Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods
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.
u''=λxu
u≠0
u(±1)=0
-1≤x≤1
λ
u
Au=λBu
A
B
th
n
Aix
1/3
λ
n
Ai(x)
λ
n
th
n
n=10
u