Robin boundary condition

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In mathematics, the Robin (or third type) boundary condition is a type of boundary condition, named after (Victor) Gustave Robin (1855-1897) who lectured in mathematical physics at the Sorbonne in Paris and worked in the area of thermodynamics.[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 should be pronounced as a French name, although some English speaking mathematicians anglicize the word.

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.

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 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 Ω, 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, Ω = [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.

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[edit] 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. ISBN 3540008896. 
  • Atkinson, Kendall E.; Han, Weimin (2001). Theoretical numerical analysis: a functional analysis framework. New York: Springer. ISBN 0387951423. 
  • Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C. (1996). Computational differential equations. Cambridge; New York: Cambridge University Press. ISBN 0521567386. 
  • Mei, Zhen (2000). Numerical bifurcation analysis for reaction-diffusion equations. Berlin; New York: Springer. ISBN 3540672966.