Theory of differential equations

Theory of differential equations;

Partial differential equations

January--March, 1998, Christer Kiselman

This semester I have taught a course on partial differential equations. It was intended both for students of the Engineering Programs (called Theory of differential equations) and the Mathematics and Natural Science Program (then called Partial differential equations).

The main book to be used was Partial Differential Equations, Springer-Verlag, by Fritz John (1910--1994). More precisely, I proposed that the curriculum comprise the following parts of this book:
Chapter 1: all of it;
Chapter 2: sections 1--5;
Chapter 3: sections 1, 2, 4, 6;
Chapter 4: all of it;
Chapter 5: sections 1--2;
Chapter 7: section 1;
Chapter 8: Hans Lewy's example.

The course was held in English because of the participation of some exchange students.

Some exercise problems were distributed, to be solved during the course. The problems were distributed on paper in the classroom but are also available on the web:
Sheet No. 1 available as a DVI file and as a PostScript file.
Sheet No. 2 available as a DVI file and as a PostScript file.
Sheet No. 3 available as a DVI file and as a PostScript file.
Sheet No. 4 available as a DVI file and as a PostScript file.
Sheet No. 5 available as a DVI file and as a PostScript file.
Sheet No. 6 available as a DVI file and as a PostScript file.
Sheet No. 7 available as a DVI file and as a PostScript file.
Sheet No. 8 available as a DVI file and as a PostScript file.
Sheet No. 9 available as a DVI file and as a PostScript file.

There is a Word list available as a DVI file and as a PostScript file. It contains translations into Swedish, Esperanto, and French of terms used during the course.

The students were invited to lecture on specified topics. This was intended to make them more familiar with some of the interesting themes in the great theory of PDE that cannot be covered in detail in the course. Giving such a lecture counted as part of the examination. A list of possible topics was distributed on paper on January 26. The list is also available on the web as a DVI file and as a PostScript file. The list of lectures actually delivered by the students is also available.

The course started on January 19 and ended on March 13.

The program was as follows:
January 19: A survey of ordinary differential equations.
January 21: A survey of first order partial differential equations. Characteristic curves. Excercise problems on the existence solutions in different domains: 1.1, 1.2.
January 23: Exercise problems on characteristics, complete description of solutions, blow-up of solutions: 1.3, 1.4, 2.1, 2.2, 2.4.
January 26: Every quasilinear equation corresponds to a linear equation in one more variable. Exercise problems 3.1, 3.5
January 28: Waves as solutions to first order equations. General nonlinear equations of the first order. Envelopes of solutions.
January 30: Characteristic strips. Exercise problem 4.1 and some other examples where both envelopes and characteristic strips can be used for solving. Which method is best?
February 2: Solving 4.2 and 4.3 using the method of characteristic strips. Second order equations: introduction. Cauchy's problem.
February 4: Second order equations, cont'd. Characteristics. Normal form. Well-posed and ill-posed Cauchy problems. Finding the Taylor series of a solution. Exercise problem 5.1.
February 6: Well-posed and ill-posed problems for different types of equations. Exercise problem 5.3.
February 9: Characteristic hypersurfaces (any dimension, any order of the equation).
February 10: Domain of dependence, range of influence. The wave equation with initial and boundary condition. Reflection of waves. Discontinuities in a solution. The hodograph method (the Legendre transformation).
February 12: Hyperbolic systems. Real analytic functions.
February 16: Real analytic functions. The Cauchy--Kovalevsky theorem.
February 18: Lecture by Bengt Eliasson on The equations of fluid mechanics (6). Proof of the Cauchy--Kovalevsky theorem. Green's formula.
February 20: Lecture by Markus Jonsson on Differential forms. (24). Fundamental solutions for the Laplacian.
February 23: Lecture by Peter Borg on Soft solutions. (15). Green's function.
February 25: Lecture by Lars Axelson: Why is the world three-dimensional? (36). Lecture by Martin Nilsson on Wavelet transforms (10). Hypoellipticity of the Laplacian.
March 2: Lecture by Lawrence McCandless: The Schrödinger equation and its relations to the Korteweg--de Vries equation (39). Dirichlet's problem for the Laplacian. The Poisson kernel. Hilbert space methods for the Laplacian.
March 3: Lecture by Karl Håkansson and Johan Edlund: Solitons and the Korteweg--de Vries equation (1, 2, 3). Hilbert space methods for the Laplacian (cont'd). Mean values of functions.
March 4: Darboux's equation for the mean values of functions. Solving the wave equation in any number of variables. Explicit solution in three variables. Hadamard's method of descent. Duhamel's principle.
March 6: Lecture by Lars Berggren: The Laplace transformation (11). Exercise problems 5.6 and 8.1. The energy integral. Mixed problems for the wave equation. Eigenfunctions for the Laplacian. Higher order hyperbolic equations.
March 9: Exercise problem 4.4. Gårding's hyperbolicity condition. Solution using the Fourier transformation of initial-value problems with a higher-order hyperbolic equation.
March 13: Lecture by Per Andersson: Representation formulas for solutions to the equation Laplacian(u) = u. Lecture by Roger Agenstam and Daniel: Geometric interpretation of the Hamilton--Jacoby equation. Parabolic equations. Fundamental solutions to the heat equation. The maximum principle for solutions to the heat equation. Regularity of solutions to the heat equation. Uniqueness if the solution is bounded from below.

The written examination took place on March 16, 1998. There were oral examinations afterwards. All results have now been reported to the university's computer system UPPDOK. Those who did not succeed can try again in June or August---the exact days are not known to me.


Last change 1998 04 18. Christer Kiselman,