MATH 6397 - Spring 2003

Special Topics in Applied Analysis

Instructor: Suncica Canic

Office Hours: Monday and Wednesday: 1-2:30pm (PGH 622)

E-mail: canic@math.uh.edu

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)