Integral Equations and the Method of Green's Functions

Evans M. Harrell II and James V. Herod*

*(c) Copyright 1996 by Evans M. Harrell II, harrell@math.gatech.edu. All rights reserved.

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):_________

________________________________________________________________

a) Find the Green function for Poisson's equation

grad²u = 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):_________

________________________________________________________________

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):_________

________________________________________________________________

See the solutions.

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