Homework 4

Due to the dropbox by 11:00pm on November 8, 2013.

Do the following exercises from ATAP:

• Exercises 12.1, 13.3, 14.1, 14.2

• Exercise 15.4. Also plot the Lebesgue function for \(n=10\).

• Exercise 15.A: (additional exercise)

  Confirm using Chebfun that the bound (15.5) holds for the following cases, with \(n=20\):

  • \(f(x) = 1/(1+25x^2)\) with both equally spaced and Chebyshev points
  • \(f(x) = 1/(1+500x^2)\) with both equally spaced and Chebyshev points

  The functions remez and lebesgue may be useful.

• Exercise 21.8

• Exercise 21.A:

  Use Chebfun to solve the singular perturbation problems

  \(\epsilon u''(x) + (1+\epsilon)u'(x) + u(x) = 0\)

  on the interval \([0,1]\) with boundary conditions \(u(0)=0\) and \(u(1)=1\).

  Solve the problem for various values of \(\epsilon\) to see how small you can take it and still get an accurate solution.

  Compare to the exact solution, which can be found at http://en.wikipedia.org/wiki/Method_of_matched_asymptotic_expansions.

  Plot the solution zoomed in around the boundary layer to see better how it behaves.

Notes: