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| [[File:Boundary value problem-en.svg|thumb|right|Shows a region where a [[differential equation]] is valid and the associated boundary values]]
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| In [[mathematics]], in the field of [[differential equation]]s, a '''boundary value problem''' is a [[differential equation]] together with a set of additional restraints, called the '''boundary conditions'''. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
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| 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 involves the [[eigenfunction]]s of a [[differential operator]].
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| 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.
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| 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]].
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| ==Explanation==
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| Boundary value problems are similar to [[initial value problem]]s.
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| A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).
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| For example, if the independent variable is time over the domain [0,1], a boundary value problem would specify values for <math>y(t)</math> at both <math>t=0</math> and <math>t=1</math>, whereas an initial value problem would specify a value of <math>y(t)</math> and <math>y'(t)</math> at time <math>t=0</math>.
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| Finding the temperature at all points of an iron bar with one end kept at [[absolute zero]] and the other end at the freezing point of water would be a boundary value problem.
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| If the problem is dependent on both space and time, one could specify the value of the problem at a given point for all time the data or at a given time for all space.
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| Concretely, an example of a boundary value (in one spatial dimension) is the problem
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| :<math>y''(x)+y(x)=0 \, </math>
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| to be solved for the unknown function <math>y(x)</math> with the boundary conditions
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| :<math>y(0)=0, \ y(\pi/2)=2.</math> | |
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| Without the boundary conditions, the general solution to this equation is
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| :<math>y(x) = A \sin(x) + B \cos(x).\,</math>
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| From the boundary condition <math>y(0)=0</math> one obtains
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| :<math>0 = A \cdot 0 + B \cdot 1</math>
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| which implies that <math>B=0.</math> From the boundary condition <math>y(\pi/2)=2</math> one finds
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| :<math>2 = A \cdot 1 </math>
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| and so <math>A=2.</math> One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is
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| :<math>y(x)=2\sin(x). \,</math>
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| ==Types of boundary value problems==
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| [[Image:Bounday value problem for a rod.PNG|frame|right|The boundary value problem for an idealised [[Dimension|2D]] rod]]
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| If the boundary gives a value to the [[normal derivative]] of the problem then it is a [[Neumann boundary condition]]. For example, if there is a heater at one end of an iron rod, then energy would be added at a constant rate but the actual temperature would not be known.
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| If the boundary gives a value to the problem then it is a [[Dirichlet boundary condition]]. For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space.
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| If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a [[Cauchy boundary condition]].
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| Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For an [[elliptic operator]], one discusses [[elliptic boundary value problem]]s. For an [[hyperbolic operator]], one discusses [[hyperbolic boundary value problems]]. These categories are further subdivided into linear and various nonlinear types.
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| ==See also==
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| {{multicol}}
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| '''Related mathematics:'''
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| *[[initial value problem]]
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| *[[differential equation]]s
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| *[[Green's function]]s
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| *[[Stochastic processes and boundary value problems]]
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| *[[Sturm–Liouville theory]]
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| *[[Dirichlet boundary condition]]
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| *[[Neumann boundary condition]]
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| *[[Robin boundary condition]]
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| *[[Sommerfeld radiation condition]]
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| *[[Cauchy boundary condition]]
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| *[[Mixed boundary condition]]
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| *[[Perfect thermal contact|Perfect thermal contact condition]]
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| {{multicol-break}}
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| '''Physical applications:'''
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| *[[wave]]s
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| *[[normal mode]]s
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| *[[electrostatics]]
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| *[[Laplace's equation]]
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| *[[potential theory]]
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| *[[Computation of radiowave attenuation in the atmosphere]]
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| *[[Black holes]]
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| {{multicol-break}}
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| '''Numerical algorithms:'''
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| *[[shooting method]]
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| *[[direct multiple shooting method]]
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| {{multicol-end}}
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| == References ==
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| * A. D. Polyanin and V. F. Zaitsev, ''Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)'', Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.
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| * A. D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9.
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| == External links==
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| * {{springer|title=Boundary value problems in potential theory|id=p/b017390}}
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| * {{springer|title=Boundary value problem, complex-variable methods|id=p/b017340}}
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| * [http://eqworld.ipmnet.ru/en/solutions/lpde.htm Linear Partial Differential Equations: Exact Solutions and Boundary Value Problems] at EqWorld: The World of Mathematical Equations.
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| * {{scholarpedia|title=Boundary value problem|urlname=Boundary_value_problem}}
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| [[Category:Ordinary differential equations]]
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| [[Category:Partial differential equations]]
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| [[Category:Boundary conditions| ]]
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| [[Category:Mathematical problems]]
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| [[fr:Problème aux limites]]
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