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Sturm-Liouville Equation


A second-order ordinary differential equation

 d/(dx)[p(x)(dy)/(dx)]+[lambdaw(x)-q(x)]y=0,

where lambda is a constant and w(x) is a known function called either the density or weighting function. The solutions (with appropriate boundary conditions) of lambda are called eigenvalues and the corresponding u_lambda(x) eigenfunctions. The solutions of this equation satisfy important mathematical properties under appropriate boundary conditions (Arfken 1985).

There are many approaches to solving Sturm-Liouville problems in the Wolfram Language. Probably the most straightforward approach is to use variational (or Galerkin) methods. For example, VariationalBound in the Wolfram Language package VariationalMethods` and NVariationalBound give approximate eigenvalues and eigenfunctions.

Trott (2006, pp. 337-388) outlines the inverse Sturm-Liouville problem.


See also

Adjoint, Self-Adjoint, Sturm-Liouville Theory

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References

Arfken, G. "Sturm-Liouville Theory--Orthogonal Functions." Ch. 9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 497-538, 1985.Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. http://www.mathematicaguidebooks.org/.

Referenced on Wolfram|Alpha

Sturm-Liouville Equation

Cite this as:

Weisstein, Eric W. "Sturm-Liouville Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Sturm-LiouvilleEquation.html

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