Spring 2007, M332: Take–Home Final
] 1: Consider the nonhomogeneous heat equation,
∂u
∂ 2u
=
+x−β,
∂t
∂x2
with the boundary conditions,
∂u
(0, t) = 0 ,
∂x
∂u
(π, t) = 0 (L = π) .
∂x
(a) Show that for β = π/2 a one parameter family of equilibrium distributions
us (x) (∂us /∂t = 0) exists and determine these distributions. Explain physically
why no equilibrium can exist if β 6= π/2.
(b) Solve the full, time-dependent heat equation (any β), subject to the initial
condition
u(x, 0) = f (x) .
Analyze the behaviour of the solution u(x, t) for t → ∞ when β < π/2, β = π/2
and β > π/2.
] 2: Consider the Laplace equation
∇2 u = 0 in R subject to
∂u
= h(x, y) on C ,
∂n
(1)
where R is a bounded, two-dimensional region, C is its boundary curve and
∂u/∂n ≡ ∇u · n̂ is the outward normal derivative along C.
H
(a) Show that (1) has a solution only if the solvability condition C hds = 0 is
satisfied. Interpret this condition by considering u as steady state solution of a
two-dimensional heat equation with prescribed heat flow on the boundary.
(b) Let R be the semi-annulus defined in polar coordinates by a < r < b (a > 0)
and 0 < θ < π (see Figure). The boundary condition in (1) is specialized as
= ∂u
(r, π) = 0 (a < r < b) ,
∂θ
∂u
∂u
(a, θ) = sin θ , ∂r (b, θ) = sin 3θ (0 < θ < π) .
∂r
∂u
(r, 0)
∂θ
Show that a solution of (1) exists only if the ratio of the radii a and b is b/a = 3.
Hint: On the semicircle r = a we have ∂u/∂n = −∂u/∂r and on the semicircle
r = b we have ∂u/∂n = ∂u/∂r.
∂u/∂r =sin(3θ)
R
b
∂u/∂r=sin(θ)
a
∂u/∂θ=0
∂u/∂θ=0
] 3: Consider equation (1) with R and the boundary conditions as specialized in
] 2(b), and assume that b = 3a so that a solution exists.
(a) By using the method of separation of variables, show that u(r, θ) is given by
u(r, θ) = A0 + B0 ln r +
∞
X
(Am rm + Bm r−m ) cos mθ .
(2)
m=1
Find equations that relate the coefficients B0 , Am , Bm in (2) to the boundary
data. You don’t need to solve these equations, but you should show that they
have unique solutions and state which of the coefficients are zero.
(b) One of the coefficients in (2) is not determined by the boundary conditions.
State an equation which relates this coefficient to the initial condition of a heat
equation,
∂w
= ∇2 w, w(r, θ, 0) = f (r, θ),
∂t
with the same boundary conditions as for
u, using the fact that w(r, θ, t) → u(r, θ)
R π R 3a
when t → ∞ and that the total heat, 0 a w(r, θ, t) rdr dθ, in the semi-annulus
is conserved.