


Fysikens Matematiska Metoder, Del 2
Loggböcker VT13
Bos log
(övningsgrupp 1)
Johans
log (övningsgrupp 2)
Vivecas log
(övningsgrupp 3)
KS= G. Sparr och A. Sparr, Kontinuerliga system,
Studentlitteratur, Lund (2000)
ÖB= Sparr och Sparr, Övningsbok till Kontinuerliga system,
Studentlitteratur, Lund (2000)
OBS If you write email to me I will answer usually in
english since this is faster for me. However, it is prefectly
fine if you write swedish. For the same reason I write here in
English.
Sammanfattning
av föreläsningarna
Vad hände egentligen på vilken föreläsning!?
 Lektion 1: (based
on [KS] Sections 1.2, 1.2.2, 1.4.1, 1.7 ) Introduction.
Discussion of "kursPM".
General remarks on modeling. Mathematical models=mathematical
problems. One important task: Derivation of partial differential
equation (partiell differentialekvation=PDE), boundary
conditions (randvillkor=RV), and initial conditions
(begynnelsevillkor=BV). Example: model of a guitar string
(intuitive discussion). Derivation of the heat equation
(discussed in some detail). Other examples: Diffusion (emphasis
that the same model can describe very different physical
situations). Remark: Note that exercise groups 1 and 2
are for SI1140 (F students) exercise group 3 is for SI1141 and
SI1143 (CL and E students). Reading instructions
for next time (in addition to the material discussed this
lecture): [KS] 1.10, 1.4
 Lektion 2: (based
on [KS] Sections 1.1, 1.3, 1.6.1, 1.6.2, 1.10, 3.2.1, 3.3.1)
About PDE problems we are going to discuss in the course
(PDE+RV+BV). Heat eq., wave eq., Laplaces eq. and applications.
Fourier's method for homogeneous problem (example heat equation
in 1D). Extension to inhomogeneous problems (main idea). Remark:
As discussed in my lecture, in this course we
mainly discuss linear PDE problems, but this does not mean at
all that nonlinear PDE problems are not interesting. As an
example, I mentioned the KdV equation. If you want to know
more about it I recommend you take a look at the Wikipedia page on
KdV (in my lecture I mainly mentioned what is
discussed in the section "History" there).
 Lektion 3: (based on [KS] 1.4.1, 3.2.3, 1.6.2) Model
for a guitar string (including external forces, friction etc.).
Fourier's method for homogeneous problem: example string model,
and general strategy (linearity, superposition principle).
Physical interpretation of solution of the string model.
Boundary conditions; Newton's law of cooling. Introduction to
the theory of Hilbert spaces (motivation).
 Lektion 4: (based in [KS] H1, H8, H9, H10, H13)
Motivation why we discuss Hilbert space theory: PDE problems in
higher dimensions and how the theory of selfadjoint operators
on a Hilbert space provide the tool to solve them using
Fourier's method. Vector space, scalarproducts, and Hilbert
spaces (examples you know from previous courses and which we
encounter in PDE problems). How to interpret the problem "Find all functions X and
konstants λ on the intervall [0,L] that satisfy
X''(x)+λX(x)=0, X(0)=X(L)=0" as eigenvalue equation of
a selfadjoint operator. Generalizations. Examples of general
arguments used to prove results about selfadjoint operators
(demonstrating that such arguments are powerful and need not be
difficult). Remarks: (i) In my lectures I can only give an overview on the
theory of Hilbert spaces, trying to motivate and summarize the
main results that we use in the course. In particular, I suppressed all
technical issues concerning convergence of infinite series and
the like.I strongly
advise you study "bilaga H" in the course book to get a better
understanding. (ii) A nice summary of the theory of
Hilbert spaces (including historic information and various
examples) can be found on Wikipedia.
 Lektion 5: (based
on [KS], H2H7, H11) Laplace equation in a disc.
Orthogonal systems, basis in Hilbert spaces, selfadjoint
operators, positive operators etc. (examples and continuation of
useful abstract arguments). Schrödinger equation (another
motivation for Hilbert space theory). Eigenvalue equation of the
Laplacian on a rectangle. Remarks: (i) Nice summaries of the theory
of Bessel functions and
other special functions (including historic information and
various examples) can be found on Wikipedia. (ii) A Wikipedia
link to complementary information about the Schrödinger
equation.
 Lektion 6: ([KS] 1.10, 3.3, Bilaga S) (i) General
remarks on how to solve inhomogeneous problems: how to make a
inhomogeneous problem homogeneous by finding a particular
solution; solving a problem with inhomogeneous PDE but
homogeneous boundary condition using Fourier's method; how to
remove inhomogeneous boundary conditions. (ii) Special
functions: Gamma function. Bessel functions and how to use them
to solve the Helmholtz equation in polar coordinates. Spherical
Bessel functions. Legendre polynomials. Remarks:
(i) You might find the
Wikipedia pages on the Gamma
function and the Bessel
functions helpful. You can find e.g. nice plots there. (ii) A good way to get acquainted
with special functions is to plot them using MAPLE. You can
find a program to start below. I also included a program
allowing to solve the heat equation numerically with MATLAB.(iii) The mathematical theory of
special functions is a beautiful subject, and my lecture gives
only an overview and a few examples. If I managed to motivate
you to want to study more about this subect I recommend Lebedev's book mentioned on the "" page. (iii) In the KS no "formelsamling" is
allowed, but if you need some formula that I cannot expect
you to know by heart (e.g. the Laplace operator in polar
coordinates) I will write it on the problem sheet.(iii) A
good way to learn about special functions is to use MAPLE.
