This article's lead sectionmay be too technical for most readers to understand. Pleasehelp improve it tomake it understandable to non-experts, without removing the technical details.(March 2024) (Learn how and when to remove this message)

Inmathematics, theRobin boundary condition (/ˈrɒbɪn/ROB-in,French:[ʁɔbɛ̃]), orthird type boundary condition, is a type ofboundary condition, named afterVictor Gustave Robin (1855–1897).^[1] When imposed on anordinary or apartial differential equation, it is a specification of alinear combination of the values of afunctionand the values of its derivative on theboundary of the domain. Other equivalent names in use areFourier-type condition andradiation condition.^[2]

Robin boundary conditions are a weighted combination ofDirichlet boundary conditions andNeumann boundary conditions. This contrasts tomixed boundary conditions, which are boundary conditions of different types specified on different subsets of the boundary. Robin boundary conditions are also calledimpedance boundary conditions, from their application inelectromagnetic problems, orconvective boundary conditions, from their application inheat transfer problems (Hahn, 2012).

If Ω is the domain on which the given equation is to be solved and ∂Ω denotes itsboundary, the Robin boundary condition is:^[3]

${\displaystyle au+b\frac{\mathrm{\partial }u}{\mathrm{\partial }n}=g\text{on }\mathrm{\partial }\mathrm{\Omega }}$

for some non-zero constantsa andb and a given functiong defined on ∂Ω. Here,u is the unknown solution defined on Ω and⁠∂u/∂n⁠ denotes thenormal derivative at the boundary. More generally,a andb are allowed to be (given) functions, rather than constants.

In one dimension, if, for example, Ω = [0,1], the Robin boundary condition becomes the conditions:

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.

Robin boundary conditions are commonly used in solvingSturm–Liouville problems which appear in many contexts in science and engineering.

In addition, the Robin boundary condition is a general form of theinsulating boundary condition forconvection–diffusion equations. Here, the convective and diffusive fluxes at the boundary sum to zero:

${\displaystyle u_x(0)c(0)-D\frac{\mathrm{\partial }c(0)}{\mathrm{\partial }x}=0}$

whereD is the diffusive constant,u is the convective velocity at the boundary andc is the concentration. The second term is a result ofFick's law of diffusion.

• Gustafson, K. and T. Abe, (1998a).The third boundary condition – was it Robin's?,The Mathematical Intelligencer,20, #1, 63–71.
• Gustafson, K. and T. Abe, (1998b). (Victor) Gustave Robin: 1855–1897,The Mathematical Intelligencer,20, #2, 47–53.
• Eriksson, K.; Estep, D.; Johnson, C. (2004).Applied mathematics, body and soul. Berlin; New York: Springer.ISBN 3-540-00889-6.

• Atkinson, Kendall E.; Han, Weimin (2001).Theoretical numerical analysis: a functional analysis framework. New York: Springer.ISBN 0-387-95142-3.

• Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C. (1996).Computational differential equations. Cambridge; New York: Cambridge University Press.ISBN 0-521-56738-6.

• Mei, Zhen (2000).Numerical bifurcation analysis for reaction-diffusion equations. Berlin; New York: Springer.ISBN 3-540-67296-6.

• Hahn, David W.; Ozisk, M. N. (2012).Heat Conduction, 3rd edition. New York: Wiley.ISBN 978-0-470-90293-6.

• Boundary conditions

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles that are too technical from March 2024
• All articles that are too technical
• Pages with French IPA