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.
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 .
 L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.