1. (a) Find the solution of the heat conduction problem:
4uxx = ut, 0 < x < 5,
u(0, t) = 40,
u(5, t) = 120, t > 0
u(x, 0) = f (x) = 16x + 40 + 13sin(πx) − 5sin(3πx).
(b) Find lim u(3, t).
t → ∞
(c) If the initial condition is, instead, u(x,0) = 8x + 60 + 13sin(πx) − 5sin(3πx), will the limit
in (b) be different? What would the difference be?
(d) Given the initial condition in (c), how would you find the particular solution? (Write
down the necessary integral(s), but you do not need to integrate.)
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