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.


  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




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



PDE Presentations