Use finite difference method to solve for eigenvalue E in Matlab

[EigenCake]

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

y_i-1 - [2 + h²(x²_i/4)] y_i + y_i+1 = - E h² y_i

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

This is my code:

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

 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?

 

[Valkarie]

Yes. You can use the shooting method to solve for the eigenvalue. This involves solving the differential equation by integrating from a given boundary condition and then using an iterative process to find the eigenvalue that produces the desired boundary condition.