18.303 Problem Set 5
Due Friday, 15 October 2010.
Problem 1: Min–max theorem
2
d
(a) Consider the operator − dx
2 on the space of functions u(x) for x ∈ [0, 1] and u(0) = u(1) = 0,
R1
with the usual inner product hu, vi = 0 ūv. Integrating by parts, the Rayleigh quotient can
be written R(u) = hu0 , u0 i/hu, ui as in class. Consider the function:
(
x/a x ≤ a
,
ua (x) = 1−x
x>a
1−a
where a is some number in (0, 1). For what a is R(ua ) minimized? How does the minimum
of R(ua ) compare with the smallest eigenvalue of −d2 /dx2 ?
(b) Consider −∇2 u = λu for functions u(x) in the 2d triangular domain Ω given by x ≥ 0, y ≥ 0,
|x| + |y| ≤ 1 (a square cut in half diagonally) with Dirichlet boundary conditions u|dΩ = 0.
Sketch contour plots of the eigenfunctions for the smallest
3 eigenvalues,
making reasonable
R
R
guesses based on the fact that these minimize R(u) = |∇u|2 / |u|2 (constrained by the fact
that they must be orthogonal). (In your plots, label peaks with a “+” and dips with a “-”.)
Problem 2: Green’s functions
2
d
2
In this problem, you will solve for the 1d Green’s function of the operator  = − dx
for some
2 +κ
real κ, on the space of functions u(x) for x ∈ [0, L] and Dirichlet boundaries u(0) = u(L) = 0. Note
that this operator is real-symmetric positive-definite. Helpful information: the ODE −y 00 +κ2 y = α,
where α is a constant, is solved by functions y(x) of the form y(x) = c1 e−κx + c2 e+κx + κα2 for
arbitrary constants c1 and c2 .
(
1 x ∈ [0, ∆x]
s(x−x0 )
(a) First, as in class, solve the “finite-delta” problem Âg(x) = ∆x where s(x) =
,
0 otherwise
and x0 ∈ [0, L − ∆x] for g(x) by breaking it up into three regions (x < x0 , x ∈ [x0 , x0 + ∆x],
x > x0 ) and enforcing continuity of g and g 0 . [Trick: you will get four equations in four
unknowns, but by dividing two of the equations by the other two you can eliminate two of
the unknowns immediately.] Plot your solution in Matlab for x0 = 0.25, 0.5, and 0.75 with
∆x = 0.1 and 0.01, with the parameters L = 1, κ = 5.
(b) Take the limit ∆x → 0 to obtain the Green’s function G(x, x0 ). Alternatively, you may
compute G(x, x0 ) directly by solving ÂG(x, x0 ) = δ(x − x0 ) as in class and the notes (breaking
G into two regions where ÂG = 0, and then matching the two regions by requiring G to be
continuous and its slope to have a discontinuity = 1 at x0 ). Plot it for L = 1, κ = 5.
(c) Verify that G(x, x0 ) = G(x0 , x).
(d) Verify that ÂG(x, x0 ) = δ(x − x0 ), using the rules for differentiating discontinuous functions
in the distribution sense.
R
(e) Optional: Using u(x) = G(x, x0 )f (x0 )dx0 , solve Âu = f for f (x) = ex .
Problem 3: Distributions
Define sε (x) by:
s∆x (x) =
(
0
1
2∆x
|x| > ∆x
.
|x| ≤ ∆x
As a distribution, s∆x {φ} → δ{φ} = φ(0) as ∆x → 0. What is the distributional derivative s0∆x {φ}?
Show that s0∆x {φ} → δ 0 {φ} = δ{−φ0 } = −φ0 (0) as ∆x → 0.
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