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.
Usually boundary value problems come with a Sturm-Liouville problem: finding the eigenvectors (more accurately, eigenfunctions) of a differential operator. The boundary value problem theory has major applications in physics, especially in problems of finding normal modes.
Example
We wish to find a function y(x) which solves the following Sturm-Liouville problem:
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and satisfy the boundary conditions
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We will use k to denote the square root of the absolute value of .
If then
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solves the ODE.
Substitute boundary conditions gives that both A and B are equal to zero.
For positive we obtain that
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solves the ODE.
Substitution of boundary conditions again yields A = B = 0.
For negative it is easy to show that
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solves the ODE.
From the first boundary condition
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Now, after the cosine is gone, we will substitute the second boundary condition:
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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.
