Mixed boundary condition

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

• Guru, Bhag S.; Hiziroglu, Hüseyin R. (2004). Electromagnetic field theory fundamentals, 2nd ed.. Cambridge, UK; New York: Cambridge University Press. p. 593. ISBN 0521830168.