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, February 25th.

A final exam will be given on Friday, March 19th, 8:00am - 11:00am in GIRV 2108.

A final exam review session is schedule for Wednesday, March 17th, 5:00pm - 6:00pm, location South Hall 4607.

Final Exam Outline:

- Method Characteristics (Advection Equation / Wave Equation)
- Solution of Wave Equation (Hyperbolic PDEs) on R.
- Solution of Wave Equation (Hyperbolic PDEs) with Dirichlet and Neumann Boundary Conditions.
- Solution of the Diffusion Equation (Parabolic PDEs) on R.
- Solution of the Diffusion Equation (Parabolic PDEs) with Dirichlet and Neumann Boundary Conditions.
- Maximum principle for the Diffusion Equation (Parabolic PDEs).
- Energy principles for Wave Equation and Diffusion Equation.
- Proof of uniqueness using energy principles.
- Steady-state diffusion equation (Elliptic PDE) on R^2.
- Green's First Identity.
- Separation of variables for Diffusion Equation and Wave Equation (Homogeneous Dirichlet and Neumann Boundary Conditions).


FINAL EXAM SOLUTIONS [PDF] Δ

Supplemental Class Notes:

(none)

Class Annoucements:

- The grader Jon Karl Sigurdsson has hours in the MATHLAB (ground floor South Hall 1607) from 3:00pm - 5:00pm on Fridays.

Homework Assignments:

Turn all homeworks into the graders mailbox (Jon Karl Sigurdsson) in South Hall 6th Floor by 5pm on the due date. Graded homeworks will be returned in class.

HW1: (Due Tue, Jan 12th) 1.1: 2ace, 3acdg, 4, 5ace, 10, 12; 1.2: 1, 4, 6, 8, 9.
HW2: (Due Tue, Jan 19th) 1.2: 2,3, 7, 8, 10; 1.3: 1,2.
HW3: (Due Tue, Jan 26th) 2.1: 1, 3, 4, 5, 8; 2.2: 1, 5, 6.
HW4: (Due Tue, Feb 2nd) 2.2: 2, 3, 4; 3.2: 1, 3, 7, 9, 11.
HW5: (Due Tue, Feb 9th) 2.3: 1, 3, 5, 6, 7; 2.4: 1, 4, 6, 7, 15.
HW6: (Due Thurs, Feb 18th) 3.1: 1, 3, 4; 3.3: 1, 2, 3; 3.5: 1, 2.
HW7: (Due Thurs, Mar. 4th) 4.1: 2, 4, 6; 6.1: 3, 4, 7, 9, 10; 7.1: 1, 4, 5, 6.

FINAL EXAM REVIEW SESSION: Wed, March 17th, 5:00pm - 6:00pm, location TBA (check-website).

FINAL EXAM SOLUTIONS [PDF] Δ.

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