Editing Boundary value problem (section)

{{Short description|Type of problem involving ODEs or PDEs}}
[[File:Boundary value problem-en.svg|300px|thumb|right|Shows a region where a [[differential equation]] is valid and the associated boundary values]]
{{Differential equations|expanded=General }}

In the study of [[differential equation]]s, a '''boundary-value problem''' is a [[differential equation]] subjected to constraints called '''boundary conditions'''.<ref name="Zwillinger2014">{{cite book|author=Daniel Zwillinger|title=Handbook of Differential Equations|url=https://books.google.com/books?id=9QLjBQAAQBAJ&q=%22boundary+value%22&pg=PA536|date=12 May 2014|publisher=Elsevier Science|isbn=978-1-4832-2096-3|pages=536–}}</ref> A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the [[wave equation]], such as the determination of [[normal mode]]s, are often stated as boundary value problems. A large class of important boundary value problems are the [[Sturm–Liouville theory|Sturm–Liouville problems]]. The analysis of these problems, in the linear case, involves the [[eigenfunction]]s of a [[differential operator]].

To be useful in applications, a boundary value problem should be [[well-posed problem|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 equation]]s 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 function]]s (solutions to [[Laplace's equation]]); the solution was given by the [[Dirichlet's principle]].