Math 781-001, Advanced Methods in Applied Mathematics (Fall 2017)

Tuesday, 7:20pm - 10:00pm, Exploratory Hall, Room 4106

Instructor: Daniel M. Anderson (4411 Exploratory Hall, 703.993.1482,
Office Hours: Tuesday/Thursday 3:00PM-4:00PM, and by appointment.

Text: There is no required text for this course. Some informal course notes will be provided. Other recommended books are Boundary Value Problems of Mathematical Physics, Vol. 1 and 2 (SIAM Classics in Applied Mathematics) by Ivar Stakgold and Green's Functions and Boundary Value Problems by Stakgold and Holst (Wiley).

Prerequisites: Familiarity with ordinary and partial differential equations.

Course Description: This course will examine Green's function problems for ordinary differential equations and a variety of partial differential equations (elliptic, parabolic and hyperbolic type) as well as examine some basic types of Integral equations. Problems arising in sciences and engineering applications will be considered.

Grading Policy: The course grade will be based on homework (50%), a midterm exam (20%) and the final exam (30%).

Topics Covered/Schedule
Distribution Theory Basics
Green's functions for ODEs and Boundary Value Problems
Green's functions for PDEs (elliptic, parabolic, hyperbolic)
Integral equations (Fredholm, Volterra type)

FINAL EXAM: Tuesday, December 19, 7:30-10:15pm

Honor System: It is expected that each student in this class will conduct himself or herself within the guidelines of the Honor Code. All academic work should be done with the level of honesty and integrity that this University demands.

Schedule - Approximate: to be updated as needed

8/29: Distribution Theory and Green's functions for ODEs ODE Notes: [ODE Notes]

9/5: Distribution Theory and Green's functions for ODEs (continued) Homework 1: [Homework 1] (Due: Tuesday, September 12)

9/12: Green's functions for ODEs (Examples with 2nd Order BVPs) Homework 2: [Homework 2] (Due: Tuesday, September 19)

9/19: Green's functions for ODEs (Fredholm Alternative and Modified Green's functions) Homework 3: [Homework 3] (Due: Tuesday, September 26 )

9/26: Modified Green's functions: Non-self adjoint problems Homework 4: [Homework 4] (Due: Tuesday, October 3)

10/3: Green's Identities in 2D and Laplace's Equation Solutions PDE Notes: [PDE Notes] Homework 5: [Homework 5] (Due: Tuesday, October 17)


10/17: Free Space Green's Function Homework 6: [Homework 6] (Due: Tuesday, October 24)

10/24: Dirichlet Problem on a Circle Midterm Exam (Due: Tuesday, October 31)

10/31: Exterior Problems: no homework this week

11/7: Half-Space Green's Function Problems Homework 7: [Homework 7] (Due: Tuesday, November 14)

11/14: Half-Space Green's Function Problems, Heat Equation Homework 8: [Homework 8] (Due: Tuesday, November 28)

11/21: Start Reading Integral Equations PDE Notes: [Integral Equation Notes]

11/28: Integral Equations Homework 9: (Due: Tuesday, December 5)

12/5: Integral Equations

12/19: Final Exam

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