Boundary value problem

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

In mathematics, a boundary value problem consists of a differential equation and the initial or boundary values required to solve the equation. The solution to the differential equation will not only satisfy the differential equation everywhere inside the boundary but will also satisfy the boundary conditions themselves. Boundary value problems may be posed for ordinary differential equations as well as partial differential equations.

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 principle.

1 Where boundary problems occur
2 Initial value problem
3 Types of boundary value problems
4 Example
5 See also
6 References
7 External links

[edit] Where boundary problems occur

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 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.

[edit] Initial value problem

Main article: Initial value problem

If the problem is dependent on both space and time, then instead of specifying the value of the problem at a given point for all time the data could be given at a given time for all space. For example, the temperature 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. Whereas in a the middle of a still pond if somebody taps the water with a known force that would create a ripple and give us an initial condition.

[edit] Types of boundary value problems

The boundary value problem for an idealised 2D rod

If the boundary gives a value to the normal derivative of the problem then it is a Neumann boundary condition. For example if one end of an iron rod had a heater at one end then energy would be added at a constant rate but the actual temperature would not be known.

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 had one end held at absolute zero then the value of the problem would be known at that point in space.

If the boundary has the form of a curve or surface that gives a value to the normal derivative and the problem itself then it is a Cauchy boundary condition.

[edit] Example

For more details on Sturm-Liouville problems, for ordinary and partial differential equations, see Examples of boundary value problems.

Consider the ordinary differential equation

$y''(x)+y(x)=0 \,$

to be solved for the unknown function $y (x)$ . Impose 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 $y (π / 2) = 2$ 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

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

[edit] See also

[edit] References

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.