Additional problem 1. Solve the following initial-boundary value problem for
the heat equation
ut = kuxx, x ∈ [0, L], t ∈ [0, ∞)
u(0, t) = A, u(L, t) = B for t ∈ [0, ∞]
u(x, 0) = f(x), for x ∈ [0, L],
where A and B are constants, in general nonzero. Since we know how to solve
problems with two homogeneous (i.e. zero) boundary conditions on a pair of
opposite sides, we would be able to solve this problem if A = B = 0. One
method to reduce the given problem to a problem of the type we like is to set
u(x, t) = v(x, t) + l(x),
where l(x) := αx + β is a linear function that - for an appropriate choice
of the constants α and β - can be made to satisfy the boundary conditions
l(0) = A, l(L) = B. Choose the constants α and β appropriately, show that the
initial-boundary value problem for the function v(x, t) is
vt = kvxx, x ∈ [0, L], t ∈ [0, ∞)
v(0, t) = 0, v(L, t) = 0 for t ∈ [0, ∞]
u(x, 0) = f(x) − l(x), for x ∈ [0, L],
and solve this problem. Assume that the sine-Fourier expansion of the function
f(x) = X∞
Be careful in formulating the problem for v(x, t); in particular, the initial condition for v(x, t) will differ from the one for u(x, t). Here are two facts that you
will need (when you expand) the function l(x)).
, x =
Show that as t → ∞, the function v(x, t) will tend to 0, so that the asymptotic
behavior of the temperature u(x, t) is determined completely by the auxiliary
function l(x) (recall that the constant k in the heat equation is strictly positive!).
Sketch limt→∞ u(x, t) as a function of x.
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