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Study Guide Introduction to Partial Differential Equations 2WA90 Luc Florack Updated June 1, 2016
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.
