18.303 Midterm, Fall 2013
Each problem has equal weight. You have 55 minutes.
Problem 1: Definite (30 points)
In class, we only defined definiteness for Hermitian operators. However, if  6= Â∗ , we can more generally define  to
be “negative-definite” if <hu, Âui < 0 for all u 6= 0, where <hu, Âui denotes the real part: <hu, Âui =
hu,Âui+hu,Âui
.
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(a) For such a negative-definite non-Hermitian operator, show that the real parts of its eigenvalues are negative.
(b) Show that this generalization of negative-definiteness is equivalent to saying that the Hermitian operator  + Â∗
is negative definite according to the definition from class.
∂v
(c) Consider the system of equations ∂u
∂t = ∂x − αu and
Dirichlet boundary conditions u(0) = u(L) = 0.
(i) Write this as equation
∂w
∂t
∂v
∂t
=
= Âw in terms of w =
u
v
∂u
∂x
− βv for some α(x) and β(x), for Ω = [0, L] with
.
(ii) Under what conditions on α and β will this system of equations
have decaying
solutions (i.e. which → 0 as
∂/∂x
t → ∞)? You can use the fact, derived in class, that D̂ =
= −D̂∗ (i.e. is anti-Hermitian)
∂/∂x
0 R
u
u
under the inner product h
,
i = ūu0 + v̄v 0 .
0
v
v
Problem 2: Finite differences (30 points)
2
m +um−1
+ O(∆x2 ) with Dirichlet boundaries
In class, we discretized Âu = ddxu2 on um = u(m∆x) by u00m ≈ um+1 −2u
∆x2
u(0) = u(L) = 0. Now, we will consider the same operator with Neumann boundaries u0 (0) = u0 (L) = 0.
(a) Describe (sketch if needed) a finite-difference grid with spacing ∆x between adjacent points, with N unknown
values of u for x ∈ (0, L), such that u0 (0) and u0 (L) can be evaluated to second-order-accuracy by center
differences (with spacing ∆x, not 2∆x) on the grid. What is ∆x in terms of L and N ?
(b) Using this grid, write the discrete version Au of the operator Âu, applying the Neumann boundary conditions
to get the first and last rows.
(c) Show that your matrix is symmetric negative semidefinite, like Â.
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