Techniques of applied maths: Eigenfunction expansions I

Eigenvalues and eigenfunctions of a BVP

Consider the eigenvalue problem on $0 \leq x \leq 1$, \begin{gather} y'' + 2y' + (1 + \lambda) y = 0, \\ y'(0) + y(0) = 0, \\ y'(1) + y(1) = 0. \end{gather}

a) Eigenvalues and eigenfunctions of the BVP

Assuming $\lambda$ to be a positive constant, what is the general solution of the homogeneous ODE. Apply the boundary conditions to determine the eigenvalues and eigenfunctions.

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b) Eigenvalues and eigenfunctions of the adjoint problem

What is the adjoint problem? Obtain the adjoint eigenfunctions

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c) Placing the problem into SL form

What are the functions $p$, $q$, $r$, that put this problem in standard Sturm-Liouville form: \[ \left( p(x) \frac{dy}{dx}\right)' + q(x) y = -\lambda r(x) y \]

Verify that the expected orthogonality conditions (i.e. the one in terms of $y$'s and $r$ and the one in terms of $y$'s and $w$'s) are satisfied by direct integration with the eigenfunctions.

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d) Eigenfunction expansion

Use the eigenfunctions and their adjoints to obtain the coefficients in the eigenfunction expansion \[ y(x) = \sum_{k = 0}^\infty c_k y_k \]

for the solution of the problem \begin{gather*} y'' + 2y' + 2y = 1 \\ y'(0) + y(0) = 2 \\ y'(1) + y(1) = 3. \end{gather*}

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