mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Robin boundary condition ( , ), or third type boundary condition, is a type of

boundary condition In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...

, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

, it is a specification of a

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

of the values of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

''and'' the values of its derivative on the boundary of the domain. Other equivalent names in use are Fourier-type condition and radiation condition.

Definition

Robin boundary conditions are a weighted combination of

Dirichlet boundary condition In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes along the boundary of the domain are fixed. The question of finding solutions to such equat ...

s and

Neumann boundary condition In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative app ...

s. This contrasts to

mixed boundary condition In mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of ...

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

electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...

problems, or convective boundary conditions, from their application in

heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...

problems (Hahn, 2012). If Ω is the domain on which the given equation is to be solved and ∂Ω denotes its boundary, the Robin boundary condition is: :

a u + b \frac =g \qquad \text \partial \Omega

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 denotes the

normal derivative In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. The directional derivative of a multivariable differentiable (scalar) function along a given vector ...

at the boundary. More generally, ''a'' and ''b'' are allowed to be (given) functions, rather than constants. In one dimension, if, for example, Ω = ,1 the Robin boundary condition becomes the conditions: :

\begin a u(0) - bu'(0) &=g(0) \\ a u(1) + bu'(1) &=g(1) \end

Notice the change of sign in front of the term involving a derivative: that is because the normal to ,1at 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–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 The convection–diffusion equation is a parabolic partial differential equation that combines the diffusion equation, diffusion and convection (advection equation, advection) equations. It describes physical phenomena where particles, energy, or o ...

s. Here, the convective and diffusive fluxes at the boundary sum to zero: :

u_x(0)\,c(0) -D \frac=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 Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the Mass diffusivity, diffusion coefficient, . Fick's first law can be used to ...

References

Bibliography

*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. * * * * *{{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 Boundary conditions