## Math 553. Partial Differential Equations Instructor Syllabus

### Text

Robert McOwen, *Partial Differential Equations: Methods and Applications*, Prentice-Hall, Inc. (2003)

#### Introduction (1 lecture)

General discussion of PDEs and background material, mainly through reading of pp 1-10.

#### Chapter 1 - First Order Equations (5 lectures)

- 1.1 Cauchy Problem for Quasilinear Equations (Characteristics, Semilinear Equations, Quasilinear Equations, General Solutions) (2)
- 1.2 Weak Solutions for Quasilinear Equations (Conservation Laws, Jump Conditions, Fans, Rarefaction Waves, Traffic Flow Problems) (2)
- Concluding Remarks on First-Order Equations (1) (1 variable version of section 2.1)

#### Chapter 2 - Principles for Higher Order Equations (11 lectures)

- 2.1 The Cauchy Problem (Normal Form, Power Series and the Cauchy-Kovaleski Thm, Definition of Well-Posed) (2)
- 2.2 Second Order Equations in Two Variables (Classification by Characteristics, Canonical Forms and General Solutions, First Order Systems) (3)
- 2.3 Linear Equations and Generalized Solutions
- Adjoints and Weak Solutions, Transmission Conditions (1)
- Distributions: Basic Concepts, Definitions and Examples (3) (based on Muncaster's notes)
- Distributions: Convolutions and Fundamental Solutions (2)

#### Solution Techniques for Bounded and UnBounded Domains (6 lectures)

- 4.4 Eigenvalues of the Laplacian (Eigenvalues and Eigenfunction Explansions, Application to the Wave Equation) (3)
- 5.2 Fourier Transforms (Basic Definitions and Properties, Examples, FTs of Distributions, Applications) (3)

#### Chapter 3 - The Wave Equation (7 lectures)

- 3.1 The One Dimensional Wave Equation (The Initial Value Problem, Weak Solutions, Initial/Boundary Value Problems, The Nonhomogeneous Equation) (2)
- 3.2 Higher Dimensions (Spherical Means, The Cauchy Problem, The Three-Dimensional Wave Equation, The Two-Dimensional Wave Equation, Huygens' Principle) (2)
- 3.3 Energy Methods (Conservation of Energy, The Domain of Dependence) (2)
- 3.4 Lower-Order Terms (Dispersion, Dissipation, The Domain of Dependence) (1)

#### Chapter 4 - The Laplace Equation (8 lectures)

- 4.1 Introduction to the Laplace Equation (Separation of Variables, Boundary Values and Physics, Green's Identities and Uniqueness, Mean Values and the Maximum Principle) (3)
- 4.2 Potential Theory and Green's Functions (The Fundamental Solution and Potentials, Green's Function and the Poisson Kernel, The Dirichlet Problem on a Half Space, The Dirichlet Problem on a Ball, Properties of Harmonic Functions) (4)
- 4.4 Eigenvalues of the Laplacian (Application to the Laplace's Equation) (1)

#### Chapter 5 - The Heat Equation (4 lectures)

- 5.1 The Heat Equation in a Bounded Domain (Existence by Eigenfunction Expansion, The Maximum Principle and Uniqueness) (1)
- 5.2 The Pure Initial Value Problem (Solution of the Pure Initial Value Problem, The Fundamental Solution, The Nonhomogeneous Equation) (2)
- 5.3 Regularity and Similarity (Smoothness of Solutions, Scale Invariance and the Similarity Method) (1)

#### Midterm Exam (1 lectures)

#### Total: 43