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Boundary value problem

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From Wikipedia, the free encyclopedia

Type of problem involving ODEs or PDEs

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

Differential equations

Scope

Fields

Classification

Types

By variable type
Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear
Dependent and independent variables Autonomous Coupled / Decoupled Exact Homogeneous / Nonhomogeneous
Features
Order Operator Notation

Relation to processes

Difference(discrete analogue)

Solution

Existence and uniqueness

General topics

Initial conditions
Boundary values
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function

Solution methods

People

List

Specific equations

Named equations

List of named differential equations

In the study ofdifferential equations, aboundary-value problem is adifferential equation subjected to constraints calledboundary conditions.^[1] 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 thewave equation, such as the determination ofnormal modes, are often stated as boundary value problems. A large class of important boundary value problems are theSturm–Liouville problems. The analysis of these problems, in the linear case, involves theeigenfunctions of adifferential operator.

To be useful in applications, a boundary value problem should bewell 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 ofpartial 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 theDirichlet problem, of finding theharmonic functions (solutions toLaplace's equation); the solution was given by theDirichlet's principle.

Explanation

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Boundary value problems are similar toinitial value problems. 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). Aboundary value is a data value that corresponds to a minimum or maximum input, internal, or output value specified for a system or component.^[2]

For example, if the independent variable is time over the domain [0,1], a boundary value problem would specify values for $y(t)$ at both $t=0$ and $t=1$ , whereas an initial value problem would specify a value of $y(t)$ and $y'(t)$ at time $t=0$ .

Finding the temperature at all points of an iron bar with one end kept atabsolute zero and the other end at the freezing point of water would be a boundary value problem.

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 or at a given time for all space.

Concretely, an example of a boundary value problem (in one spatial dimension) is

y''(x)+y(x)=0

to be solved for the unknown function $y(x)$ with the boundary conditions

y(0)=0,\ y(\pi /2)=2.

Without the boundary conditions, the general solution to this equation is

y(x)=A\sin(x)+B\cos(x).

From the boundary condition $y(0)=0$ one obtains

0=A\cdot 0+B\cdot 1

which implies that $B=0.$ From the boundary condition $y(\pi /2)=2$ one finds

2=A\cdot 1

and so $A=2.$ One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is

y(x)=2\sin(x).

Types of boundary value problems

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The 3 standard classes^[3] of conditions are Dirichlet, Neumann and Cauchy^[4] or Robin,^[3] with also mixed,^[5] boundary as infinity.^[6]

Summary of boundary conditions for the unknown function, $y {\displaystyle y}$ , constants $c_{0}$ and $c_{1}$ specified by the boundary conditions, and known scalar functions $f {\displaystyle f}$ and $g {\displaystyle g}$ specified by the boundary conditions.

Name	Form on 1st part of boundary	Form on 2nd part of boundary
Dirichlet	$y=f$
Neumann	${\partial y \over \partial n}=f$
Robin	$c_{0}y+c_{1}{\partial y \over \partial n}=f$
Cauchy	both $y=f$ and ${\partial y \over \partial n}=g$
Mixed	$y=f$	$c_{0}y+c_{1}{\partial y \over \partial n}=g$

Boundary value conditions

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Finding a function to describe the temperature of this idealised 2D rod is a boundary value problem withDirichlet boundary conditions. Any solution function will both solve theheat equation, and fulfill the boundary conditions of a temperature of 0 K on the left boundary and a temperature of 273.15 K on the right boundary.

A type 1 boundary condition,Dirichlet boundary condition,^[7] specifies the value of the function itself. 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.

A type 2 boundary condition,Neumann boundary condition,^[7] specifies the value of thenormal derivative of the function. 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.

A type 3 boundary condition is the Robin condition.^[7]

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 aCauchy boundary condition.

A type 0 boundary condition has no physical boundary.^[7]

Differential operators

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Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For anelliptic operator, one discusseselliptic boundary value problems. For ahyperbolic operator, one discusseshyperbolic boundary value problems. These categories are further subdivided intolinear and various nonlinear types.

Applications

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Electromagnetic potential

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Inelectrostatics, a common problem is to find a function which describes theelectric potential of a given region. If the region does not contain charge, the potential must be a solution toLaplace's equation (a so-calledharmonic function). The boundary conditions in this case are theInterface conditions for electromagnetic fields. If there is nocurrent density in the region, it is also possible to define amagnetic scalar potential using a similar procedure.

References

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^Daniel Zwillinger (12 May 2014).Handbook of Differential Equations. Elsevier Science. pp. 536–.ISBN 978-1-4832-2096-3.
^ISO/IEC/IEEE International Standard - Systems and software engineering. ISO/IEC/IEEE 24765:2010(E). pp. vol., no., pp.1-418.
^^a ^bCarsten Gundlach (2021)."MATH3083/MATH6163 Advanced Partial Differential Equations"(PDF).University of Southampton. p. 12.
^M. Stone; P. Goldbart (2004)."Chapter 6 Partial Differential Equations"(PDF).Mathematics for Physics.Cambridge University Press: gatech.edu/~predrag -Predrag Cvitanović atGeorgia Tech. p. 194.ISBN 9780521854030.
^Zhen Liu."Multiphysics Learning & Networking - Boundary Conditions: Basics".University of Virginia.
^Colm Connaughton."Chapter 2 Partial Differential Equations (PDE's) - 2.1.2 Types of Boundary Conditions"(PDF).University of Warwick Complexity DTC program. p. 21.
^^a ^b ^c ^dKevin D. Cole (21 July 2003)."Boundary Condition Types".University of Nebraska–Lincoln - Engineering (ENGR).

Bibliography

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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.
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.