Problem Statement: Find the eigenvalues and eigenvectors of a double derivative operator D2 defined in x=[-1 1].
D2y = d2y/dx2If we apply boundary condition y(-1)=y(1)=1, how many eigenfunctions are there? (We should eliminate half of the eigenfunctions from the original set with this boundary condition.) Finally, represent any twice-differentiable function as a a linear combination of the eigenfunctions.
f(x) = S aj*vjFind the coefficients aj. Find the second derivative of any twice-differentiable function. Try f(x)=1 as an example. Another example: f(x)=1/(1+25*x*x). How about an odd function such as f(x)=x in x=[-1 1]? (Answer: No, we get 0. Try it with the Mathcad worksheet.) How about f(x)=x in x=[0 1]? (Answer: OK, we can approximate. Try it.)
(This is rather silly, because, as far as functions go, "1" is as simple as it gets. However, the point of this problem is to demonstrate how we can represent any twice-differentiable function as a a linear combination of the eigenfunctions. Once we do that, we can find the second derivative of any twice-differentiable function from the eigenvalue-eigenvector idea. I choose a simple function 1 so that we don't kill ourselves with the computation. In practice, there is no good reason why we should turn something as simple as 1 into a mess, except that we can use this trick to derive all sorts of relationships between different functions. In the case of f(x)=1, we can show that a certain combination of the eigenfunctions add up to 1 -- interesting theoretically but useless in practice.)
Solution:
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