Welcome to the class website for Partial Differential Equations (PDEs). The theory of PDEs provides an important mathematical foundation for studying a wide variety of phenomena arising in the physical sciences, finance, and engineering. This class will discuss both fundamental models based on PDEs and mathematical techniques for their solution and 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 Identities
- Energy Methods
- Maximum Principle
- Harmonic Functions
- Green's Functions
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 exam as follows:
Homework Assignments 30%
Midterm Exam 30%
Final Exam 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 Tuesday, February 5th.
The final exam will be given according to the [registrar exam information here.]
Midterm Exam Outline:
- Fourier Series
- Determining the coefficients of the Fourier Series when expanded in cosine and sine functions
- Determining the coefficients of the Fourier Series when expanded using the complex exponential
- Definition of L2-inner product of two functions
- Meaning of the statement that two functions are orthogonal (condition this requires when using the L2-inner product).
- Use of orthogonality to determine the coefficients in a series expansion
- Elliptic PDEs (Laplace's Equation)
- Solution of Laplace's equation on a rectangle
- Solution with homogeneous and inhomogeneous Dirichlet boundary conditions
- Solution with homogeneous and inhomogeneous Neumann boundary conditions
- Maximum principle
- Uniqueness of solutions
Final Exam Outline:
- Elliptic Equations
- Laplace Equation
- Solution on rectangle using Fourier methods.
- Solution using Green's functions.
- Harmonic Functions
- Mean value property.
- Representation formula for arbitrary domain D.
- Green's functions
- Green's first identity.
- Green's second identity.
- Laplace equation with Dirichlet boundary data.
- Method of images.
- Laplace equation on half-plane.
- Laplace equation on disk.
- Hyperbolic Equations
- Wave equation in R^3 (Kirchoff formula).
- Wave equation in R^2.
- Wave equation with a source.
- Parabolic Equations
- Diffusion equation in R^2 and R^3.
- Diffusion equation on rectangle using Fourier methods.
- Fourier Series
- Determining the coefficients of the Fourier Series (trigonometric and complex exponential forms).
- L2-inner product of two functions.
- Orthogonality.
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:
- Jon Lo Kim Lin is TA for the class with office hour Mondays 4pm - 5pm, 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 Mondays 4pm-5pm, Graduate Tower, Office 6431W. Solution keys for the homework were prepared by Jon Lo Kim Lin.
HW1: (Due Thurs, Jan 17th) 5.1: 1, 2, 5; 5.2: 1, 5, 6, 10, 15, 16; 5.3: 2, 3, 10. Solution Key. Δ
HW2: (Due Tues, Jan 29th) 5.3: 1, 4, 7, 9, 12; 6.1: 12, 13. Solution Key. Δ
HW3: (Due Thurs, Jan 31st) 6.1: 3, 4, 5, 7, 9, 10; 6.2: 1, 4. Solution Key. Δ
HW4: (Due Thurs, Feb 7th) 6.3: 1, 2; 6.4: 1, 5, 7, 8, 13. Solution Key. Δ
HW5: (Due Tues, Feb 19th) 7.1: 1, 2, 5, 6, 7. Solution Key. Δ
HW6: (Due Thurs, Feb 21th) 7.2: 1, 2, 3. Solution Key. Δ
HW7: (Due Thurs, Feb 28th) 7.3: 1, 2; 7.4: 1, 3, 6, 11, 13, 21. Solution Key. Δ
HW8: (Due Thurs, Mar 7th) 9.2: 1, 3, 6, 8, 10, 14, 20. Solution Key. Δ
HW9: (Due Thurs, Mar 12th) (optional) 9.3: 5, 6, 8; 10.1: 1, 2, 3.
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