Test 3
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Integral Equations and the Method of Green's Functions
Evans M. Harrell II and James V. Herod*

SAMPLE TEST

At Georgia Tech, this was used as the final exam after a short term (less than 10 weeks). Students were allowed 2 hours, 50 minutes, which was adequate.

1. (This problem is taken almost directly from Herod's notes.)

Suppose that L[u] = u_{xx} - u_{yy} - u_{zz} -

M = {u: u(0,y,z) = 0, u(1,y,z) = 0,

u(x,y,0) = u(x,y,1),

u_{z}(x,y,0) = u_{z}(x,y,1)

u_{y}(x,0,z) = 2 u(x,0,z)

u_{y}(x,1,z) = -1 u(x,1,z) }

Give L*. Find F such that v L[u} - L*[v] u = --.F. What is M*?

ANSWERS:

a) L* v = _______________________________________

b) F = _______________________________________

b) M* = _______________________________________

_______________________________________

_______________________________________

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

________________________________________________________________

2.
a) Find the Green function for Poisson's equation

grad^{2}u = f

in the three-dimensional region z > 0, x > 0, with boundary conditions

u_{z}(x,y,0) = 0, u(0,y,z) = 0.

ANSWER:

G(**P**,**Q**) = _______________________________________________

Is the solution to Poisson's equation unique? Y_____N_____.

Briefly explain:

EXPLANATION:

__________________________________________________

__________________________________________________

__________________________________________________

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

________________________________________________________________

3. **Background**. The operator to consider in this problem is defined by
L(u) := u''(x) - u' (x) - 2 u.

Impose boundary (initial) conditions that u(0) = u(1) = 0 for L.

Find the following:

a) L*(u) = _______________________________________________

with conditions on u: __________________________________

b) The Green function for L is

G(x,t) = _______________________________________________

c) The Green function for L* is

G^{#}(x,t) = _______________________________________________

d) The solution of L(u) = e^{x} , u(0) = u(1) = 0, is:

u(x) = _______________________________________________

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

________________________________________________________________

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