Spring 2007, M332: Take–Home Exam 1
] 1: Consider the heat equation for a one dimensional rod of length π with Dirichlet
boundary conditions and a heat source Q(x) = sin(x),
∂u
∂ 2u
=
+ sin(x), 0 < x < π, u(0, t) = u(π, t) = 0.,
∂t
∂x2
(1)
where we assume that cρ = 1 and K0 = 1.
(a) Determine the total heat per unit time that is generated by the source.
(b) Find the equilibrium heat distribution us (x) and determine the amount of heat flowing
out of the rod per unit time across the boundaries when u(t, x) = us (x). What relation
should exist between this answer and the answer obtained in (a)? Explain!
(c) Set v(x, t) = u(x, t) − us (x). Show that v(x, t) satisfies the standard heat equation
(without source)
∂v
∂ 2v
=
, 0 < x < π, v(0, t) = v(π, t) = 0.
(2)
∂t
∂x2
(d) Using (b) and (c), find the solution of (1) for the initial condition
u(x, 0) = sin x + sin 2x
] 2: Let
∞
X
nπx
x ∼
bn sin(
)
L
n=1
2
2
and
x ∼ a0 +
(3)
∞
X
an cos(
n=1
nπx
)
L
(4)
be the sine- and cosine–expansions of x2 in 0 < x < L. The following questions should be
answered without making use of the explicit forms of the expansion coefficients an , bn .
(a) Which of the following two formulae, obtained via term by term differentiation of the
series in (4), is correct and which one is not correct?
2x ∼
∞
X
nπ
n=1
L
bn cos(
nπx
) (i),
L
2x ∼ −
∞
X
nπ
n=1
L
an sin(
nπx
) (ii).
L
(5)
Explain in terms of smoothness properties of periodic extensions and sketch these extensions.
(b) Let An and Bn be the cosine– and sine–expansion coefficients of 2x = (x2 )0 . Derive
relations between an and Bn , and between bn and An , by performing a partial integration
in the integral expressions
bn =
nπx
2ZL 2
x sin(
)dx,
L 0
L
an =
2ZL 2
nπx
x cos(
)dx (n ≥ 1).
L 0
L
How do these results relate to your answer of (b)?
(c) Find an appropriate function g(x), as simple as possible, such that the series of
f (x) = x2 − g(x) can be differentiated also in the case in which it is not allowed to
differentiate the series of x2 . Sketch both the sine and the cosine series of f (x).
] 3: Consider the nonhomogenous (forced) wave equation with Dirichlet boundary
conditions,
2
∂ 2u
2∂ u
=
c
+ cos(2ct) sin(x), 0 < x < π, u(0, t) = u(π, t) = 0,
∂t2
∂x2
(6)
and initial conditions
∂u
(x, 0) = g(x).
(7)
∂t
(a) The boundary conditions suggest that the solution can be represented as a sine–series,
u(x, 0) = 0,
u(x, t) =
∞
X
Bn (t) sin(nx).
(8)
n=1
Substitute the expansion (8) in the PDE (6) and assume that term by term differentiation
with respect to t and twice with respect to x is allowed. As a result, both the left hand
sides and the right hand sides (you have to include the source term in the series) take the
form of a sine-series. If both series coincide, their coefficients must be the same for each
n. Use this to derive 2nd order ODEs for the coefficients Bn (t).
(b) Set up initial conditions for the Bn ’s from the initial conditions for u. To this end
assume that g(x) is also expanded in a sine series,
g(x) =
∞
X
gn sin(nx),
n=1
with known coefficients gn .
(c) Solve the ODEs derived in (a) for the initial conditions obtained in (b). This yields
unique solutions Bn (t) that depend on the coefficients gn . The solution is completed when
the Bn (t) are determined.
Hint: One of the ODEs for the expansion coefficients is a forced harmonic oscillator ODE
of the form (subscript suppressed)
B 00 + ω 2 B = a cos(νt),
(9)
with ω 6= ν. To solve this ODE, find first a particular solution Bp (t) by substituting the
ansatz Bp (t) = α cos(νt) in the ODE. The requirement that Bp be a solution fixes the
“undetermined coefficient” α. Next add the general solution to the homogenous ODE,
B 00 + ω 2 B = 0, to Bp to obtain the general solution of (9),
B(t) = c1 cos(ωt) + c2 sin(ωt) + α cos(νt).
Finally determine the coefficients c1 , c2 by invoking the initial conditions. You may look
in an ODE book for the “method of undetermined coefficients”.
(d) Specify the general result obtained in (c) for the case when g(x) = 0 and write down
the resulting solution u(x, t).