Editing Boundary value problem (section)

===Boundary value conditions===
[[Image:Bounday value problem for a rod.PNG|frame|right|Finding a function to describe the temperature of this idealised 2D rod is a boundary value problem with [[Dirichlet boundary condition]]s. Any solution function will both solve the [[heat 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 boundary condition which specifies the value of the function itself is a [[Dirichlet boundary condition]], or first-type 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.

A boundary condition which specifies the value of the [[normal derivative]] of the function is a [[Neumann boundary condition]], or second-type 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.

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]].

====Examples====
Summary of boundary conditions for the unknown function, <math>y</math>, constants <math>c_0</math> and <math>c_1</math> specified by the boundary conditions, and known scalar functions <math>f</math> and <math>g</math> specified by the boundary conditions.
{| class="wikitable" style="text-align: center"
|-
! Name
! Form on 1st part of boundary
! Form on 2nd part of boundary
|-
| [[Dirichlet boundary condition|Dirichlet]]
| colspan="2" |<math>y=f</math>
|-
| [[Neumann boundary condition|Neumann]]
| colspan="2" |<math>{\partial y \over \partial n}=f</math>
|-
| [[Robin boundary condition|Robin]]
| colspan="2" |<math>c_0 y + c_1 {\partial y \over \partial n}=f</math>
|-
| [[Mixed boundary condition|Mixed]]
| <math>y=f</math>
| <math>c_0 y + c_1 {\partial y \over \partial n}=g</math>
|-
| [[Cauchy boundary condition|Cauchy]]
| colspan="2" |both <math>y=f</math> and <math>{\partial y \over \partial n}=g</math>
|}