PDE II

AMCS 331. Applied Partial Differential Equations II, Spring 2011, KAUST
Instructor: Aslan Kasimov
Lectures:  9:00-10:30 every Monday and Wednesday in Room 9-3224

Prereqs: Multivariate calculus, elementary complex variables, ordinary differential equations.
Recommended: AMCS 231 or AMCS 201. 
Second part of a sequence of courses on partial differential equations (PDE) emphasizing theory and solution techniques for nonlinear equations. Quasi-linear and nonlinear PDE in applications. Conservation laws, first-order equations, the method of characteristics. Burgers’ equation and wave breaking.  Weak solutions, shocks, jump conditions, and entropy conditions. Hyperbolic systems of gas dynamics, shallow-water flow, traffic flow, and bio-fluid flow. Variational principles, dispersive waves, solitons. Nonlinear diffusion and reaction-diffusion equations in combustion and biology. Traveling waves and their stability. Dimensional analysis and similarity solutions. Perturbation methods. Turing instability and pattern formation. Eigenvalue problems. Stability and bifurcation.

Texts:

  1. D. Logan, Introduction to nonlinear PDE
  2. L. Debnath, Nonlinear PDE for scientists and engineers
  3. L. Evans, Partial Differential Equations
  4. S. Smoller, Shock waves and reaction-diffusion equations
  5. G. Whitham, Linear and nonlinear waves

The final grade will be based on homework (~30%), one midterm exam (~30%), and a project paper (~40%). There will be no final exam. The project paper is a crucial part of the course. Students will be expected to write a 10-15 page paper and give an end-term presentation on a topic selected from the following list or one proposed by the student.

Sample topics: Nonlinear water waves, shock stability, continuum models of traffic flow, non-classical shock waves, Non-linear Schroedinger equation, fractional Burgers equation, Turing instability,...

I highly recommend that students who take this course get friendly (if they are not already) with the following software:

  1. Matlab/Mathematica/Maple for symbolic/numerical computations and visualization
  2. LyX/LaTex/TeXShop for writing papers
  3. BidDesk for organizing references
  4. Keynote/Powerpoint for presentations

Week

Topics

Notes

1: Feb 7,9

Introduction: Review of the origin of nonlinear PDE and systems of PDE: Burgers, KdV, Euler, reaction-diffusion, etc.
Variational principles: Euler-Lagrange equations, applications to water waves, Plateau problem, waves in elastic media, Klein-Gordon equation

 

2: Feb 14, 16

Weak solutions: of hyperbolic equations, shock waves and shock conditions, entropy conditions

Feb 11 - last day to add

3: Feb 21, 23

Hyperbolic systems: Euler equations of gas dynamics, shallow-water equations, traffic flow equations, Rankine-Hugoniot conditions, Riemann invariants, the Riemann problem

- Feb 18 - last day to drop w/out W

4: Feb 28, Mar 2

Hyperbolic systems: Hodograph transformation, wavefront expansions, weakly nonlinear approximations

 

5: Mar 7, 9

Reaction-diffusion phenomena: nonlinear diffusion, porous medium equation, similarity solutions, traveling wave solutions and their stability

 

6: Mar 14, 16

Reaction-diffusion systems: applications in combustion, chemotaxis, and population dynamics; traveling wave solutions, existence and uniqueness of solutions

 

7: Mar 21, 23

Reaction-diffusion systems: Energy estimates and asymptotic behavior, Turing instability, pattern formation

Mid-term exam

8: Apr 2, 6

Spring Break

 

9: Apr 11, 13

Dispersive PDE: linear and nonlinear dispersion, Whitham’s equations, integrability

Apr 9 - last day to drop with a W

10: Apr 18, 20

Solitons and inverse scattering transform: solitons in Korteweg-de Vries equation, conservation laws

 

11: Apr 25, 27

Nonlinear Schrödinger equation

 

12: May 2, 4

Equilibrium problems: Elliptic models, maximum principle, eigenvalue problems

 

13: May 9, 11

Equilibrium problems:  Stability and bifurcation

 

TBD:

PDE Presentations