#jsDisabledContent { display:none; } My Account |  Register |  Help

# Robin boundary condition

Article Id: WHEBN0009945920
Reproduction Date:

 Title: Robin boundary condition Author: World Heritage Encyclopedia Language: English Subject: Collection: Boundary Conditions Publisher: World Heritage Encyclopedia Publication Date:

### Robin boundary condition

In mathematics, the Robin boundary condition (; properly French: ), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897).[1] When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain.

Robin boundary conditions are a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. This contrasts to mixed boundary conditions, 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 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 \partial\Omega denotes its boundary, the Robin boundary condition is:

a u + b \frac{\partial u}{\partial n} =g \qquad \text{on} ~ \partial \Omega\,

for some non-zero constants a and b and a given function g defined on \partial\Omega. Here, u is the unknown solution defined on \Omega and {\partial u}/{\partial 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, \Omega = [0,1], the Robin boundary condition becomes the conditions:

a u(0) - bu'(0) =g(0)\,
a u(1) + bu'(1) =g(1).\,

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 solving Sturm–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 equations. Here, the convective and diffusive fluxes at the boundary sum to zero:

u_x(0)\,c(0) -D \frac{\partial c(0)}{\partial x}=0\,

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

1. ^ Gustafson, K., (1998). Domain Decomposition, Operator Trigonometry, Robin Condition, Contemporary Mathematics, 218. 432–437.
• Gustafson, K. and T. Abe, (1998a). (Victor) Gustave Robin: 1855–1897, The Mathematical Intelligencer, 20, 47–53.
• Gustafson, K. and T. Abe, (1998b). The third boundary condition – was it Robin's?, The Mathematical Intelligencer, 20, 63–71.
• Eriksson, K.; Estep, D.; Johnson, C. (2004). Applied mathematics, body and soul. Berlin; New York: Springer.
• Atkinson, Kendall E.; Han, Weimin (2001). Theoretical numerical analysis: a functional analysis framework. New York: Springer.
• Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C. (1996). Computational differential equations. Cambridge; New York: Cambridge University Press.
• Mei, Zhen (2000). Numerical bifurcation analysis for reaction-diffusion equations. Berlin; New York: Springer.
• Hahn, David W.; Ozisk, M. N. (2012). Heat Conduction, 3rd edition. New York: Wiley.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.