Eigenvalues of Ordinary Differential EquationsJake BlanchardUniversity of Wisconsin

IntroductionFinite Difference TechniquesMatlab

Model ProblemA simple eigenvalue problem

Solution

d y

dxy

2

22 0

y y( ) ( )0 1 0

n n ,

y A n xn n sin( )

Finite Difference Solution

y y y

hyi i ii

1 1

222

0

y h y yi i i 12 2

12 0( )

Choosing a MeshDivide range 0<x<1 into 8

regionsThis produces 9 mesh pointsBoundary conditions eliminate

two unknownsWe’re left with 7 unknowns (the 7

internal mesh points)

Matrix Equation

2 1 0 0 0 0 0

1 2 1 0 0 0 0

0 1 2 1 0 0 0

0 0 1 2 1 0 0

0 0 0 1 2 1 0

0 0 0 0 1 2 1

0 0 0 0 0 1 2

02 2y h y

i ih e2 2

Coden=7;h=1/(n+1);voffdiag=ones(n-1,1);mymat=-2*eye(n)

+diag(voffdiag,1)+diag(voffdiag,-1);D=sort(eig(mymat),'descend');lam=sqrt(-D)/h;check=lam/pi;myint=(1:n)';plot(myint,check,myint,myint)myerr=abs(check-myint)./(myint);figuresemilogy(myint,myerr)

Results – n=7

1 2 3 4 5 6 70

1

2

3

4

5

6

7

Results (error) – n=7

1 2 3 4 5 6 710

-3

10-2

10-1

100

Results – n=2000

0 200 400 600 800 1000 1200 1400 1600 1800 20000

200

400

600

800

1000

1200

1400

1600

1800

2000

Results (error) – n=2000

0 200 400 600 800 1000 1200 1400 1600 1800 200010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100