Math 553 Partial Differential Equations

Professor Richard S. Laugesen, Fall 2007

Syllabus

Text: Robert McOwen, Partial Differential Equations: Methods and Applications, second edition, Prentice-Hall. On reserve at the Mathematics Library.

Some parts of the course will follow L. C. Evans Partial Differential Equations, American Mathematical Society. This book is also on reserve at the Mathematics Library, and you need not purchase it. Note: Evans' book is the text for Math 554.

Introduction (1 lecture)

General discussion of PDEs and background material; read pp. 7-10 for the prerequisites, and pp. 1-6 for a survey of famous PDEs and concepts.

Chapter 1 - First-Order Equations (6 lectures)

• 1.1 Cauchy Problem for Quasilinear Equations (Characteristics, Semilinear Equations, Quasilinear Equations, General Solutions) (2.5)
• 1.2ab Weak Solutions for Quasilinear Equations (Conservation Laws and Jump Conditions, Fans and Rarefaction Waves) [including EVANS 3.4.1b, Entropy Condition] (1.5)
• EVANS 3.4.3b and 3.4.4 (results only), 3.4.1b continued, 3.4.5 (results only) (Riemann Problem, Asymptotics in L^infty and L¹ norms) (1.4)
• 1.2c (Traffic Flow Problems) (0.3)
• 1.3 Fully Nonlinear Equations (0.3)

Chapter 2 - Principles for Higher Order Equations (4 lectures)

• 2.1 The Cauchy Problem (Normal Form, Power Series and the Cauchy-Kovalevski Theorem, Definition of "Well Posed") (1)
• 2.2 Second Order Equations in Two Variables (Classification by Characteristics, Canonical Forms and General Solutions; omit: First Order Systems, The Telegraph System) (1)
• 2.3 Linear Equations and Generalized Solutions (2.3ab: Adjoints and Weak Solutions, Transmission Conditions) (2)

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) (3)
• 3.2 Higher Dimensions (Spherical Means, Cauchy Problem, Three-Dimensional Wave Equation, Two-Dimensional Wave Equation, Huygens' Principle) (2)
• 3.3 Energy Methods (Conservation of Energy, Domain of Dependence) (1)
• 3.4 Lower-Order Terms (Dispersion, Dissipation, Domain of Dependence) (1)

Chapter 4 - The Laplace Equation (10 lectures)

• 4.1 Introduction to the Laplace Equation (Boundary Values and Physics, Green's Identities and Uniqueness, Mean Values and the Maximum Principle) (2.5)
• 2.3c Distributions and Fundamental Solutions (1)
• 4.2a Potential Theory and Green's Functions (Fundamental Solution and Potentials, including EVANS 2.2.1b (Poisson's Equation)) (2)
• 4.2bcde Potential Theory and Green's Functions (Green's Function and the Poisson Kernel, Dirichlet Problem on a Half Space, Dirichlet Problem on a Ball, Properties of Harmonic Functions) (3)
• EVANS 2.2.5 Energy Methods (Dirichlet's Principle) (0.5)
• 4.4 Eigenvalues of the Laplacian (Eigenvalues and Eigenfunction Explansions, Application to the Wave Equation) (1)

Chapter 5 - The Heat Equation (7 lectures)

• 5.1 The Heat Equation in a Bounded Domain (Existence by Eigenfunction Expansion, Maximum Principle and Uniqueness) (1)
• 5.2 The Pure Initial Value Problem (Fourier Transform, Solution of the Pure Initial Value Problem, Fundamental Solution, Nonhomogeneous Equation) (3)
• 5.3a Regularity (Smoothness of Solutions), EVANS 2.3.2 Mean-value Formula, 2.3.3a Strong maximum principle, 2.3.4 Energy Methods (Uniqueness, Backwards Uniqueness), MCOWEN 5.3b Similarity (Scale Invariance and the Similarity Method) (3)

EVANS Chapter 4 - Other Ways to Represent Solutions (3 lectures)

• EVANS 4.2.2 and 4.2.1b Similarity Solutions (Traveling Waves, Solitons, Similarity Under Scaling) (1)
• EVANS 4.3.1b Fourier Transform Revisited (Schrodinger equation) (1)
• EVANS 4.4.1 and 4.5.2 (Burger's equation with viscosity) (1)

MATLAB Period (1 lectures)

Leeway (2 lecture)

Test and Debriefing (2 lectures)

Total: 43