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Eigenvalues of Ordinary Differential Equations

Eigenvalues of Ordinary Differential Equations. Jake Blanchard University of Wisconsin. Introduction. Finite Difference Techniques Matlab. Model Problem. A simple eigenvalue problem Solution. Finite Difference Solution. Choosing a Mesh. Divide range 0<x<1 into 8 regions

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Eigenvalues of Ordinary Differential Equations

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  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

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