540-00889-6
}}

*{{cite book
 | last       = Atkinson
 | first      = Kendall E.
 |author2=Han, Weimin
  | title      = Theoretical numerical analysis: a functional analysis framework
 | publisher  = New York: Springer
 | year       = 2001
 | pages      = 
 | isbn       = 0-387-95142-3
}}

*{{cite book |last       = Eriksson
 |first      = K.
 |author2    = Estep, D.
 |author3    = Hansbo, P.
 |author4    = Johnson, C.
 |title      = Computational differential equations
 |publisher  = Cambridge; New York: Cambridge University Press
 |year       = 1996
 |pages      = 
 |isbn       = 0-521-56738-6
 |url-access = registration
 |url        = https://archive.org/details/computationaldif0000unse
}}

*{{cite book
 | last       = Mei
 | first      = Zhen
 | title      = Numerical bifurcation analysis for reaction-diffusion equations
 | publisher  = Berlin; New York: Springer
 | year       = 2000
 | pages      = 
 | isbn       = 3-540-67296-6
}}

*{{cite book
 | last       = Hahn
 | first      = David W.
 |author2=Ozisk, M. N. 
  | title      = Heat Conduction, 3rd edition
 | publisher  = New York: Wiley
 | year       = 2012
 | pages      = 
 | isbn       = 978-0-470-90293-6
}}

[[Category:Boundary conditions]]