TMA4305 Partial Differential Equations

# TMA4305 Partial Differential Equations, spring semester 2007

#### Messages

• The exam: pdf.
• Solutions for the first problem set (week 3): pdf.
• Solution of exercise 2.2.1(b) (from week 4): pdf.
• Solutions for the supplementary problem set (from the midterm weeks): pdf. (May 11: typo in the solution of ex. 11 has been corrected.)
• Solutions for the last problem set (from lecture notes): pdf.
• Lecture notes for April 30 and May 7: pdf. See the notes for problem assignment. Solutions will be posted here soon. (There will be no problem session on Friday May 11.)
The solution set for the "extra" problem set for the midterm weeks will also be posted here this week.
Here is the exam from last year: pdf and the solution: pdf. NB! Problem 5b on last year's exam is not included in the course syllabus this year.
• Lecture notes for April 23 and 24: pdf.
Solutions of problems for this week (from McOwen: 6.1:5, 6.2:4, 6.3: 3,7) pdf. NB! No problem session Friday April 27.
• Solutions of problems for April 20: pdf.
• Lecture notes for April 16 and 17: pdf. On Friday April 20 we will do the exercise given in these notes.
• Solutions of problems for March 23: pdf.
• Solutions of problems for March 16: pdf.
• Handout with proof of the Proposition in Section 4.2.b of McOwen: pdf.
• Next lecture is Monday, March 12. Extra problem set for "tiltaksukene": pdf. Solutions will be posted here later.
• Solutions of problems from week 8: pdf.
• Solutions of problems from week 7: pdf.
• Handout on the local energy method for wave equation with lower order terms (replaces Section 3.4c from McOwen): pdf.
• NB! Starting Feb. 16 (and then every Friday) the problem session will be at 0915 in room 922 in the Math. Dept. (SB2).
• Solutions of problems from week 6: pdf.
• Solutions of problems from week 5: pdf.

#### Course information

A mathematical model for a physical phenomenon often takes the form of a partial differential equation (PDE). PDEs are therefore important in a wide variety of applications. In this course we give an introduction to the basic mathematical techniques used to analyze PDEs. The main emphasis is on the three classical PDEs from mathematical physics: wave equation, heat equation and Laplace equation, for which we investigate existence and uniqueness of solutions, and various interesting properties of the solutions, such as finite speed of propagation for the wave equation, and the maximum principle for the heat equation. We also study first order PDEs. At the end of the course we will look at some more modern, functional analytic methods: Specifically we will cover the rudiments of Sobolev spaces and apply this theory to the Dirichlet problem for elliptic equations (more general than the Poisson equation). This includes questions of existence, uniqueness and regularity.
• Lecturer: Sigmund Selberg
• Schedule: Lectures Mon 8-10 in room 734 Sentralbygg II, and Tue 8-10 in room F4 Gamle fysikk.
Problem solving: Fri 14-15 in room F3.
• Textbook: Robert C. McOwen, Partial differential equations: Methods and applications 2nd ed., Prentice Hall
• Exam: June 1, 2007, from 09.00 to 13.00. You can bring: Approved calculator (HP30S) and a sheet of A4 paper stamped by the math. dept., on which you can write what you want (must be handwritten by you). You can get this from the department office on the 7th floor of SB2.

#### Calendar/plan

(Subject to change!)

Week Dates Chapter/Section Subject Problems
(from McOwen unless stated otherwise)
Solutions Remarks
2 9.1, 11.1 1.1a,b,c,d 1st order PDE We skip section 1.1e
3 15.1, 16.1 1.2 (all); 2.1a 1.1: 4a, 6a.
1.2: 2, 7, 8, 9.
In 2.1a we skip Example 2.
4 22.1, 23.1 2.1b; 2.2a,b,c Higher order PDE: General principles 2.2: 1b, 6, 9
5 29.1, 30.1 2.3a; 3.1a,d; 3.2a,b,c Definition of weak solutions of linear PDE.
Wave equation
3.2: 2,3,6 pdf
6 5.2, 6.2 3.2d; 3.3a,b; 3.4a,b Wave equation, dispersive equations 3.2: 5
3.3: 1,2
3.4: 2,3
pdf Instead of 3.4c we do domain of dependence for the wave equation with lower order terms by the local energy method, as in class. Here is a handout: pdf.
7 12.2, 13.2 2.3c,d Distributions, fundamental solutions of PDE 2.3: 4, 8, 10, 11c pdf We skipped p. 70; it may be covered later, in connection with chapter 5
8 19.2, 20.2 4.1, 4.2 Laplace equation 4.1: 3, 5, 6, 7 pdf
9 and 10 26.2–9.3 "Tiltaksuker", no lectures. Special problem set: pdf.
11 12.3, 13.3 4.2 Laplace equation 4.2: 5, 6, 7, 8 pdf Section 4.2.f: skip Theorem 10.
Handout with proof of the Proposition in Section 4.2.b of McOwen: pdf.
12 19.3, 20.3 4.3, 4.4 Laplace equation 4.2: 10, 11
4.3: 2,3
pdf We skip Section 4.3.c
13 26.3, 27.3 ch. 5 Heat equation no problem session this week Skip 5.2.c.
In 5.2.d., skip Theorem 2.
In 5.3.a., skip the proof of the Theorem.
Section 5.4 we skip entirely.
14 and 15 2.4–14.4 Easter holidays
16 16.4, 17.4 ch. 6 Sobolev spaces,
Poincare inequality,
weak solutions of Dirichlet problem for elliptic equations
see lecture notes pdf Lecture notes: pdf
17 23.4, 24.4 ch. 6 Weak convergence and compactness results 6.1: 5
6.2: 4
6.3: 3, 7
pdf Lecture notes: pdf
18 30.4 7.1 Variational method no problems this week No lecture Tue May 1
19 7.5 7.1 See lecture notes pdf Lecture notes: pdf

#### Syllabus

This is the final syllabus:
• Chapter 1: 1.1a,b,c,d; 1.2 (all)
• Chapter 2: 2.1a,b (minus Example 2 on pp. 45-46); 2.2a,b,c; 2.3a,c,d
• Chapter 3: 3.1a,d; 3.2:a,b,c,d; 3.3a,b; 3.4a,b
• Chapter 4: 4.1b,c,d; 4.2a,b,c,d,e,f (in 4.2.f we skip Theorem 10); 4.3a,b,c; 4.4a,b
• Chapter 5: 5.1a,b; 5.2:a,b,d (in 5.2.d we skip Theorem 2); 5.3a,b (in 5.3.a we skip the proof of the Theorem)
• Chapter 6: Selected topics. See lecture notes: pdf#1 and pdf#2.
• Chapter 7: Selected topics from 7.1. See lecture notes: pdf