You can plot the Bessel function J_2(x) by the command
"plot(BesselJ(2,x),x=0..10)", for example. Use "?BesselJ" to
get information on Bessel functions. I suggest you study
Appendix S in [KS] and test the claimed properties using
MAPLE. ADVISE ON HOW TO
PREPARE FOR THE MIDTERM TEST ("KS") IN TWO WEEKS: (i)
The material for the "KS" is what was done up to and including
"Ovn7" and "Frl7" (but of course you are expected to know how
to solve an integral containing a Dirac deltafunction). (ii)
If you need the Laplace operator in spherical or cylinder
coordinate to solve a problem in the KS the formula will be
given (i.e. you are not expected to know it by heart). (iii)
As I said, my aim is to make the "KS" such that those who have
done the problems as suggested on the course
planning page
and have followed the course and the exercise class should be
rewarded by bonus points. Thus a good way to prepare for the
"KS" is to do as many homework problems ("Hemuppgifter") on the course
planning page as possible. If you have not yet solved
all problems I suggest you start with the ones in bold face. I
recommend that you first make sure that you know how to solve
problems where eigenfunctions can be expressed in terms of elementary
functions (sin, cos, exp). (iii) The
file
Summary
of
the course.pdf contains
a priority list of the course material (this is for a whole
course, but I decided this year to it available already now).
The topics for the KS are 14 and 89 on this summary.
 Lektion 7: (based on [KS] H.11, 4.1.1, Appendix D.9)
Fourier transform as limit of Fourier series. Fourier
transformations and how to use them to solve PDE problems:
generalities and examples. Green's function for the heat
equation in 1D. Remark:
The Wikipedia article on Fourier
transformation
contains a lot of useful information (including the
derivation of FT as a limit of Fourier series that I sketched
in my lecture).
 Lektion 8: based on [KS] Appendix S4.B, 4.1.2,
Appendix D) Remarks on Legendre polynomials. Dirichlet problem
on half place. Distributions: What are test functions? The
mathematical definition of distributions. Delta and Heaviside
distributions. How to differentiate distributions, e.g. what
does δ'(xa) or δ''(xa) mean? Principle value and other
examples of distributions. Applications and examples. Fourier
transform of distributions. Remark: As you certainly
have seen on previous courses, distributions like the delta
"functions" are an important tool in theoretical physics etc.
I tried to give you an introduction to the mathematical theory
of distributions mainly to give you tools to determine the
precise meaning of mathematical expressions involving
distributions. I highly recommend that you study "Bilaga D" in
the book carefully.
 Lektion 9: (based on [KS] Appendix D, 4.2, 5.8, 5.1)
Distribution theory cont. (how to solve f(x)U(x)=0 in
distribution sense; distributions for test functions in several
variables). Inhomogeneous problems on the real line.
Problems on halfline: "spegling". Green's functions
(introduction).
 Remarks:
The "tilläggskurs" SI1142
will be this year
identical with Part 1 of a new course SI1145. In SI1145 Part 1 we plan to
discuss (i) Variational calculus, (ii) more on Green's
functions. In SI1145 Part 2 we plan to discuss: (iii)
Analytical mechanics (in particular Hamilton formalism and
advanced topic that are helpful for a better understanding of
quantum physics), (iv) relativistic mechanics and introduction
to field theory (which hopefully will better prepare you for
advanced theoretical physics courses). If you want to
participate in SI1142 I recommend you still register for
SI1145  if you decide to only do part 1 you will be able to
change your registration to SI1142 (you should be able to
register via "mina sidor" and also in the first few
lectures).
 Lektion 10: (based on [KS] 4,5, and Bilaga H.11,
H.13) Dimensional
analysis: what does it mean to set "L=a=1" in a heat
equation problem (choosing convenient physical units, adapted to
a problem at hand; I also mentioned natural
units which you will encounter in more advanced
theoretical physics courses). How to solve problems in this
course in a more efficient way: Green's function method. Basic
idea and other examples. Green's functions and Poisson kernels
for Dirichlet problems. Examples: Fundamental solutions 3D.
Green's functions and Poisson kernels for half spaces
("spegling"). Poisson kernel for disc. Fouriers method and
Green's functions for abstract operators. How other chapters in
the course (SturmLiouville problem, special functions) are
related to our discussion on Green's functions.
 Lektion 11: (partly based in [KS] 2 and 7)
Discrete models (difference equations) and relation of
analytical and numerical methods. Intuitive explanation of
spectrial theorem (selfadjoint operator can be obtained as a
limit of a selfadjoint matrix). Propagation of
waves,
damping and dispersion,
group velocity and phase velocity etc. Heuristic derivation of
the Schrödinger
equation (see the part "Historical background...." further
down this Wikipedia page). Remark: (i)
The
file Summary
of
the course.pdf contains
a
priority list of the course material which, I hope,
should be helpful to prepare for the exam. I recommend that
you study
the problems suggested there rather than spending most of the
time on
old exams. Many
more old exam problems (with
solutions) are available on
the
course literature page [EX] (ii) There
will be a "räknestuga"
(RS) this Friday where
you can get help
(some of us will be there to answer questions). (iii) NOTE that there is a "tillägskurs"
for those who are interested in getting a broader
perspective. (iv) If
you did not pick up your
midterm test yet, you can do in the
"studentexpeditionen". If you have questions on your marking
you can ask these in the RS.
Please check the course homepage regularly: if there is anything
I will put out information there.
That's it for
this year. Thank you!

