Course Outline : Tentaive
1. Differential Equations and Their Approximations
- Boundary Value Problems for Second Order Partial Differential Equations
- Initial-Boundary Value Problems ( Heat Conduction/Transfer and Wave Equations )
- Coordinate Transformation
- Finite Difference Methods at a Curve Fitted Domain
- Extended Domain Method
- Wavelet Based Galerkin Method
- Element Free Galerkin Method
- Error Analysis for Adaptive Finite Element Methods
2. Equation Solvers
- Direct Linear Equation Solvers : LU Decomposition and Wave Front Method
- Iteration Methods for Linear Equations
- Newton's Method and Its Variations for Nonlinear Systems
- Iteration Methods for Nonlinear Systems
- Linear & Nonlinear Systems with Constraints
- Constrained Optimization Problems
3. Time Dependent Problems
- Transient Heat Conduction Problems
- Heat Transfer Problems with Adapative & Stream-Line Up-Wind Methods
- Wave Propagation Problems
4. Analysis of Images and Signals
- FFT for Image/Signal Processing
- Wavelet for Images and Signals
Reading Materials
I will prepare several reading materials from the series (Part 1 - Part 8) of
monographs
- Avner Friedman's "Mathematics in Industrial Problems," from Springer, New York
such as :
- Linear Algebra
- iterative solution methods on the Cray YMP/90
- Geometry, Design, and Optimization
- geometric path planning in rapid prototype
- robot localization using landmark
- coordinates for mechanism configuration spaces
- surface modeling in design and manufacturing
- experimental design and quality loss function
- Approximation Methods
- computation of volume integrals in potential theory
- element-free Galerkin method
- Homogenization Methods
- predicting properties of composite materials
- averaged equations for layered and blockly media
- micromechanics effects in creep of metal matrix composites
- Free and Moving Boundary Problems
- fluid flows in a porous media
- Differential Equations
- ideal forming theory : rigid plastic flow
- aeroacoustics in automotive industry
- Image and Signal Processing
- feature detection and tracking in three dimensional image
- topics in tomography
Textbook & References
This course is the second course of Applied Engineering Analysis in Mechanical
Engineering and Applied Mechanics Programs. We shall not assign a specific textbook for
this course, I suggest the book
- Gilbert Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Box 157,
Wellesley, Massachusetts 02181, (617) 235-9573, and also, Room 2-240, M.I.T., Cambridge,
MA 02139, (617) 253-4383
for the course. This book might be the most appropriate for engineering students who
are interested in applied mathematics. I also strongly suggest Greenberg's book as a basic
reference to this course, although we shall cover more advanced materials than Greenberg's
coverage in this course.
- M.D. Greenberg, Foundations of Applied Mathematics, Prentice-Hall International, 1978
The following books should help students to find more deeper and advanced discussions
on the subjects I will cover in this course. I sall give many applied examples of the
mathematics in mechanics and mechanical engineering programs.
Ingrid Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992
J.J. Benedetto and M.W. Frazier, Wavelets : Mathematics and Applications, CRC Press,
Boca Raton, 1994
S.D. Stearns and R.A. David, Signal Processing Algorithms in MATLAB, Prentice Hall, 1996
T.W. Korner, Fourier Analysis, Cambridge University Press, Cambridge, 1988
B. Dacorogna, Direct Methods in Calculus of Variations, Springer-Verlag, Berlin, 1989
M.R. Hestenes, Calculus of Variations and Optimal Control theory, Robert E. Krieger
Publishing Company, 1980
R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and
Technology, Volume 2 : Functional and Variational Methods, Springer-Verlag, Berlin, 1988
J.L. Lions, Sur quelques questions d'analyse, de mecanique et de controle optimal, LES
Press de L'Universite de Montreal, 1976
E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in
Physics #127, Springer-Verlag, Berlin, 1980
W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipies : The
Art of Scientific Computing, Cambridge University Press, Cambridge, 1986
N.S. Bakhvalov, Numerical Methods, Mir Publishers, Moscow, 1977
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,
Springer-Verlag, New York, 1990