Ph.D. Comprehensive Examination: Applied Mathematics
January 12, 2004.
Instructions: Attempt all four questions
1. Find the y(x) ∈ C 2 [0, 1] which minimizes the functional
J(y) =
Z
1
0
1 02
y + yy 0 + y 0 + y dx
2
stating the natural boundary conditions y must satisfy.
2. Use a Green’s function g(x, y) to find the general solution of
d
Lu ≡
dx
1 du
x dx
!
= f (x) ,
x ∈ (1, 2)
u(1) = 0
u(2) = 1
3. Let the integral operator K defined on L2 (0, 1) be given by
Ku ≡
Z
1
k(x, y)u(x)dx .
0
Further suppose that K is compact, self adjoint and has a complete set of mutually orthonormal
eigenfunctions {φn (x)}, n = 1, 2, . . . with associated eigenvalues {λn }n≥1 . Consider the following
coupled integral equations:
u + Kw = f
(1)
βw − Ku = g
(2)
where f, g ∈ L2 (0, 1) and β ∈ IR.
a) Compute series solutions of u(x) and w(x) using {φn (x)} as a basis. For what β is such a
solution valid?
b) When (1)-(2) has a nonunique solution, what conditions must f (x) and g(x) satisfy?
Q UESTION F OUR ON BACK
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4. The Fourier transform û(λ, t) of the scalar function u(x, t) is defined by:
û(λ, t) =
Z
u(x, t)eiλx dx
IR
with associated inverse
1 Z
û(λ, t)e−iλx dλ
û(x, t) =
2π IR
Now, consider the following initial value problem for u(x, t)
ut = uxx − tu ,
x ∈ IR , t > 0
(3)
u(x, 0) = f (x)
(4)
where u is bounded.
a) Determine the Fourier transform û(λ, t) of the solution u(x, t) of (3)-(4).
b) Use the Fourier convolution theorem and the identity
1 Z
1
2
2
e−aλ /4 e−iλx dλ = √ e−x /a
2π IR
πa
,
a>0
to determine an integral representation of the solution u(x, t) of (3)-(4).
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