Boundary value problem

For people who check facts

In mathematics, a boundary value problem consists of a differential equation to be satisfied at all points in the interior of an interval or a region and a set of boundary conditions specifying the values of the solution or some of its derivatives everywhere on the boundary of the interval or region. Boundary value problems may be posed for ordinary differential equations as well as partial differential equations.

Boundary value problems arise in several branches of physics. 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.

Example

We wish to find a function y(x) which solves the following Sturm-Liouville problem:

and satisfy the boundary conditions

We will use k to denote the square root of the absolute value of .

If then

solves the ODE. Substitute boundary conditions gives that both A and B are equal to zero.

For positive we obtain that

solves the ODE. Substitution of boundary conditions again yields A = B = 0.

For negative it is easy to show that

solves the ODE. From the first boundary condition

Now, after the cosine is gone, we will substitute the second boundary condition:

So either A = 0 or k is an integer. Thus we get that the eigenfunctions which solve the "boundary value problem" are:

One may easily check they satisfy the boundary conditions.