Function field of an algebraic variety: Difference between revisions
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[[File:Mixed boundary conditions.svg|right|thumb|Green: Neumann boundary condition; purple: Dirichlet boundary condition.]] | |||
In [[mathematics]], a '''mixed boundary condition''' for a [[partial differential equation]] indicates that different [[boundary condition]]s are used on different parts of the [[boundary (topology)|boundary]] of the [[domain (mathematics)|domain]] of the equation. | |||
For example, if ''u'' is a solution to a partial differential equation on a set <math>\Omega</math> with [[piecewise]]-[[smooth function|smooth]] boundary <math>\partial\Omega</math>, and <math>\partial\Omega</math> is divided into two parts, <math>\Gamma_1</math> and <math>\Gamma_2</math>, one can use a [[Dirichlet boundary condition]] on <math>\Gamma_1</math> and a [[Neumann boundary condition]] on <math>\Gamma_2</math>: | |||
: <math>u_{\big| \Gamma_1} = u_0</math> | |||
: <math>\left. \frac{\partial u}{\partial n}\right|_{\Gamma_2} = g</math> | |||
where ''u₀'' and ''g'' are given functions defined on those portions of the boundary. | |||
[[Robin boundary condition]] is another type of hybrid boundary condition; it is a [[linear combination]] of Dirichlet and Neumann boundary conditions. | |||
==See also== | |||
*[[Dirichlet boundary condition]] | |||
*[[Neumann boundary condition]] | |||
*[[Cauchy boundary condition]] | |||
*[[Robin boundary condition]] | |||
==References== | |||
*{{cite book | |||
| last = Guru | |||
| first = Bhag S. | |||
| coauthors = Hızıroğlu, Hüseyin R. | |||
| title = Electromagnetic field theory fundamentals, 2nd ed. | |||
| publisher = Cambridge, UK; New York: Cambridge University Press | |||
| date = 2004 | |||
| isbn = 0-521-83016-8 | |||
| page = 593 | |||
}} | |||
{{Mathapplied-stub}} | |||
[[Category:Boundary conditions]] | |||
[[Category:Partial differential equations]] | |||
Revision as of 19:12, 14 March 2013
In mathematics, a mixed boundary condition for a partial differential equation indicates that different boundary conditions are used on different parts of the boundary of the domain of the equation.
For example, if u is a solution to a partial differential equation on a set with piecewise-smooth boundary , and is divided into two parts, and , one can use a Dirichlet boundary condition on and a Neumann boundary condition on :
where u₀ and g are given functions defined on those portions of the boundary.
Robin boundary condition is another type of hybrid boundary condition; it is a linear combination of Dirichlet and Neumann boundary conditions.
See also
- Dirichlet boundary condition
- Neumann boundary condition
- Cauchy boundary condition
- Robin boundary condition
References
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