Question: Determine the thermal conductivity components a(y) > 0 and b(y) > 0 and the tempera- ture distribution u(x, y) of a rectangular orthotropic heterogeneous y-layered conductor,

i.e. depending on the single variable y, but independent on x, subject to the diffusion of heat, satisfying



a(y)uxx + (b(y)uy)y = 0, (x, y) ? (0, ?) × (0, L),

where the length L > 0 is given, the boundary conditions

u(0, y) = u(?, y) = 0, y ? (0, L),

u(x, 0) = g0 sin(x), u(x, L) = gL sin(x), x ? [0, ?],

where g0 and gL are given positive constants, and the additional conditions

a(y)ux(0, y) = k(y), y ? (0, L),





and

b(y)ux(?, y) = l(y), y ? (0, L),where k > 0 and l < 0 are given functions.

(Hint: You may start by seeking the solution for the temperature in the separable form

u(x, y) = ?(y) sin(x), (x, y) ? (0, ?) × (0, L),

where ? is an unknown function to be determined (you may also wish to work with ? := ln(?)). Your solutions for u(x, y), a(y) and b(y) will be given in terms of some integral involving the input data k(y) and l(y).)

  Edit

Answer: a(y)uxx + (b(y)uy)y = 0, (x, y) ? (0, ?) × (0, L)--------------------------(1)
u(0, y) = u(?, y) = 0, y ? (0, L) ---------------------------------------------(2)
u(x, 0) = g0 sin(x), u(x, L) = gL sin(x), x ? [0, ?]-------------------------(3)
a(y)ux(0, y) = k(y), y ? (0, L)-------------------------------------------------(4)
b(y)ux(?, y) = l(y), y ? (0, L)--------------------------------------------------(5)
Suppose that the following assumptions hold:
(A1) ? ? C 2+? (D), µi ? C 2+?,1+?/2 ([0, L] × [0, ?]), i ? {1, 2}, µk ? C 2+?,1+?/2 ([0, h] × [0, T]), k
? {3, 4}, µ5 ? C ?,?/2 ([0, L] × [0, ?]), µ6 ? C ?,?/2 ([0, h]×[0, T]), f ? C ?,?/2 (QT ) for some ? ?
(0, 1);
(A2) ?x(x, y) > 0, ?y(x, y) > 0,(x, y) ? D, µ5(y, t) > 0,(y, t) ?([0, L] × [0, ?]), µ6(x, t) > 0,(x, t)
? [0, h] × [0, T];
(A3) consistency conditions of the zero and the first orders.
a formal elimination of a(y) and b(x) in (5) and (6), respectively, and substitution into (1) result
in the nonlinear partial differential equation u(x, y) = k(y) /ux(0, y) uxx + l(y)/( ux(?, y) uy)y
? QT
to be solved for the temperature u(x, y) subject to the given conditions.
by the superposition
u(x, y) = v(x, y) + ?(x, y) + ?(x, y)
where
v= a(y)vxx + b(x)vyy + F(x, y) + a(y)(?xx(x, y) + ?xx(x, y)) + b(x)(?yy(x, y) + ?yy(x, y)), (x,
y) ? QT
v(x, y, 0) = 0, (x, y) ? D
v(0, y) = v(h, y) = 0, (y) ? [0, L]
v(x, 0) = v(x, l) = 0, (x, t) ? [0, h] × [0, L]
The assumptions (A1)-(A3) ensure the existence of a Green function. Then the solution u of
the problem is given by the formula
u(x, y) = ?(x, y) + ?(x, y) + ? ? D G(x, y, ?, ?, ? ) [ F(?, ?, ? ) + a(?, ? )(???(?, ?) + ???(?, ?, ?
)) + b(?, ? )(???(?, ?) + ???(?, ?, ? ))] d?d?d?, (x, y) ? QT .
a(y) = k(y) { ?x(0, y) + ?x(0, y) + ? ? D Gx(0, y, t, ?, ?, ? ) [ F(?, ?, ? ) + a(?, ? )(???(?, ?) +
???(?, ?, ? )) + b(?, ? )(???(?, ?) + ???(?, ?, ? ))] d?d?d?}?1
,
b(y) = l(y) { ?y(x, 0) + ?y(x, 0) + ?? D Gy(x, 0, t, ?, ?, ? ) [ F(?, ?, ? ) + a(?, ? )(???(?, ?) +
???(?, ?, ? )) + b(?, ? )(???(?, ?) + ???(?, ?, ? ))] d?d?d?}?1  Edit