## Math 400 (Applied Partial Differential Equations)

• Course Details:
• Section 101, Fall 2012
• Lectures 9-10 MWF in room Klinck 460.

• Instructor:
• Brian Wetton
• wetton@math.ubc.ca
• office: MATX 1107
• office hours:
• Mondays 10-11
• Tuesdays 11-12
• Fridays 2-3

• News:
• Final exam notes are posted below, along with a practice exam with solutions.
• Assignment #12 posted, not collected. Solutions are posted below.

• Final Exam Material:
• Outline of exam material: here and a practice final: here. Solutions to the practice final are here.
• Exam period office hours: Monday 11-1 in Math Annex 1118, Tuesday 10-12 in Math Building 103.

• Midterm Material:
• Midterm solutions. There are a couple of errors in the solutions:
• In B2a the axis labels x and t should be switched.
• In B2b in the right equations on the first two lines, u and u_x should be evaluated at (0,t) rather than (x,0).
• Outline of midterm material: here and a practice midterm: here with solutions. Note that there is an error in B1b in the second form listed for x(s), which should read s/(1-st).

• Course Resources:
• Course outline
• There is no required text for the course.
• Optional text: Elementary Applied Partial Differential Equations by Richard Haberman.
• Other text books to look at:
• S.J. Farlow. PDE's for Scientists and Engineers, Dover.
• E. Zauderer, Partial Differential Equations of Applied Mathematics, Wiley.
• J.Kevorkian, Partial Differential Equations, Springer.
• Ockendon et al, Applied Partial Differential Equations, Oxford.
• Handwritten notes will be posted on this page. Some notes from previous intructors of the course will also be posted.

• Assignments:
• Assignment #1, and solutions.
• Assignment #2, and solutions.
• Assignment #3, and solutions. For question #1 you may assume that a(x) is positive, piecewise smooth and bounded away from zero.
• Assignment #4, and solutions. In the solutions at the top of page 4, the statements should have |V| instead of V. I would have taken marks off from myself for this. Notes:
• In question #2, "standard form" for the second order derivative terms for elliptic equations is Uxx + Uyy and hyperbolic is Uxx - Uyy, where here the lower case letters represent derivatives in new variables x and y. As a hint, the equation in #2 is not parabolic.
• The notation in #4 is standard for this equation (for transonic potential flow) but may be a bit confusing. The subscripts x and y on Phi are partial derivatives, but the subscripts on M are just subscripts defining Mx as the scaled partial derivative of Phi with respect to x and similarly My.
• Assignment #5, and solutions. There is an error in the solution to question #6, the corrected version is here.
• Assignment #6. Intended for midterm review, will not be collected or graded. Notes:
• In problem 1e, I forgot to define the oprator L, it is defined by L phi = phi'' with the given boundary conditions.
• Assignment #7, and solutions. Notes:
• Problem #2 was meant to be more straightforward, with initial conditions 3 for x<0 and -1 for x>0. You may answer the question with these initial data but if you have already done the question with the data as written in the problem, that is OK as well.
• Problem #6 should refer to problem #5, not #4.
• Assignment #8 and solutions. Note that the equation in problem #1 should read phi''(x) = lambda phi (x). In Problem #4, it should read "replaced by u(1,t)=0".
• Assignment #9 and solutions. Notes:
• Question #4 should be referring to question #3 and the problem is to show that the quantity at the bottom of the first page is zero when n is not equal to m.
• Question #5 should be referring to questions #3 and #4. Also, the boundary conditions should be homogeneous Neumann conditions, that is du/dx(0,t) = 0 and du/dx(1,t) = 0 [where the d/dx are partial derivatives].
• In question #6 you will be faced with a complicated integral for the perturbation to the coefficient of the first term. It is not necessary to evealuate this integral.
• Assignment #10 and solutions.
• Assignment #11 and solutions.
• Assignment #12 and solutions. Note the error in the scaling of question #2 corrected in the solutions.

• Course Notes:
• If you find any mistakes or omissions in the notes, please let me know and I will post corrections here.
• Part I. Errata:
• At the top of page 20, the listed inequality is true at all points in the domain t > 0, 0 < x < 1. The discussion below this inequality presents contradictory conditions that must hold at points where v attains a maximum value.
Some notes from previous instructors on D'Alembert's solution of the wave equation are given here: extra notes #1, extra notes #2, and extra notes #3. There is some review of ODE and linear algebra material. If any of this seems unfamiliar, you can look up this material in the following texts:
• (Linear Algebra) Gilbert Strang, Linear Algebra and its Applications.
• (ODE) Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems.
• Part II. Errata:
• At the bottome of p.3 all the k parameters should be K instead.
Some notes from previous intructors that are userful in this section are given here: extra notes #1 (derivation of the heat equation), extra notes #2 (derivation of the heat and wave equations), and extra notes #3 (derivation of the wave equation).
• Part III. Some notes from previous intructors that are useful in this section are given here: extra notes #1 and extra notes #2.
• Part IVa and Part IVb. Corrections:
• In notes 4b at the top of page 10 I have right and left states mixed up.
Some notes from previous intructors that are userful in this section are given here: method of characteristics, shocks and expansion fans, shock waves and traffic flow, shocks and Burger's equation, characteristics and shocks, traffic flow modelling.
• Part Va, Part Vb, and Part Vc. An appendix with a table of zeros of integer order Bessel J functions is given here. Corrections:
• In Va top of p.3 the frequencies should be n/2.
Some notes from previous intructors that are userful in this section are given here: fourier series (complex form) and fourier transform, heat equation notes, heat equation and Sturm-Liouville problems, Sturm Liouville theory.
• Part VIa (Fourier Transform) and Part VIb (Laplace Transform). Corrections:
• In part a, page 5 middle the uhat(alpha,t) should be uhat(y,t). This occurs in two places.
• In part a, page 6, the integral should be over s (add ds to the end) and the integrand should be u0(x-s)y/(s^2 + y^2).
• In part a, page 2 at least some of the convolution rules written have the wrong scaling. See the solutions to assignment #12 above.
• Part VII Time harmonic problems.