Exercises
Number 1
Find the eigenfunction expansion of Sin[x]+Exp[x] on -1<x<1 with Legendre
polynomials. Go up to n=8 and compare the graphs with the graph of the original.
Number 2
Find the eigenfunction expansion of f(x) on -1<x<1 where
0 -1<x<0
f(x)=
x 0<x<1
with Legendre polynomials. Go up to n=8 and compare the graphs with the graph
of the original.
Number 3
Verify that kn=nPi/ln(b) and sin[kn ln(x)] are eigenvalues and eigenfunctions
of
(xy')'+L^2 1/x y 1<x<b
y(1)=0 y(b)=0
(You can verify these using Mathematica up to n=6 or so.)
Find the eigenfunction expansion of f(x)=x in terms of these
eigenfunctions.
To what values does the series converge at x=1 and x=b?
To get graphs you will need to pick a value for b>1. Try b=3.
Log is the Mathematica function for the natural logarithm.
Number 4
Verify that Ln=(nPi/a)^2 +1/4 and Exp[-x/2]Sin[nPix/a] are eigenvalues and
eigenfunctions of
(Exp[x]y')'+LExp[x] y 0<x<a
y(0)=0 y(a)=0
(You can verify these using Mathematica up to n=6 or so.)
Find the eigenfunction expansion of f(x)=1 in terms of these eigenfunctions.
For the graphs, you will need to pick an a>0. Use a=4
Number 5
Find the eigenfunction expansion of f(x) on -1<x<1 where
0 -1<x<0
f(x)=
1 0<x<1
with Chebyshev polynomials. Go up to n=10 and compare the graphs with the graph
of the original.
Up to Sturm Liouville Theory and Orthogonal Polynomials