Applied Mathematics Comprehensive Exam
August 2011
Instructions: Answer 3 of the problems from Part A, and answer 3 of the problems from
Part B. Indicate clearly which questions you wish to be graded.
Part A
A.1 (a) Determine conditions on f, α and β for which there are solutions of
u00 = f (x),
0<x<π
u0 (0) = α,
u0 (π) = β
(b) Find the best possible (least squares) solution of
u00 = sin2 (x),
u0 (0) = 1,
u0 (π) = 1
A.2 Let A ∈ lRN ×N be a Symmetric Positive Definite matrix. Recall the quadratic form
q(x) = hAx, xi. Show that the level surface q(x) = 1 is an ellipsoid.
A.3 Use the Green’s function to solve
u00 = f (x), 0 < x < 1
Z 1
u(x)dx = 4.
u(0) = 3,
0
A.4 Find solvability conditions for
u00 + u = f (x),
0 < x < 2π
u(0) − u(2π) = α, u0 (0) − u0 (2π) = −β
A.5 (a) Find the resolvent (or pseudo-resolvent) kernel for the problem
Z 1
u(x) = f (x) + λ
u(y)dy
0
and for any choice of λ.
(b) Solve the integral equation for f (x) = x.
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Part B
B.1 Define the bounded linear operator T on L2 (0, 1) by
Z 1
e−|x−y| u(y) dy
T u(x) =
0
(a) Show that if T u = v then v(x) satisfies
v 00 − v = −2u,
0 < x < 1 and v(0) − v 0 (0) = v(1) + v 0 (1) = 0
(b) Show that if λ is a nonzero eigenvalue of T , with eigenfunction u(x) then
2
u00 + ( − 1)u = 0,
λ
0 < x < 1 and u(0) − u0 (0) = u(1) + u0 (1) = 0
(c) Show that the eigenvalues of T are real and lie in the interval (0, 2).
B.2 Let p(x) > 0 be smooth and define the Sturm-Liouville operator L with domain D(L)
as follows:
du
d
(p(x) ),
Lu ≡
dx
dx
2
2
D(L) ≡ {u ∈ L (0, π): Lu ∈ L (0, π), u(0) = u(π) = 0}
(a) Assume that L has a complete orthonormal set of eigenfunctions φn (x), n =
1, 2, . . . with associated eigenvalues λn , for n = 1, 2, . . .. Show that the Green’s
function g(x, y) solving Lg = δ(x − y) has the representation
g(x, y) =
∞
X
φn (x)φn (y)
λn
n=1
(b) For the case p(x) = 1, compute g(x, y) and then use the result in part a) to find
a formula for the following sum:
∞
X
sin nx sin ny
n2
n=1
B.3 Find the modified Green’s function for
u00 = f (x),
u(0) = u(1), u0 (0) = u0 (1)
0 < x < 1,
B.4 Let f (x) be a continuously differentiable function except at a discrete set of points
x1 , x2 . . . , xn , where f has finite jump discontinuities ∆f1 , ∆f2 , . . . , ∆fn . Show that
the derivative of f (x) in the sense of distributions is given by
n
df X
+
f =
∆fj δ(x − xj ),
dx j=1
0
where df /dx is the classical derivative of f wherever it exists.
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B.5 Use Neumann iterates to solve the following integral equation using u0 (x) = 1.
Z x
(y − x)u(y)dy
u(x) = 1 +
0
B.6 (a) Consider the variational problem of minimizing J(y1 (t), y2 (t)), where
Z t1
f (s, y1 (s), y2 (s), ẏ1 (s), ẏ2 (s)) ds
J(y1 , y2 ) =
t0
subject to
y1 (t0 ) = y1,0
y1 (t1 ) = y1,1
y2 (t0 ) = y2,0
y2 (t1 ) = y2,1
Assume that f (s, y1 , y2 , v1 , v2 ) is a C 2 function. Derive the set of Euler-Lagrange
equations for this problem.
(b) Find extremals of the functional
π/2
Z
[y 0 ]2 + [z 0 ]2 + 2yz ds
J(y, z) =
0
subject to y(0) = 0, y(π/2) = 1, z(0) = 0, z(π/2) = 1.
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