Explanation
Boundary value problems are similar to initial 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).
For example, if the independent variable is time over the domain, a boundary value problem would specify values for at both and, whereas an initial value problem would specify a value of and at time .
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 the data or at a given time for all space.
Concretely, an example of a boundary value (in one spatial dimension) is the problem
to be solved for the unknown function with the boundary conditions
Without the boundary conditions, the general solution to this equation is
From the boundary condition one obtains
which implies that From the boundary condition one finds
and so One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is
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