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Use finite difference method to solve for eigenvalue E in Matlab

  1. Dec 6, 2012 #1
    Use finite difference method to solve for eigenvalue E from the following second order ODE:

    - y'' + (x2/4) y = E y

    I discretize the equation so that it becomes

    yi-1 - [2 + h2(x2i/4)] yi + yi+1 = - E h2 yi

    where xi = i*h, and h is the distance between any two adjacent mesh points.

    This is my code:
    Code (Text):
    clear all
    n = 27;
    h = 1/(n+1);
    voffdiag = ones(n-1,1);
    for i = 1:n
        xi(i) = i*h;
    end
    mymat = -2*eye(n)-diag(((xi.^2).*(h^2)./4),0)+diag(voffdiag,1)+diag(voffdiag,-1);
    D=sort(eig(mymat),'descend');
    lam= -D/(h^2);
    spy(mymat)
    fprintf(1,' The smallest eigenvalue is %g \n',lam(1));
    fprintf(1,' The second smallest eigenvalue is %g \n',lam(2));
    fprintf(1,' The 3rd eigenvalue is %g \n',lam(3));

     
    it returns
    Code (Text):

     The smallest eigenvalue is 9.92985
     The second smallest eigenvalue is 39.3932
     The 3rd eigenvalue is 88.0729
     
    Obviously, something wrong here, since the analytic solution should be
    E = n + 1/2 (for n = 0, 1, 2, 3...)
    The smallest eigenvalue should be 0.5, instead of 9.92985.

    I don't know whether my numerical solution agrees with the analytic solution or not, if I impose a boundary condition (ie. when x goes to infinity, y(x) should vanish to 0). And I don't know how to impose boundary condition. Please help, thank you very much!

    By the way, is there any another way to find the eigenvalue E please?
     
    Last edited: Dec 6, 2012
  2. jcsd
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