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.
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 (D. Gurarie)
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
Tentative Syllabus
Modeling with DE: balance- conservation/ laws; variational principles
(ch. 1, 13, additional notes)
:
Jan. 17-22-24
Traffic flow; diffusion; vibrations: ch. 1.3
Fluids, electromagnetism, and elasticity: 13.1-2
First-order equations and characteristics (Ch. 1,14): Jan.29-31-Feb.5
Linear and quasilinear: 1.2
Shocks: 14.1
Initial and boundary conditions; well-posed
problems; classification: 1.4-6
Diffusions in 1D and multi-D (ch.2-3)
: Feb. 7-12-14
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)
Waves in 1D and several D (ch.2-3):
Feb. 19-21-26-28; March 5
The wave equation: causality and energy: 2.1-2
d'Alembert solution, traveling waves
Reflection and sources: 3.1-4; Scattering: 13.3
Multi-D wave equation: energy and causality; Rays, singularities and sources: : 9.1-2-3
Wave propagators in 2D and 3D
Boundary-value problems and eigenfunction expansion (Sturm-Liouville and
Fourier series) (ch.4-5)
: March 7-19-21-26
Separation of variables: Dirichlet, Neumann and
mixed condition:
4-1-3
Trigonometric - Fourier series (orthogonality,
completeness, convergence): 5.1-4
Application to boundary-value problems: 5.6
1D Green’s functions
Boundary-value problems and multi-D eigenvalues; Special functions
(ch.10-11): March 28; Apr. 2-4-9
Fourier method: 10.1
Vibrations of drums and the ball: 10.2-3
Bessel and Legendre functions: 10.5-10.6
Variational form and computation (Rayleigh-Ritz), completeness
and orthogonality: 11.1-3
Application to diffusion and waves: 11.4-5
Laplaces equation and potential theory; Harmonic functions; Green's
functions (ch.6-7): Apr. 11-16-18-23
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
Nonlinear models: equilibria, stability and bifurcations (ch. 14): Apr 25-30
Nonlinear string and elastic rod
Bifurcations: Kolmogorov-Fisher equation
Course-work
includes regular homework assignments (graded selectively), and the final project, or term-paper.
Projects could be based on student research
topics that involve differential equations. It combines a 20-30 min
(power-point et al) presentation at the end of semester, and the accompanied
5-6 pp report. The project requires:
(i)
Explain the physical model(s) and
problem(s), their context and significance. Set them up as differential
equation problem(s)
(ii)
Outline mathematical solution methods, if
available, or special cases that would allow such solutions
(iii)
Outline numerical procedures and results,
illustrated with tables, plots etc.
(iv)
Future results, directions and comments.
Term paper is a comprehensive overview of the
class material (see details)
The grade will be based on homework (60%), and
project/term paper (40%).