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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints For a different notion of boundary related to Manifolds see that article In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined

For example, if u is a solution to a partial differential equation on a set Ω with piecewise-smooth boundary $\partial \Omega,$ and $\partial \Omega$ is divided into two parts, Γ1 and Γ2, one can use a Dirichlet boundary condition on Γ1 and a Neumann boundary condition on Γ2,

$u_{\big| \Gamma_1} = u_0$
$\frac{\partial u}{\partial n}\bigg|_{\Gamma_2} = g$

where u0 and g are given functions defined on those portions of the boundary. In Mathematics, a piecewise-defined function (also called a piecewise function) is a function whose definition is dependent on the value of the Independent In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, the Dirichlet (or first type) boundary condition is a type of Boundary condition, named after Johann Peter Gustav Lejeune In Mathematics, the Neumann (or second type) boundary condition is a type of Boundary condition, named after Carl Neumann.

Robin boundary condition is another type of hybrid boundary condition; it is a linear combination of Dirichlet and Neumann boundary conditions. In Mathematics, the Robin (or third type) boundary condition is a type of Boundary condition, named after Victor Gustave Robin (1855-1897 In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics