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Study Guide
Introduction to Partial Differential Equations2WA90
Luc Florack
Updated June 1, 2016
Recommendations for Exam Preparation
2WA90 credit: 5ECTS ∼ 140 hours, divided over 8 weeks:
8× 4 = 32 oral classes (attendance)8× 2 = 16 supervised learning (attendance)8× 10 = 80 homework study (theory and preparation for supervised learning)
12 homework study (interim and final exam preparation)
140 (∼ 5ECTS)
Supervised learning: Problem Companion: 1–51.
2
Preliminaries
April 18 2016:
• Notational conventions and definitions of number fields, function classes, multi-indices.
• Definition of an oriented region with boundary.
• Stokes’ Theorem, notably the classical forms given in the Corollary.
4
1. Introduction
April 18 2016:
• Section 1.1: Partial Differential Equations and Boundary Conditions.
• Section 1.2: Examples.
• Section 1.3: Issues.
• Section 1.4: Ill-Posed versus Well-Posed Problems.
• Section 1.5: Generalised Solutions.
April 20 2016:
• Section 1.6: Classification of Partial Differential Equations.
6 Introduction
2. Existence and Uniqueness
April 20 2016:
• Cauchy-Kowalewska.
8 Existence and Uniqueness
3. Calculus of Variations
April 20 2016:
• Section 3.2: Basic Technique.
• Section 3.3: Examples.
10 Calculus of Variations
4. Distribution Theory
April 25 2016:
• Section 4.1: Motivation.
• Section 4.2: Distributions Formalised.
May 2 2016:
• Section 4.3: Examples.
12 Distribution Theory
5. Fourier Transformation
May 2 2016:
• Section 5.1: Introduction.
• Section 5.2: The Fourier Transform on S (Rn).
• Section 5.3: The Fourier Transform on S ′(Rn).
• Section 5.4: The Fourier Transform on L2(Rn).
May 4 2016:
• Section 5.5: Fourier Theorems.
14 Fourier Transformation
6. Complex Analysis
16 Complex Analysis
7. The Fourier Method
May 4 2016:
• Section 7.1: Basic Technique.
• Section 7.2: Examples.
18 The Fourier Method
8. The Method of Characteristics
May 18 2016:
• Section 8.1: Basic Technique.
• Section 8.2: Examples.
20 The Method of Characteristics
9. Separation of Variables
May 9 2016:
• Section 9.1: Basic Technique.
• Section 9.2: Examples.
May 18 2016:
• Section 9.1: Basic Technique.
• Section 9.2: Examples.
22 Separation of Variables
10. Fundamental Solutions & Green’s Functions
May 23 2016:
• Section 10.1: Basic Technique.
• Section 10.2: Construction.
May 25 2016:
• Section 10.2: Construction.
24 Fundamental Solutions & Green’s Functions
11. First Order Systems
May 25 2016:
• Section 11.2: The Dirac Equation.
26 First Order Systems
12. Second Order Hyperbolic Systems
May 30 2016:
• Section 12.1: The Wave Equation.
• Section 12.2: Examples: n = 1 and n = 3.
June 1 2016:
• Section 12.2: Examples: n = 2 and n = 3.
• Formal substitution to convert (solutions of) the (1+1)-dimensional wave equation into (solutions of) the(1+1)-dimensional Laplace equation. Harmonic functions.