For each IBVP, (1) solve the associated eigenvalue problem; (2) derive the formu

Question

For each IBVP, (1) solve the associated eigenvalue problem; (2) derive the formulas for the coe?cients
in the Fourier series of the functions f (x, t), g (x), and h (x), whichever are involved in the IBVP;
(3) state the initial value problems for the coe?cients in the Fourier series of the solution u (x, t).
(a)

PDE ut ? uxx = f (x, t) 0 < x < 1, t > 0,
BCs ux (0, t) = 0, u (1, t) = 0 t > 0,
IC u (x, 0) = g (x) 0 < x < 1.

(b)
PDE utt ? uxx = f (x, t) 0 < x < 1, t > 0,
BCs ux (0, t) = 0, ux (1, t) = 0 t > 0,
ICs u (x, 0) = g (x) , ut (x, 0) = h (x) 0 < x < 1.

Explanation / Answer

For each IBVP, (1) solve the associated eigenvalue problem; (2) derive the formu

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