Boundary value problem

A boundary value problem asks for a differential equation solution together with conditions on the boundary of a specified domain. These can prescribe the value, a normal derivative, a flux, or a combination. For an ordinary differential equation on an interval, the conditions are often distributed between its endpoints; a partial differential equation uses the spatial boundary and, for evolution, also needs appropriate initial data.

For the Dirichlet boundary conditions intwice integrating gives $u(x)=x(1-x)/2$. The boundary conditions determine the two constants. By contrast, for $-u^{\prime \prime }=f$ with derivative-zero conditions at both ends, integrating the equation requires ${\int }_0^{1}f=0$ and leaves an additive constant undetermined. This is the Neumann boundary-value problem compatibility and normalization issue.

Existence, uniqueness and continuous dependence on forcing belong to the whole operator/domain/boundary specification. A formal differential formula alone does not establish that a boundary value problem is solvable.^[ref]