Introduction to Partial Differential Equations
Professor: Paul J. Atzberger
124A Fall 2012 in NH 1105
TR 3:30pm - 4:45pm

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Welcome to the class website for Partial Differential Equations (PDEs). The theory of PDEs provides an important mathematical approach for studying a wide variety of phenomena arising in the physical sciences, engineering, and finance. This class will discuss both fundamental models based on PDEs and mathematical techniques for their study. For more details see the syllabus and the topics listed below.

Please be sure to read the prerequisites and grading policies for the class.

Selection of Topics

  • Methods of Solution for Parabolic, Hyperbolic, and Elliptic PDE's
  • Separation of Variables
  • Fourier Transform
  • Fourier Series
  • Poisson's Formula
  • Green's Functions
  • Linear Stability Analysis
  • Introduction to Finite Element Method for PDE's

Prerequisites:

Calculus I, II, Ordinary Differential Equations, and Linear Algebra.

Grading:

The grade for the class will be based on the homework assignments (see policy below), midterm exam, and final project as follows:

Homework Assignments 30%
Midterm Exam 30%
Final Project 40%

Homework Policy:

Assignments will be made weekly and posted on the class website. Prompt submission of the homework assignments is required. While no late homework submissions will be accepted, one missed assignment will be allowed without penalty. While it is permissible and you are encouraged to discuss materials with classmates, the submitted homework must be your own work. The assignments will consist of a combination of analytic problems and numerical simulations. Some basic programming in Matlab/Octave may be required.

Exams:

A midterm exam will be given in the class on Thursday, November 1st.

The final exam will be given on Thursday, December 13th in NH 1105 from 4:00pm - 7:00pm. [based on registrar exam information here]

Final Exam Outline:

  • Practice Exam Δ
  • Solutions Δ
  • First Order Constant Coefficient PDEs in Two Variables
    • Method of characteristic for solutions
    • Initial value problem with data either on x-axis or y-axis
    • Uniqueness of solutions
  • First Order Variable Coefficient PDEs in Two Variables
    • Method of characteristic for solutions
    • Initial value problem with data either on x-axis or y-axis
    • Uniqueness of solutions
  • Classification of linear second order constant coefficient PDEs in two variables (Parabolic, Hyperbolic, Elliptic)
  • Wave Equation
    • Solution method using method of characteristics
    • Initial value problems with data for initial configuration and/or initial velocity
    • Domain of dependence of solution u at location (x,t)
    • Domain of influence of data at location (x,0)
    • Energy associated with wave equation
    • Uniqueness of solutions
  • Diffusion Equation
    • Solution representation using the heat kernel
    • Initial value problems with data for initial concentration
    • Energy associated with diffusion equation
    • Uniqueness of solutions
  • Solutions beyond the real-line for Diffusion/Wave Equations
    • Neumann/Dirichlet boundary conditions for the half-line
    • Inhomogeneous right-hand side
    • Non-zero Neumann and Dirichlet boundary conditions.
    • Solution on an interval

A final exam will date will be announced near the end of the quarter in accordance with the university exam schedule.

Supplemental Class Notes:

(none posted at this time)

Class Annoucements:

  • Special office hours for Jon Lo Kim for questions in preparation for the final from 10am - 12pm on Tuesday, Graduate Tower, Office 6431W.

Homework Assignments:

Turn all homeworks into the TA mailbox (Jon Lo Kim Lin) in South Hall 6th Floor by 5pm on the due date. Graded homeworks will be returned in class. TA office hours Tuesdays 9:30am-10:30am, Graduate Tower, Office 6431W. Solution keys for the homework were also prepared by Jon Lo Kim Lin.

HW1: (Due Thurs, Oct 4th) 1.1: 2ace, 3acdg, 4, 5ace, 10, 12; 1.2: 1, 4, 6, 8, 9. Solution Key Δ
HW2: (Due Tue, Oct 16th) 1.2: 2,3, 7, 8, 10; 1.3: 1,2. Solution Key Δ
HW3: (Due Tue, Oct 23th) 2.1: 1, 3, 4, 5, 8; 2.2: 1, 5, 6. Solution Key Δ
HW4: (Due Thurs, Nov 8th) 2.2: 2, 3, 4; 2.3: 1, 3, 7. Solution Key Δ
HW5: (Due Thurs, Nov 22nd) 3.1: 1, 3, 4; 3.2: 1, 2, 3, 4; 3.3: 1, 2, 3. Solution Key Δ
HW6: (Due Thurs, Nov 29th) 3.4: 1, 2, 4, 6, 9, 15; Solution Key Δ
HW7: (optional) (Due Thurs, Dec 13th) 4.1: 1, 4; 4.2: 1, 2; 5.1: 2, 9; 6.3: 2, 3.


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