MATH 4335-6 - PARTIAL DIFFERENTIAL EQUATIONS

University of Houston
Department of Mathematics
MATH 4335-6 - Partial Differential Equations

Prerequisites: Math 2433 and either Math 3321 or Math 3331.

Course Description: Partial differential equations and boundary value problems, Fourier series, the heat equation, vibrations of continuous systems, the potential equation, spectral methods.

Text: Partial Differential Equations, First Edition, by Walter Strauss, John Wiley & Sons, Inc. Pub.

Suggested Syllabus

4335

Chapter 1: Where PDEs come from

1.1 What is a Partial Differential Equation?
1.2 First-Order Linear Equations
1.3 Flows, Vibrations, and Diffusions
1.4 Initial and Boundary Conditions
1.5 Well-Posed Problems

Chapter 2: Waves and Diffusions

2.1 The Wave Equation
2.2 Causality and Energy
2.3 The Diffusion Equation
2.4 Diffusion on the Whole Line
2.5 Comparison of Waves and Diffusions

Chapter 3: Reflections and Sources

3.1 Diffusion on the Half-Line
3.2 Reflections of Waves
3.3 Diffusion with a Source
3.4 Waves with a Source
3.5 Diffusion Revisited

Chapter 4: Boundary Problems

4.1 Separation of Variables, the Dirichlet Condition
4.2 The Neumann Condition
4.3 The Robin Condition

Chapter 5: Fourier Series

5.1 The Coefficients
5.2 Even, Odd, Periodic, and Complex Functions
5.3 Orthogonality and General Fourier Series
5.4 Completeness
5.5 Completeness and the Gibbs Phenomenon

4336

Chapter 6: Harmonic Functions

6.1 Laplace's Equation
6.2 Rectangles and Cubes
6.3 Poisson's Formula

Chapter 7: Green's Identities and Green's Functions

7.1 Green's First Identity
7.2 Green's Second Identity
7.3 Green's Functions
7.4 Half-Space and Sphere

Chapter 9: Waves in Space

9.1 Energy and Causality
9.2 The Wave Equation in Space-Time
9.3 Rays, Singularities, and Sources

Chapter 10: Boundaries in the Plane and in Space

10.1 Fourier's Method, Revisited
10.2 Vibrations of a Drumhead
10.3 Solid Vibrations in a Ball

Chapter 11: General Eigenvalue Problems

11.1 The Eigenvalues Are Minima of the Potential Energy
11.2 Computation of Eigenvalues
11.3 Completeness
11.4 Symmetric Differential Operators
11.5 Completeness and Separation of Variables
11.6 Asymptotics of the Eigenvalues

Syllabus by David Wagner

1.1 What is a Partial Differential Equation? 1.2 First-Order Linear Equations 1.3 Flows, Vibrations, and Diffusions 1.4 Initial and Boundary Conditions 1.5 Well-Posed Problems

2.1 The Wave Equation 2.2 Causality and Energy 2.3 The Diffusion Equation 2.4 Diffusion on the Whole Line 2.5 Comparison of Waves and Diffusions

3.1 Diffusion on the Half-Line 3.2 Reflections of Waves 3.3 Diffusion with a Source 3.4 Waves with a Source 3.5 Diffusion Revisited

4.1 Separation of Variables, the Dirichlet Condition 4.2 The Neumann Condition 4.3 The Robin Condition

5.1 The Coefficients 5.2 Even, Odd, Periodic, and Complex Functions 5.3 Orthogonality and General Fourier Series 5.4 Completeness 5.5 Completeness and the Gibbs Phenomenon

6.1 Laplace's Equation 6.2 Rectangles and Cubes 6.3 Poisson's Formula

7.1 Green's First Identity 7.2 Green's Second Identity 7.3 Green's Functions 7.4 Half-Space and Sphere

9.1 Energy and Causality 9.2 The Wave Equation in Space-Time 9.3 Rays, Singularities, and Sources

10.1 Fourier's Method, Revisited 10.2 Vibrations of a Drumhead 10.3 Solid Vibrations in a Ball

11.1 The Eigenvalues Are Minima of the Potential Energy 11.2 Computation of Eigenvalues 11.3 Completeness 11.4 Symmetric Differential Operators 11.5 Completeness and Separation of Variables 11.6 Asymptotics of the Eigenvalues