Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods

Consider the Airy differential equation, , where , , and . Values of and (the eigenvalues and eigenfunctions) can be determined by solving the generalized eigenvalue problem , where the matrices and are given in the details section. The eigenfunction is given by , where is the classic Airy function and is the eigenvalue. This Demonstration approximates the values of the eigenvalues and eigenfunctions (up to ) numerically using spectral methods. When the number of grid points is large, the numerical values of at the grid points match the Airy function very closely.

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DETAILS

The interior points, the Chebyshev–Gauss–Lobatto points, are given by . These points are the extremums of the Chebyshev polynomial of the first kind .
The Chebyshev derivative matrix at the quadrature points , , is given by
, , for , and for , and ,
where for and .
The matrix is equal to (without its first row and first column) and .
Reference
[1] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.
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