| Course Description: |
This course is concerned with the classical methods of Applied Mathematics.
The objective is to introduce the student to a number of fundamental
mathematical ideas and techniques that lie at the core of the applied
mathematician's approach. For this, the course will focus on the development,
analysis and interpretation of mathematical models from the natural sciences.
Special attention will be given to the formulation ("mathematical
modelling") of selected natural processes, including planetary motion,
Brownian dynamics, heat conduction, osmotic flow, acoustic and elastic
wave propagation, electrostatics and, finally, to the general aspects
of thermomechanics of continua.
Besides being important in their own right these events demand a variety of techniques for their mathematical description. Accordingly, some basic mathematical methods will be discussed and used in the analyses of their motivating applications. These procedures will range from simple dimensional analysis, scaling and linearization to (introductory material in) transform methods, perturbation theory, integral equations and the calculus of variations. |
| Prerequisites: | Advanced undergraduate mathematics. |
| Instructor: | Fernando Reitich. |
| Office: VinH538, Telephone: 626-1324, E-mail: reitich@math.umn.edu | |
| Text: | Mathematics Applied to Deterministic Problems in the Natural Sciences (Third Printing, 1994), C. C. Lin and L. A. Segel. |
| Classes: | MWF 09:05-09:55am, Vincent Hall 209. (Spring 2006) |
| Homework: | There will be 9 (nine) homework assignments during the year. These will consist of exercises from the textbook supplemented with problems to be provided by the instructor. |
| Exams: | Final Exam, Fall 2005: Wednesday, December 14, 09:05-09:55am, VinH 301. |
| Final Exam, Spring 2006: Friday, May 5, 09:05-09:55am, VinH 209. | |
| Grading Policy: | The course grade will be computed according to the following weighting system: |
| Homework ........................... 65% | |
| Final Exam ........................... 35% |
| SYLLABUS FOR Math 8401: Fall 2005 | |||
|---|---|---|---|
| Week | Dates | Sections | Topics | 1 | W 9/7 F 9/9 |
2.1 |
Planetary Orbits: Kepler's laws and gravitation N-body problems |
| 2 | M 9/12 W 9/14 F 9/16 |
2.2 |
Elementary perturbation theory
Linear ODE's Linear ODE's (cont'd) |
| 3 | M 9/19 W 9/21 F 9/23 |
2.2 2.3 |
The pendulum Poincare's perturbation theory Nonlinear ODE's: Existence |
| 4 | M 9/26 W 9/28 F 9/30 |
2.3 |
Nonlinear ODE's:Uniqueness Nonlinear ODE's: Stability Nonlinear ODE's: Finite differences |
| 5 | M 10/3 W 10/5 F 10/7 |
11.3 6.1 |
The phase plane The phase plane (cont'd) The basic simplification procedure |
| 6 | M 10/10 W 10/12 F 10/14 |
6.1 6.2 |
Conditioning and sensitivity Dimensional analysis Dimensional analysis (cont'd) |
| 7 | M 10/17 W 10/19 F 10/21 |
6.3 7.1 |
Scaling Scaling (cont'd) Regular perturbation theory: introduction |
| 8 | M 10/24 W 10/26 F 10/28 |
7.1 7.2 |
Regular perturbation theory: series method Regular perturbation theory: parametric differentiation method Regular perturbation theory: method of successive approximations |
| 9 | M 10/31 W 11/2 F 11/4 |
9.1 |
Singular perturbation theory: introduction
Singular perturbation theory: boundary value problems for ODE's Singular perturbation theory: matched asymptotics |
| 10 | M 11/7 W 11/9 F 11/11 |
9.1 11.2 |
Singular perturbation theory: matched asymptotics (cont'd)
Singular perturbation theory: multiple-scale expansions Singular perturbation theory: multiple-scale expansions (cont'd) |
| 11 | M 11/14 W 11/16 F 11/18 |
3.1 3.2 |
1-D Brownian motion 1-D Brownian motion (cont'd) Asymptotic series |
| 12 | M 11/21 W 11/23 F 11/25 |
3.2 3.3 |
Asymptotic series (cont'd) 1-D diffusion: derivation from Brownian motion Thanksgiving Holiday |
| 13 | M 11/28 W 11/30 F 12/2 |
3.4 4.1 |
A 2-D example: probabilistic derivation of Laplace's eqn.
