Math 436/536
Partial Differential Equations
Spring 2016

Course Info

Overview

We will cover quasi-linear first order equations, hyperbolic, parabolic, and elliptic equations and their applications. Topics will include method of characteristics, separation of variables, Fourier series, orthogonal functions, Sturm-Liouville theory, special functions, Green’s functions, and Fourier and Laplace transforms.

Please see the syllabus for further details about the course.

Text Book

Tyn Myint U and Lokenath Debnath, Linear Partial Differential Equations for Scientists and Engineers, Fourth Edition, Birkhäuser 2006

Software

I will use Mathematica throughout the course to demonstrate some key concepts. You will also be asked to use this or similar software, such as WolframAlpha, Maple, MATLAB (through the symbolic toolbox), or Chebfun, on the homework assignments. The purpose here is to help you build a qualitative understanding of partial differential equations and to aid you with some tedious computations for the homework problems. All of this software is available in most computing labs around the university, including the mathematics computing lab. WolframAlpha is available through most web browsers and there is even an app for it for your phones or tablets.

Lectures

Topics that will be covered in a particular lecture will appear a day or two before the lecture. I expect you to have read the relavent sections in the book before the lecture and to be prepared to answer questions about the material.
Day Topics Notes
13-Jan-2016 Section 1.2: Basic concepts and definitions
Section 1.3: Mathematical problems
Section 1.4: Linear operators
Please also read section 1.1
15-Jan-2016 Section 1.5: Superposition principle
Section 2.2: Classification of first-order equations
Section 2.5: Construction of a first-order equation
20-Jan-2016 Section 2.4: Geometrical interpretation of a first order equation
Section 2.5: Construction of a first-order equation
Section 2.6: Canonical forms of first-order linear equations
22-Jan-2016 Section 2.6: Canonical forms of first-order linear equations
Section 2.7: Method of separation of variables
27-Jan-2016 First-order non-linear PDEs and shocks
29-Jan-2016 First-order non-linear PDEs and shocks
Section 3.1: Classical equations
3-Feb-2016 Section 3.2: Wave equation
Section 3.5: Heat (diffusion) equation
5-Feb-2016 Section 3.5: Heat (diffusion) equation (continued)
Common boundary conditions
Section 4.1: Second order equations in two independent variables
Section 4.2: Canonical forms
10-Feb-2016 Section 4.2: Canonical forms
Section 4.3: Equations with constant coefficients
12-Feb-2016 Section 4.4: General solutions
Section 4.5: Summary and further simplifications
Section 5.3: Cauchy problem for the homogeneous wave equation
D'Alembert solution example
17-Feb-2016 No class. Instead go to the Research computing days talks.
19-Feb-2016 Section 5.3: Cauchy problem for the homogeneous wave equation
Section 5.4: Initial and boundary-value problems
D'Alembert solution on semi-infinite domains example
26-Feb-2016 Section 6.1-6.7, 6.11: Fourier series
02-Mar-2016 Section 6.7, 6.8: Fourier series Fourier sine series example
04-Mar-2016 Section 6.13: Fourier transform
Section 7.1-7.3: Vibrating string and separation of variables
Fourier series and transforms in Mathematica
09-Mar-2016 Section 7.1-7.3: Vibrating string and separation of variables
11-Mar-2016 Midterm exam
16-Mar-2016 Section 7.1-7.4: Existence and uniqueness of the wave equation Wave equation demo in Mathematica
30-Mar-2016 Section 7.5-7.6: Heat conduction problem
Section 7.8: Non-homogeneous problems
Heat equation demo in Mathematica
01-Apr-2016 Section 7.8: Non-homogeneous problems
Example and plot of solution
06-Apr-2016 Section 8.1-8.2: Sturm-Liouville problems
13-Apr-2016 Section 8.10-8.12: Green's function for Sturm-Liouville problems Green's function example
15-Apr-2016 Section 8.12: Constructing the Green's function for a Sturm-Liouville problem
Section 9.1: Boundary value problems: Laplace's and Poisson's equation
Section 9.7-9.8: Solving Poisson's equation on a rectangle
Green's function example
20-Apr-2016 Section 9.7: Solving Poisson's equation on a rectangle
22-Apr-2016 Section 12.1-12.4: Fourier transforms
27-Apr-2016 Section 12.5: Fourier transforms
Section 12.6: Fourier sine and cosine transforms
Section 12.8: Laplace transforms
Understanding convolution
Heat equation: Fourier transform solution
Heat equation: Fourier sine transform solution

Homework

Your submitted homework should show all necessary work you used to solve the problems. Mathematical statements should be complete (or nearly complete) sentences and the reasoning and logic underlying all arguments should be clearly spelled out. Any work in Mathematica (or other software) should be noted and included in a readable format (e.g., only the necessary work with concise codes). Do not include pages and pages of printouts from the computer of results that are not important to the problem at hand. Homework assignments will involve a mix of analytical and computational work.

Format

The ideal way to write up your homework assignments is to use LaTeX, but legible, hand-written assignments are also acceptable.

You are also encouraged to check out these suggestions from the Mathematics Department at Harvey Mudd on formatting homework assignments in mathematics.

Assignments

Due date Problem set Notes
22-JAN-2016 HW01  
05-FEB-2016 HW02 Select solutions
26-FEB-2016 HW03 Solutions
9-MAR-2016 HW04  
18-MAR-2016 HW05 For plotting help see WaveEquationEx.nb (Mathematica) or PlottingWaveEquation.m (Matlab). Solutions
08-APR-2016 HW06 Solutions
02-MAY-2016 HW07 (This will count as extra credit) Solutions

References

The following are some other good references for the material covered in this course:

Please e-mail me regarding any problems with the links on this page.