Boundary value problem

Shows a region where a differential equation is valid and the associated boundary values

In mathematics, a boundary value problem is a problem to solve a set of differential equations. In addition, there are a set of constraints which are called the boundary constraints. A solution must both solve the differential equations, and the boundary constraints.

Boundary value problems are common in many fields of science.

Boundary value problems can be found in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, for example the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator.

To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.

Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle.