PDE II
AMCS 331. Applied Partial Differential Equations II, Spring 2011, KAUST
Instructor: Aslan Kasimov
Lectures: 9:0010:30 every Monday and Wednesday in Room 93224
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. Quasilinear and nonlinear PDE in applications. Conservation laws, firstorder equations, the method of characteristics. Burgers’ equation and wave breaking. Weak solutions, shocks, jump conditions, and entropy conditions. Hyperbolic systems of gas dynamics, shallowwater flow, traffic flow, and biofluid flow. Variational principles, dispersive waves, solitons. Nonlinear diffusion and reactiondiffusion 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:
 D. Logan, Introduction to nonlinear PDE
 L. Debnath, Nonlinear PDE for scientists and engineers
 L. Evans, Partial Differential Equations
 S. Smoller, Shock waves and reactiondiffusion equations
 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 1015 page paper and give an endterm 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, nonclassical shock waves, Nonlinear 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:
 Matlab/Mathematica/Maple for symbolic/numerical computations and visualization
 LyX/LaTex/TeXShop for writing papers
 BidDesk for organizing references
 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, reactiondiffusion, etc. 

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, shallowwater equations, traffic flow equations, RankineHugoniot 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 
Reactiondiffusion phenomena: nonlinear diffusion, porous medium equation, similarity solutions, traveling wave solutions and their stability 

6: Mar 14, 16 
Reactiondiffusion systems: applications in combustion, chemotaxis, and population dynamics; traveling wave solutions, existence and uniqueness of solutions 

7: Mar 21, 23 
Reactiondiffusion systems: Energy estimates and asymptotic behavior, Turing instability, pattern formation 
Midterm 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 Kortewegde 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 
