Boundary value problem
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions.
The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator.
To be useful in applications, a boundary value problem should be well posed.
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.
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.
Concretely, an example of a boundary value problem (in one spatial dimension) is to be solved for the unknown function
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.
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.
Summary of boundary conditions for the unknown function,
specified by the boundary conditions, and known scalar functions
These categories are further subdivided into linear and various nonlinear types.
In electrostatics, a common problem is to find a function which describes the electric potential of a given region.
If the region does not contain charge, the potential must be a solution to Laplace's equation (a so-called harmonic function).
If there is no current density in the region, it is also possible to define a magnetic scalar potential using a similar procedure.
Related mathematics: Physical applications: Numerical algorithms:
