Matrix Solutions to Airy's Eigenvalue Problem

This Demonstration treats the homogeneous boundary case of the Sturm–Liouville eigenvalue problem by solving Airy's differential equation expanded around an ordinary point. The roots of this differential equation are called eigenvalues, and the corresponding functional solutions are known as eigenfunctions. Since solving for these eigenfunctions involves finding an infinite-dimensional matrix, algebra can be used to express solutions of the differential equation. After the eigenfunctions are generated, they are then plotted over a specified interval. The periodicity of these eigenfunctions stems from the fact that the boundary conditions used in this Demonstration were homogeneous.
The differential equation can also be solved analytically, making use of the Airy functions and . The eigenvalues are then determined from the zeros of these functions.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The general form of Airy's set of differential equations used here is .
This can be expressed using infinite matrices in order to solve for the eigenfunctions. To do this, we treat the equation in terms of the derivative operator , the vector operator , and the identity operator . Set
so that
[1] U. Siedlecka, "Sturm–Liouville Eigenvalue Problems with Mathematica," Journal of Applied Mathematics and Computational Mechanics, 10(2), 2011 pp. 217–223. www.srimcs.im.pcz.pl/get.php?article=2011_ 2/art_23.pdf.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+