1-D diffusion: derivation from a macroscopic perspective The heat equation: initial-boundary value problem |
| 14 | M 12/5 W 12/7 F 12/9 |
4.1 |
The heat eqn. in 2 and 3-D: generalizations
The heat eqn.: uniqueness, max. pple. , sub- and supersolutions Existence in 1-D: separation of variables |
| 15 | M 12/12 W 12/14 |
|
Review Final Exam <-- SOLUTIONS |
| SYLLABUS FOR Math 8402: Spring 2006 | |||
|---|---|---|---|
| Week | Dates | Sections | Topics |
| 1 | M 1/16 W 1/18 F 1/20 |
4.2, 4.3 |
Martin Luther King's Birthday
Fourier series: introduction, definition and convergence Fourier series: integration, differentiation and Gibbs phenomenon |
| 2 | M 1/23 W 1/25 F 1/27 |
5.3 |
Fourier series: mean square approx., Bessel's ineq., Parseval's thm.
Fourier transform: definition Fourier transform: inversion formula |
| 3 | M 1/30 W 2/1 F 2/3 |
|
Solution of the heat eqn. in 2 and 3-D
The heat eqn. in 2 and 3-D: initial conditions; the Dirac delta The heat eqn.: steady state and Laplace's equation |
| 4 | M 2/6 W 2/8 F 2/10 |
16.1 |
Electrostatics: introduction, the electric field
Electrostatics: Coulomb's and Gauss' laws Electrostatics: electric potential and electrostatic energy |
| 5 | M 2/13 W 2/15 F 2/17 |
|
Electrostatics: Poisson's and Laplace's eqns.
Electrostatics: conductors and boundary conditions Potential theory: Green's identities and uniqueness theorems |
| 6 | M 2/20 W 2/22 F 2/24 |
16.2 |
Potential theory: fundamental solutions and integral representations
Potential theory: Green's function and Poisson's kernel Potential theory: the volume potential |
| 7 | M 2/27 W 3/1 F 3/3 |
|
Potential theory: the double-layer potential
Potential theory: the single-layer potential Potential theory: integral equations |
| 8 | M 3/6 W 3/8 F 3/10 |
|
Introduction to the calculus of variations: strings and membranes
Pple. of minimum potential energy, Hamilton's and Fermat's principles Introduction to Hilbert spaces |
| | M 3/13 W 3/16 F 3/17 |
|
Spring Break |
| 9 | M 3/20 W 3/22 F 3/24 |
|
Linear functionals, Riesz representation theorem, weak convergence
Sobolev spaces: H^1 and H^1_0 The Ritz-Galerkin method: introduction |
| 10 | M 3/27 W 3/29 F 3/31 |
12.1 |
The Ritz-Galerkin method: existence and convergence of approx. solutions
Longitudinal motion of a bar: derivation of the governing eqns. Longitudinal motion of a bar: the material derivative, conservation of mass |
| 11 | M 4/3 W 4/5 F 4/7 |
12.1 |
Longitudinal motion of a bar: force, stress and balance of linear momentum
Longitudinal motion of a bar: strain and stress-strain relations Longitudinal motion of a bar: initial and boundary conditions; linearization |
| 12 | M 4/10 W 4/12 F 4/14 |
12.2 12.4 13.1 |
Longitudinal motion of a bar: 1-D elastic wave propagation
Longitudinal motion of a bar: work and energy The continuous model |
| 13 | M 4/17 W 4/19 F 4/21 |
13.2 13.3 14.1 |
Kinematics of deformable media: Eulerian and Lagrangian descriptions
The material derivative Field eqns. of continuum mechanics: conservation of mass |
| 14 | M 4/24 W 4/26 F 4/28 |
14.2 14.3 15.1 |
Field eqns. of continuum mechanics: balance of linear momentum
Field eqns. of continuum mechanics: balance of angular momentum Inviscid fluid flow |
| 15 | M 5/1 W 5/3 F 5/5 |
15.3 |
Compression waves in gases: linear and nonlinear waves
Review Final Exam <-- SOLUTIONS |
Dr. Reitich's weekly schedule (Spring 2006) | |||||
|---|---|---|---|---|---|
| Time | Monday | Tuesday | Wednesday | Thursday | Friday |
| 9:05-09:55 | Class MA8401 (VinH209) | Class MA8401 (VinH209) | Class MA8401 (VinH209) | ||
| 10:10-11:00 | Office Hour (VinH538) | Office Hour (VinH538) | Office Hour (VinH538) | ||
| 11:15-12:05 | Applied Math Seminar | ||||
| 12:20-13:10 | |||||
| 13:25-14:15 | Industrial Problems Seminar | ||||
| 14:30-15:20 | |||||
| 15:35-16:25 | Financial Math Seminar | ||||
| 16:40-17:30 | |||||
| 17:45-18:35 | |||||