1. Eigenvalues of Ordinary Differential Equations Jake Blanchard University of Wisconsin

2. Introduction Finite Difference Techniques Matlab

3. Model Problem A simple eigenvalue problem Solution

4. Finite Difference Solution

5. Choosing a Mesh Divide range 0<x<1 into 8 regions This produces 9 mesh points Boundary conditions eliminate two unknowns We’re left with 7 unknowns (the 7 internal mesh points)

6. Matrix Equation

7. Code n=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); figure semilogy(myint,myerr)

8. Results – n=7

9. Results (error) – n=7

10. Results – n=2000

11. Results (error) – n=2000