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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.

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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
.
Reference
[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.
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