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Thread: Eigenfunctions of Linear Operator

  1. #1
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    Eigenfunctions of Linear Operator

    Hello,

    Find all the eigenvalues and eigenfunctions of the operator

    ----------> where the domain D is given by D{f belongs to : f is twice continuously differentiable and f(0)=f'(1)

    I can sove differential equation such as
    but the above notations are confusing me. Could you please get me started.

    Thanks
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  2. #2
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    Quote Originally Posted by charikaar View Post
    Hello,

    Find all the eigenvalues and eigenfunctions of the operator

    ----------> where the domain D is given by D{f belongs to : f is twice continuously differentiable and f(0)=f'(1) =0 (?)

    I can solve differential equation such as
    but the above notations are confusing me. Could you please get me started.
    A second order differential operator usually needs two initial conditions. I'm guessing that f(0) and f'(1) should both be 0?

    In that case, if T is the differential operator given by Tf = -f'', and f is an eigenfunction for the operator T, with eigenvalue \lambda, then Tf = \lambda f (just as in linear algebra). So you need to solve the differential equation -f'' = \lambda f, together with the initial conditions f(0) = f'(1) = 0.

    The general solution is f(t) = A\cos ct + B\sin ct, where c = \sqrt{-\lambda}. The initial condition f(0)=0 tells you that A=0. The condition f'(1)=0 then tells you that cB\cos c = 0. So the equation will only have a nonzero solution if c=0 or \cos c=0. In terms of \lambda = -c^2, that implies that \lambda = 0 or \lambda = -(k+\tfrac12)^2\pi^2, for some integer k. So those are the eigenvalues (an infinite sequence of them, unlike in linear algebra where you only have finitely many eigenvalues for a matrix).
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