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) = ∫₀^π k(x,t)u(t)dt, where k(x,t) = 2cos(x)cos(t) + sin(x)sin(t)
  
  1. 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 = 5^½π/2.
  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) = ∫₀^π k_n(x,t)u(t)dt, where
     k_n(x,t) = 2μ₁^n-1 cos(x)cos(t) + μ₂^n-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.
  3. 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.
  4. Section 3.6, problem 3, page 131.

Updated 11/22/06 (fjn).