Some info in English

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 non-linear 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 self-adjoint 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 self-adjoint operator. Generalizations. Examples of general arguments used to prove results about self-adjoint 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], H2-H7, H11) Laplace equation in a disc.  Orthogonal systems, basis in Hilbert spaces, self-adjoint 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 "Kurslitteratur" 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 delta-function). (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 1-4 and 8-9 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'(x-a) or δ''(x-a) 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 half-line: "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 (Sturm-Liouville 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 (self-adjoint operator can be obtained as a limit of a self-adjoint 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!

Senast uppdaterad: Mar 5, 2013