Use separation of variables to find a series solution of
utt = c 2uxx subject to u(0, t) =
0,
ux(l, t) + u(l, t) = 0, u(x, 0) = φ(x), &
ut(x, 0) = ψ(x) over the domain 0 < x < `, t >
0. Provide an equation that identifies the eigenvalues and sketch a
graph depicting this equation. Clearly identify the
eigenfunctions
If friction is present, the wave equation takes the form utt −
c2 uxx = −r ut, where the resistance r > 0 is a constant.
Consider a periodic source at one end: u(0, t) = 0, u(l, t) = Aeiωt
.
(a) Show that the PDE and the BC are satisfied by u(x, t) =
Aeiωt sin βx sin βl , where β2c2 = ω2 − irω.
(b) No matter what the IC, u(x, 0) and ut(x, 0), are,...
Using the method of separation of variables and Fourier series,
solve the following heat
conduction problem in a rod.
∂u/∂t =∂2u/∂x2
, u(0, t) = 0, u(π, t) = 3π, u(x, 0) = 0