MATH 6397 - Spring 2003
Special Topics in Applied Analysis
Instructor: Suncica Canic
Office Hours: Monday and Wednesday: 1-2:30pm (PGH 622)
Texts:
- Walter A. Strauss: Partial Differential Equations: An
Introduction, #
ISBN: 0471548685
- Alexandre J. Chorin, Jerrold E. Marsden: A Mathematical
Introduction to
Fluid Mechanics, # ISBN: 0387979182
- Randall J. Leveque: Numerical Methods for Conservation Laws, #
ISBN:
0817627235,
- Y.C. Fung: Biomechanics: Circulation, ISBN: 0387943846.
Course Description
Please note that this course is listed as a graduate course, however a
motivated undergraduate student who has completed Calculus III,
a course on differential equations and an introductory course on
partial differential equations should be able to complete the course
succesfully. Students with majors in Mathematics, Biology,
Engineering/Bioengineering or Physics
are welcome.
This course introduces students to the basic tools from mathematics,
fluid dynamics, mechanics and biology,
necessary to study problems in cardiovascular fluid dynamics by
covering the following topics.
Part I: Linear Partial Differential Equations (Overview) (W.
Strauss: Intro to PDEs):
- Introduction; Well Posedness; Linear v.s. Nonlinear PDEs
- First order linear PDEs; characteristics, transport equation
- Classification of second order linear PDEs
- The Wave Equation: causality, energy, characteristics, D'Alambert
formula
- Diffusion: Maximum principle. Energy, Green's function
- Comparison between waves and diffusion
- Laplace's equation, Poisson formula, maximum principle,
separation of
variables
Part II: Nonlinear Partial Differential Equations: Conservation
Laws (LeVeque)
- Introduction; examples, characteristics, shock formation
- Weak formulation; Weak solutions; Examples.
- Basic Finite Difference Methods for Linear Conservation Laws
- Finite Difference Methods for Nonlinear Conservation Laws
Part III: Basic Fluid Mechanics (Chorin-Marsden: Fluid Mechanics)
- Equations of Motion
- Transport Theorem, Conservation of Energy, Bernoulli's Theorem
- Navier-Stokes equations I
- Navier-Stokes equations II
Part IV: Cardiovascular Fluid Mechanics (Fung: Circulation)
- Laminar flow in a Channel or Tube
- Composition and Properties of Blood Vessel Walls;
- Linear Elasticity; Modeling of Vessel Walls Dynamics;
- Blood flow properties and modeling (large versus small vessels)
- Wave propagation in blood vessels
Part V: Axi-Symmetric Models (Canic)
- Derivation; Asymptotic analysis; Initial and Boundary Data;
- Analysis of the reduced equations (characteristics,
hyperbolicity)
- Numerical Methods for the Reduced Equations
- Research Projects
- Guest lectures (Dr. Rosenstrauch; Dr. Metcalfe)
Grading: based on presentation of book sections and on the final
research project.