Sample Midterm Questions Math 121, Fall 2004

1. Use Fourier series to find the solution u(x, y) of the following boundary value problem for Laplace’s equation in the semi-infinite strip 0 < x < 1, y > 0:

\frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂y^{2}} = 0,

u(0, y) = u(1, y) = 0, u(x, 0) = 1, u(x, y) → 0 as y → ∞.

2. Use Fourier series to find the solution u(x, t) of the following initialboundary value problem for the wave equation in 0 < x < 1 and t > 0:

\frac{∂^{2}u}{∂t^{2}} − \frac{∂^{2}u}{∂x^{2}} = 0,

\frac{∂u}{∂x}(0, t) = \frac{∂u}{∂x}(1, t) = 0,

u(x, 0) = 0,

\frac{∂u}{∂t}(x, 0) = x.

3. Use Fourier transforms to solve the following initial value problem for u(x, t) in −∞ < x < ∞, t > 0:

\frac{∂u}{∂t} = −\frac{∂^{4}u}{∂x^{4}},

u(x, 0) = f(x).

Write the solution for u(x, t) as a convolution, but do compute any inverse transforms explicitly. How smooth is the solution for t > 0?

4. (a) Give the formulas for the Fourier transform f̂(k) of a function f(x) and the inverse Fourier transform.

(b) Compute the Fourier transform of e^{−|x|}.

(c) State Parseval’s theorem, and use it to evaluate

∫_{0}^{∞} \frac{1}{(1 + k^{2})^{2}} dk.

5. Use Laplace transforms to solve the following initial value problem:

y′′ + 2y′ + 2y = 1,
y(t) = 0, y′(0) = 1.

6. (a) Say what jump conditions the solution of y(t) of the following initial value problem satisfies at t = 0, and find the solution directly (do not use Laplace transforms):

y′′ − 4y = δ(t),
y(t) = 0 for t < 0.

(b) Write the solution of the following initial value problem, where f(t) is an arbitrary function, as a convolution (you don’t need to derive your answer):

y′′ − 4y = f(t),
y(0) = y′(0) = 0.