TMA4305 Partielle Differensialligninger 2015

TMA4305 Partial Differential Equations Fall 2015


Office hours before the Exam

Tuesday 14–16
Wednesday 15–16
Thursday 14–16.

The first lecture will take place on Tuesday the 18th of August (14:15 - 16:00 in K26) The first exercises will take place on Monday, the 24th of August (17:15 - 18 in F3)

See below about the Exam.

* The lectures are in K26. (Kjemi 4, first floor)

Week Section Comments
34 Ch.1, Ch.2 Different kinds of equations and problems. Characteristics. First order eqs.
35 Ch.2, Ch. 10.4 Weak Solutions. Rankine-Hugoniot Shock Condition Notice Lemma 10.12
36 Ch.2, Ch 4.1, 4.2 One dimensional Wave and Heat Equations. Duhamel"s Principle. Energy considerations ex.4.2.35 &36 Duhamel's Principle not in the textbook
37 Ch. 4 Equations of 2nd order in two variables. Characteristic curves. Poisson's formula Characteristics
38Ch. 6Distributions. Fundamental solutions?
39Ch. 7, Ch. 8Fourier transforms, the Heat EquationOnly selected parts
40Ch. 6, Ch. 8, Sec. 12.3Green's function. Duhamel's Principle. Newtons potential. Green's formulasSection 12.3 is helpful (in space). Potential of a ball
41Sec. 8.3, Sec. 12.6Parabolic Maximum Principle, Kirchhoff's formula, Huygens's PrincipleSimple proof, One dimension, parabolic boundary
42Ch. 12 Duhamel's principle and the retarded potential for wave eqn., Nonlinear Diffusion, DispersionAn example with Cole-Hopf cole.pdf. Local energy consideration. Uniqueness. Light cone
43See notes below. Ch. 9.3Dirichlet's Principle. Variational Integrals. Euler-Lagrange EquationsSee the "direct method"
44Sobolev Spaces. Existence Proofs Weak compactness, weak convergence
45Ch. 9.4 Eigenvalues, Rayleigh quotient Merely for Helmholtz"s equationVibrating Membranes
46Some minor additions. Repetition?
47Repetition. Exercises?

Information about the course


Peter Olver: "Introduction to Partial Differential Equations", Springer 2014. Also available on Springer link.

The lectures are usually more succinct than the book. (Habent sua fata libelli pro captu lectoris.)


Tuesday 14–16 in K26
Thursday 14–16 in K26


Peter Lindqvist (SB II room 1152)


*Monday 17 -18 aud. F3.

Week Problems Comments
351.12, 1.13, 1.14, 2.1.4, 2.1.6, 2.2.4, 2.2.11aread ex. 1.16 for your information.Some answers
362.3.2, 2.3.3, 2.3.5, 2.3.15, 2.3.17, 2.3.22?
374.1.5, 4.1.16a,b, 4.1.17, 4.1.18, 4.2.9a,b,c, 4.2.35, 4.2.36Notice that 4.2.35&36 were done in class. Slns
38Problems 4.3: 1a, 25a, 26, 32, 46;. 4.4.5Comments
39 6.1: 1f, 2b, 5b, 7, 10 ExamplesNO CLASS 21.IX. Solutions
40 6.3: 9, 17, 29. 8.1: 17, 20.Solution
416.3.16, 8.1.14, 8.1.19 (misprint?), ExercisesReplace i by -i in 8.1.19. Scrodinger
428.3: 4,5,6. 12.6.1 a,d, 12.6.2, 12.6.3Solutions
43 Exercises Misprint in 1, WAVE eqn meant. Use rather cos(az)v(x,y,t) in 3. Solutions
448.4.1, 8.4.11, 8.5.3a,b. Exercises?
459.3: 18, 19, 20, 25.Some Solutions
469.4.5c, 9.4.6, 9.5.18 ExercisesTorsional Creep. More solutions
47ExercisesLast ex. too long; do it for f = 1. Misprint: - sign in the root


Preliminary Syllabus (Pensum)

Chapter 1
Chapter 2
Chapter 4
Chapter 6
Chapter 8
Chapter 9.3, 9.4
Chapter 10.4
Chapter 12

How to find the Euler-Lagrange Equation of a Variational Integral. Dirichlet's Principle.
Weak solutions (with test functions under the integral sign). Shocks.

The End

Extra literature

L. C. Evans:"Partial Differential Equations". –One of the most used standard texts today. Advanced.
W. Strauss: "Partial Differential Equations: an Introduction". –Clear and easy to read.
A. Tveito & R. Winther:" Introduction to Partial Differential Equations (A Computational Approach)". –An excellent, but very elementary, account.
S. Salsa: "Partial Differential Equations in Action". –Contains relevant information but is sometimes confusing.

Exam 30.XI.2015

The exam (written) is in English only. The students may answer in Norwegian or English. To the exam you are allowed to bring one A4-sized sheet of yellow paper on which you may write whatever you want in advance. It must be pre-stamped at the Department of Mathematics (SB II, 7th floor), where empty sheets can be acquired. (A simple calculator is permitted.) No other aids are permitted in the exam.

Exam. 30.Xi.2015 Solutions 30.XI.2015
Exam 1.XII.2014.Misprint 3b. Read u(x,y,z,0) = f(x,y,z). Solutions 1.XII.2014


  • Advanced Notes on Sobolev Spaces Sobolev



2015-12-01, Lars Peter Lindqvist