Show that the problem u t = ku xx , 0 < x < l, t > 0, u(0, t) = g(t), u(l, t) =

Question

Show that the problem

ut = kuxx, 0 < x < l, t > 0,

u(0, t) = g(t), u(l, t) = h(t), t>0,

u(x, 0) = u0(x), 0 x l,

with nonhomogeneous boundary conditions can be transformed into a problem with homogeneous boundary conditions. Hint: Introduce a new dependent variable w via w(x, t) = u(x, t) L(x, t) by subtracting from u a linear function L(x, t) of x that satisfies the boundary conditions at any fixed time t. In the transformed problem for w, observe that the PDE picks up a source term, so you are really trading boundary conditions for source terms.

Transform the entire problem to the new variable.

Explanation / Answer

Solution.

Let s(x, t) = h(t) x/l + (1 x /l )g(t)..

Then, s(0, t) = g(t) and s(l, t) = h(t).

Note that st = h'(t) x/l + (1 x/l )g'(t) and sxx = 0. Let w(x, t) = u(x, t) s(x, t).

Then wt = ut st = ut h'(t) x/l + (1 x/l )g'(t)

and wxx = uxx..So ut = kuxx if and only if wt = kwxx st

= kwxx h'(t) x/l + (1 x/l )g'(t)

Also w(0, t) = u(0, t) s(0, t) = 0 and w(l, t) = u(l, t) s(l, t) = 0.

So w(x, t) satisfies an nonhomogeneous diffusion equation, with homogeneous boundary conditions

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