Test 1

## Integral Equations and the Method of Green's Functions Evans M. Harrell II and James V. Herod*

SAMPLE TEST

At Georgia Tech, this test took 50 minutes (calculators allowed but no notes).

Background. The operator to consider in this test is defined by

L(u) := x2 u''(x) - 2 u(x),

for 1 < x < 3. (It begins at 1 because the equation becomes singular when x=0.) Perhaps it is helpful to notice that L(x2) = L(1/x) = 0. You do not have to show this.

Impose boundary (initial) conditions that u(1) = u'(1) = 0 for L on this page.

Find the following:

1. L*(u) = _______________________________________________

with conditions on u: __________________________________

2. The Green function for L is

G(x,t) = _______________________________________________

3. The Green function for L* is

G#(x,t) = _______________________________________________

4. The solution of L(u) = x3 , u(1) = u'(1) = 0, is:

u(x) = _______________________________________________

FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________

In problem 5, we consider different boundary conditions, but still

L(u) := x2 u''(x) - 2 u(x).

The boundary conditions are of the strange form:

9 u(3) = u(1)

9 u'(3) - u'(1) = 12 u(3)

You may take as given that L*(1) = 0. You do not have to show this, and you do not have to get involved in the strange boundary conditions.

5.

a) Give an example of a function f(x) for which

L(u) = f(x)

has a solution.

b) What specific differential equation must be satisfied by a function G(x,t) such that a solution to part a) is given by

u(x) = the integral from 1 to 3 of G(x,t) f(t) dt ?