Function field of an algebraic variety

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 $\Omega$ with piecewise-smooth boundary $\partial \Omega$, and $\partial \Omega$ is divided into two parts, $\Gamma _{1}$ and $\Gamma _{2}$, one can use a Dirichlet boundary condition on $\Gamma _{1}$ and a Neumann boundary condition on $\Gamma _{2}$:

$u_{{\big |}\Gamma _{1}}=u_{0}$

$\left.{\frac {\partial u}{\partial n}}\right|_{\Gamma _{2}}=g$

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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