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.