Mathematical Methods (10/24.539)
IX. The Sturm-Liouville Problem
Overview and General Terminology
As outlined above, many of the special functions we have discussed previously represent specific cases of a more generalized Sturm-Liouville problem. In particular, a 2-point boundary value problem having the form
on some given interval, 
, with homogeneous boundary conditions given by
where c1, c2, and k1, k2 are constants, the p(x), q(x), and r(x) coefficients are differentiable functions of the independent variable (with p(x) > 0), and 
is a parameter, is called a
Important Notes:
1. Homogeneous 2-point boundary value problems with homogeneous boundary conditions have an
infinite number of solutions.2. The values of 
that give non-trivial solutions are referred to as
3. The set of eigenfunctions, {yn(x)}, form an
orthogonal system with respect to the weight function, p(x), over the interval
.
4. If p(x), q(x), and r(x) are real, the eigenvalues are also real (see your text by Kreyszig and Problem 11.24 in the Schaum’s Outline,
Advanced Mathematics, for a general proof).It is the orthogonal eigenfunction solutions and their application within Generalized Fourier Series that make the Sturm-Liouville Problem so important.
Example 9.1 represents a good illustration of the solution techniques utilized for typical eigenvalue problems and it identifies, via example, much of the terminology from above. It is a simple demonstration that hopefully clarifies many of the basic concepts and notation associated with the so-called Sturm-Liouville Problem.|
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10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated November 1998).
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