Homework 1

Submit via the Canvas page by 4:00pm on October 9, 2015. Homework can be submitted up to 24 hours late with a 10% reduction in points possible. If special circumstances warrant turning in assignments late, please make arrangements in advance.

See Homework format for a discussion of formatting. Please use latex if possible for the analytical questions and turn in m-files (or Python scripts) for the coding parts, along with any results such as plots and required discussion of these. Using matlab publish or an IPython notebook is a nice way to combine them. Some samples of how you might do this will appear soon at Homework format.

There are some e-books listed in the Complex analysis section of the bibliography that may be useful if you need a review.

• Download the Matlab codes from Spectral Methods in Matlab (SMM) and make sure these work for you. See Software for the course for links and options.
• Read Chapters 1-4 of SMM.
• Do the following exercises from SMM:

• Exercise 1.1: Note that on page 2 of SMM the functions \(a_{-1}(x), a_0(x), a_1(x)\) are special cases of the Lagrange basis functions. Use a similar approach for the pentadiagonal case where each of the functions \(a_{-2}(x)\) through \(a_2(x)\) will now be quartics. The formulas simplify a bit if you consider the case \(x_j=0\), which is sufficient since the resulting method is translation invariant.
• Exercise 1.5. For both parts (a) and (b), and for both the finite difference and spectral approximations, also plot the error D*u - uprime as a function of x for the case N = 50.
• Exercise 2.1 (a,b,e,f) and familiarize yourself with the other properties.
• Exercise 2.2 (a,b) and familiarize yourself with the other properties.
• Exercise 2.4 - You need only derive (2.12).
• Exercises 2.7 and 2.8.
• Exercise 3.6. Also use this idea to modify Program p5.m to produce the same set of plots for both functions, but using only fft and one ifft.