# Math 136: General Course Outline

## Course Description

136. Partial Differential Equations. Lecture, three hours; discussion,one hour. Prerequisites: courses 33A, 33B. Linear partial differential equations, boundary and initial value problems; wave equation, heat equation, and Laplace equation; separation of variables, eigenfunction expansions; selected topics, as method of characteristics for nonlinear equations.

General Information. Math 136 is offered once each year, in the Spring. Together with 135A in the Fall and 135B in the Winter, it is the third of a natural sequence of courses in differential equations. Note however that the courses 135AB are not required for 136.

Enrollments in Math 136 have oscillated between 30 and 100 over the past several years.

## Textbook(s)

W.A. Strauss, Partial Differential Equations, 2nd Edition, John Wiley and Sons.
The course covers Chapters 1, 2, parts of 3, and most of 4-6.

## Schedule of Lectures

Lecture Section Topics

1

1.1-1.2

The notion of a partial differential equation (PDE), the order of a PDE, linear PDE, examples. First order linear PDE.

2-3

1.2

Homogeneous first order linear PDE with constant coefficients. The method of characteristics (geometric method) and the coordinate method. First order linear PDE with variable cofficients. Characteristic curves and the geometric method in the case of variable cofficients. The solvability of the Cauchy problem for a first order linear PDE (the statement only).

4

1.3

PDE from Physics. Examples: the heat equation (derivation using Fourier's law), vibrating strings and drumheads, the wave equation and the Laplace equation. Schrodinger's equation.

5

1.4, 1.6

Initial and boundary conditions for PDE. Classification of second order linear PDE with constant coefficients. Elliptic and hyperbolic PDE.

6-7

2.1

The wave equation on the real line.Traveling waves. The Cauchy problem for the wave equation and the d'Alembert formula. Examples.

8

2.2

The causality principle for the wave equation. The domain of dependence and the domain of influence. Conservation of energy.

9-10

2.3-2.5

The diffusion/heat equation on the real line. The maximum principle and the uniqueness of the Dirichlet problem for the heat equation. The heat kernel and the solution of the initial value problem for the heat equation on the real line. The smoothing property of the heat flow and the comparison of the main properties of the wave and heat equations.

11

3.1

The heat equation on the half-line. The Dirichlet and Neumann boundary conditions. The method of reflections.

12

3.2

The wave equation on the half-line. Reflected waves. (The first part of Section 3.2).

13-14

3.3, 3.4

The inhomogeneous heat equation on the real line. The inhomogeneous wave equation on the real line and the operator method. Duhamel's principle. (Section 3.4: the proof of Theorem 1 using the operator method)

15

Review before the midterm.

16

Midterm.

17

4.1

Spectral methods for boundary problems on finite intervals. Separation of variables and the wave equation with Dirichlet boundary conditions. The eigenvalues and eigenfunctions on a bounded interval with Dirichlet boundary conditions. The heat equation with Dirichlet boundary conditions. Formal eigenfunction expansions.

18

4.2

The Neumann boundary conditions for the wave and the heat equations. The eigenvalues and eigenfunctions of on a bounded interval with Neumann boundary conditions.

19

4.3

The eigenvalues and eigenfuctions on a bounded interval with Robin boundary conditions: a cursory discussion.

20-21

5.1-5.2

Fourier series and Fourier coefiicients of periodic functions in real and complex form. Fourier series expansions for functions defined on an interval of the form via even and odd extensions. Since and cosine expansions. Examples.

22-24

5.3-5.4

Symmetric boundary conditions and the orthogonality of eigenfunctions. Convergence theorems for Fourier series, the notions of uniform and L^2-convergence. The least square approximation, Bessel's inequality, and Parseval's identity. One word about the pointwise convergence of Fourier series.

25-26

6.1

The Laplace equation and harmonic functions. The maximum principle and the uniqueness of the Dirichlet problem. The Laplace operator in polar coordinates and the Newtonian potential in 2D and 3D.

27

6.2

The Laplace equation and separation of variables in a rectangle. (Section 6.2, may be omitted due to time constraints).

28-29

6.3

The Dirichlet problem in the disc and Poisson's formula. The mean value property for harmonic functions and their differentiability properties.

30

Review.