Tuesdays &
Thursdays 1:00-2:15pm, B211 Van Vleck Hall
Lecturer: Gheorghe Craciun
Office: 405 Van Vleck
Tel.
(608)265-3391
E-mail: craciun at
math dot
wisc dot
edu
Office hours: TR 2:15-3:00pm.
Textbook: J. Kevorkian, “Partial
Differential Equations: Analytical Solution Techniques” (Texts in Applied
Mathematics #35), Second Edition.
Background and
Goals: Study
of partial differential equations from classical physics, including parabolic
equations (diffusion equation), elliptic equations (Laplace equation), and
hyperbolic equations (wave equation). We will also study nonlinear hyperbolic
equations, including shock waves in traffic flow and gas dynamic models. The
emphasis will be on solution techniques such as self-similarity, transform
methods, and Green's function representations of solutions.
Course Topics:
1. Diffusion
equation (derivation, fundamental solution, Green’s functions, infinite,
semi-infinite, and finite domains, maximum principle)
2. Laplace
equation (conformal maps, fundamental solution, dipole potential, Green’s
formula, maximum principle)
3. Wave
equation (derivation, method of characteristics, infinite, semi-infinite, and
finite domains)
4. Scalar
hyperbolic conservation laws (method of characteristics, shocks, rarefactions,
entropy solutions)
5. Quasilinear
hyperbolic systems (the Riemann problem, shallow water waves, Riemann
invariants, gas dynamics)
6. Time
permitting: approximate solutions by perturbation methods
Grading: The grade will be based
on homework and exams as follows: Homework and Class Participation 33%, Midterm
Exam 33%, Final Exam 33%.
Homework: Homework will be due on
some Thursdays (as listed below), before class starts. Discussion with fellow
students is allowed on homework assignments; however, every student must write
his/her own assignment. Please use separate sheets of paper for each homework
problem, and label them clearly with the homework number and the problem
number. No late homework will be accepted.
Homework#1 (due Thursday
Feb 9)
1. Write a short summary of
the derivation of the heat equation in section 1.1 in the textbook (including
the three-dimensional case at the end of the section).
2. Use the definition from
Appendix A.1.1 to explain formula (1.2.12) on page 7.
3. True or false: for any
function F(x,t) that satisfies the equation (1.2.15) there must exist some
function f(z) as discussed on page 8.
4. Solve
problem 1.2.1 on page 12.
5. Solve problem 1.2.2 on
page 12.
Homework#2 (due Thursday
March 22)
Choose 5 of the
following 6 problems:
1. Write a short summary of
the derivation of Laplace’s equation in section 2.1 in the textbook.
2. Solve problem 1.5.1 on
page 41.
3. Solve problem 1.5.2 ,
parts (a)-(c) on page 42.
4. Solve
problem 2.3.2 on page 88.
5. Solve problem
2.6.2(a)-(b) on page 126.
6. Solve problem 2.6.5(a)
on page 127.
The Midterm exam will be
on Tuesday March 27 during class.
Review topics for
midterm exam:
1. Diffusion equation on
infinite domain: finding the fundamental solution using similarity structure
(pages 6-9), solution of inhomogenous problems by superposition (pages 13-15).
2. Diffusion equation on
semi-infinite domain: finding Green’s function using the method of images
(pages 16-18).
3. Diffusion equation on
finite domain: finding Green’s function using the method of images (pages
32-34).
4. Laplace’s equation:
fundamental solution in 3D (pages 75-76) and in 2D (pages 76-78).
5. Laplace’s equation:
Green’s formula, Gauss Integral Theorem, Energy Theorem, Uniqueness Theorems,
Mean Value Theorem (pages 106-110).
6. Laplace’s equation:
Green’s functions on upper-half plane and upper-half space (pages 117-118) and
Green’s functions in the interior or exterior of a circle or a sphere (pages
121-122).
7. Wave equation on
infinite domain: finding fundamental solution using a change of variables
(pages 173-175).
Take-home final exam:
solve 3 of the 6 problems listed below (due date is May 19):
1. Problem 3.6.3, page
209.
2. Problem 3.9.1(a),
page 239.
3. Problem 5.2.4, page 326.
4. Problem 5.2.5(a)&(b),
page 326.
5. Problem 5.3.1, page
361.
6. Problem 5.3.2, page
362.
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