Boundary value problem

Solving a differential equation such that certain constraints are satisfied

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.

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 (in one spatial dimension) is the problem

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

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.

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.