Gurarie's Classes, Home

Math. 445: Introductory Partial Differential Equations: Modeling and applications

Spring 2009

General outline

This course is designed for students in Applied Mathematics, Sciences, Engineering. It covers a broad range of topics and applications, where PDEs are used as a modeling tool for physical, chemical, engineering, biological and other systems and phenomena. Examples include solids and fluids, transport processes, vibrations, heat and wave propagation, acoustics, optics, nonlinear dynamics. We develop the basic mathematical methods and techniques for formulating and solving such problems.

Mathematical methods and techniques are supplemented by computational tools (mostly Mathematica), for graphic visualization, symbolic manipulations and numeric procedures. Both Mathematica and Matlab have a set of built-in packages for solving and analyzing various types of differential equations and systems. In general, computer allows one to greatly extend the scope of problems and models, where analytic methods fail. Students are encouraged to use them throught this curse (lectures, homework problems, projects, presentations).

A brief introduction to Mathematica 6 & 7 (and many demonstration) is provided by Wolfram site. it requires no preliminary experience.

Lectures, supplementary material and Mathematica notebooks

Instructor: David Gurarie, Yost 331, Ext. 2857, e-mail: dxg5@case.edu

Text: W. A. Strauss, Partial differential equations: An introduction, J. Wiley &Sons

Additional sources: Class notes, Books

Haberman, Elementary Applied PDEs, Prentice-Hall, 1998

G. L. Lamb, Introductory Applications of PDE’s, Whiley&Sons

J. Kevorkian, PDEs: analytic solution techniques,  Chapman & Hall

G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience Publ.

Drazin and Reid, Hydrodynamic stability, Cambridge UP

 

Math 445: Tentative Syllabus

Topic

 

1.Basic DE Models: Balance-Conservation laws, mechanical systems; Geometry

PDE: Traffic flow, mass conservation; Diffusion/Heat; Vibrations 1.3

Systems: Fluids, Electromagnetism, and Elasticity: 13.1-2

 Jan. 12-19

 

2.First-order equations and characteristics

Linear and quasilinear equations

Shocks

Initial and boundary conditions; Classification and well-posed problems  

 Jan.21-26

3.Diffusion

The diffusion equation on line (2.3-4)

Heat-diffusion equation in space, Schrodinger equation (9.4-5)

Half-life diffusion: reflection and sources (3.1-2)

Jan. 28

Feb. 2-4

4.Waves equation 1D and several D

Causality and energy conservation

d'Alembert solution, traveling waves

Reflection and sources

Scattering

Multi-D waves: energy, causality; rays, singularities, sources

*Wave propagators in 2D and 3D

Feb.9-23

5.Boundary-value problems and eigenfunction expansion method

Sturm-Liouville problem and Fourier series

Separation of variables: Dirichlet, Neumann and mixed condition

Fourier expansion (orthogonality, completeness, convergence)

Application to boundary-value problems

1D Green’s functions

 

Feb. 25 - March 18

Project proposals

March 4

 

6.Boundaries in space: multi-D eigenvalue problem and special functions

Fourier expansion

Vibration of drums, spheres, balls

Bessel and Legendre functions

Eigenfunction expansion: completeness and orthogonality

*Variational methods and computation (Rayleigh-Ritz),

Application to diffusion and waves

March 16-25

March 23-Apr. 1

7.Laplaces equation and potential theory; Harmonic functions; Green's functions

Laplace's equation in rectangles, boxes, cubes: 6.1-2

Poisson kernel: circles, wedges, annuli: 6.3

Green's identities: 7.1-2

Green's functions in the half-space and the sphere: 7.3-4

Potential theory, applications to fluids and electrostatics

March 30-Apr. 8

Apr. 6-15

Nonlinear problems

Equilibria, stability and bifurcations

Nonlinear string and elastic rod

Kolmogorov-Fisher equation (traveling waves)

Apr. 20-27

Course-work

1.       Regular homework assignments for 7 topics (graded selectively)

2.       Midterm project proposal (presentation and draft 1-2 pp), or summary of topics 1-4 (3-4pp)

3.       Final project or term-paper.  

Projects can be based on student’s research topics that involve differential equations. It combines a 20-30 min presentation (slides and/or report) at the end of semester.

The midterm proposal requires

(i)                  Explain the physical meaning (model, problem, context, significance).

(ii)                Set them up as differential equation problem(s)

The final presentation/report

(iii)               Along with model and mathematical setup, one should outline solution method/ analysis (analytic, if available, possibly in special cases )

(iv)              Numerical procedures and results, illustrated with tables, plots etc.

(v)                Comments and future development

Term paper is a comprehensive overview of 5 selected topics of the class material

Grade will be based on homework (60%), and midterm/final project/term paper (40%).