Function field of an algebraic variety

File:Mixed boundary conditions.svg

Green: Neumann boundary condition; purple: Dirichlet boundary condition.

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 $\mathrm{\Omega }$ with piecewise-smooth boundary $\partial \mathrm{\Omega }$, and $\partial \mathrm{\Omega }$ is divided into two parts, ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$, one can use a Dirichlet boundary condition on ${\mathrm{\Gamma }}_1$ and a Neumann boundary condition on ${\mathrm{\Gamma }}_2$:

$u_{|{\mathrm{\Gamma }}_1}=u_0$

${\frac{\partial u}{\partial n}|}_{{\mathrm{\Gamma }}_2}=g$

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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