Math 641-600 Assignments
Assignment 8 - Due Wednesday, November 29.
- Read sections 3.6, 4.1
- Do the following problems. For problems 1-3, the operator K is given by
Ku(x) = ∫0π k(x,t)u(t)dt, where k(x,t) =
2cos(x)cos(t) + sin(x)sin(t)
- Verify that K is a compact, self-adjoint operator on
L2[0, π]. Find the nonzero eigenvalues and corresponding
eigenvectors for K. (There are two nonzero eigenvalues,
μ1 > μ2 > 0.) Also, show that
||K||op = π, and that ||k||L2 =
5½π/2.
- Use problem 1 and the appropriate formula from p. 119 of the text
to find the resolvent for L = I - λK. In addition, show that
Knu(x) = ∫0π
kn(x,t)u(t)dt, where
kn(x,t) =
2μ1n-1 cos(x)cos(t) +
μ2n-1sin(x)sin(t). Use Kn
and the Neumann expansion from section 3.6 to calculate the resolvent
when λπ < 1. Sum the series that you get and compare it
with your previous expression for the resolvent.
- Galerkin's method. Use your favorite software to do the following
problem. Solve u - ¼Ku = f, where f(x) = exp(x), using the
resolvent from problem 2. Then, for n = 5, 10, 30, find the
approximate solution to the same problem using the finite element
Galerkin approach outlined in section 3.6. Plot the results.
- Section 3.6, problem 3, page 131.
Updated 11/22/06 (fjn).