Test 1

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.

SAMPLE TEST

At Georgia Tech, students were allowed to take either 1 d) or 2 e) home to work out overnight. The rest of the test took 50 minutes (calculators allowed but no notes). Most students did reasonably well, but few finished early.

1. (Essentially same as Herod Section 1.5, #4)

Let K(x,t) = 1.2, if 0 < t < x < 1; = 0, if 0 < x < t <1

In this problem we wish to solve y = K y + x for y(x), 0 <= x <= 1.

a) Is this equation separable? Y____ N____ (check one).

b) If there is a solution, it will be a continuous function on 0 < x < 1. Is this, however, guaranteed by one of the conditions for K to be "small"? Y____ N____ (check one)

Give the required calculation here: _________________________________________

c) If there is a solution, it will be square-integrable on 0 < x < 1. Is this, however, guaranteed by one of the conditions for K to be "small"? Y____ N____ (check one)

Give the required calculation here: _________________________________________

d) Solve the problem or show that it cannot be solved:

ANSWER: y(x) = ____________________________________________

(If you use an iterative method, it is sufficient to give three iterations. Extra credit will be given for the exact answer.)

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

________________________________________________________________

2. Let

[<b>K</b> h](x) := (1/5)Proj_f(h) + (x/2) Proj_1(h) for 0 < x < 1, where f(x) = x<sup>2</sup>.

Background. Recall that for any function h, the orthogonal projection onto another function f is defined by

Projf(h) := (<f . h>/||f||<sup>2</sup>) f(x).

In the first contribution to K, f is the function x2 and in the second contribution it is the constant function 1.

In this problem we wish to solve y = K y + x2 for y(x), 0 <= x <= 1.

a) Write the explicit expression for the kernel of K:

K(x,t) = ________________________________________

b) Find the adjoint K*:

K*(x,t) = ________________________________________

c) What is the condition to check for the Fredholm alternative theorem?

________________________________________________

d) What does it tell us here?

________________________________________________

e) Find all possible solutions, or show that there are no solutions.

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

________________________________________________________________

See the solutions.

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