Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods

number of grid points

46

th

n

eigenfunction

3

Consider the Airy differential equation,

u''=λxu

, where

u≠0

,

u(±1)=0

, and

-1≤x≤1

. Values of

λ

and

u

(the eigenvalues and eigenfunctions) can be determined by solving the generalized eigenvalue problem

Au=λBu

, where the matrices

A

and

B

are given in the details section. The

th

n

eigenfunction is given by

Ai

1/3

λ

n

x

, where

Ai(x)

is the classic Airy function and

λ

n

is the

th

n

eigenvalue. This Demonstration approximates the values of the eigenvalues and eigenfunctions (up to

n=10

) numerically using spectral methods. When the number of grid points is large, the numerical values of

u

at the grid points match the Airy function very closely.