Mathematical Modeling and Analysis in Engineering

Instructor:  Wu-ting Tsai  wttsai@ntu.edu.tw
There is no fixed office hour, and you can track me down  in my office or after the class (preferable). 

Course Description:

This is a graduate course offered at the National Taiwan University. It was taught in the Institute of Hydrological and Oceanic Sciences (course number HS6016) and Institute of Geophysics at the National Central University and the Department of Civil Engineering at the National Chiao Tung University (course number 5228).
The course gives a utilitarian survey of elementary and intermediate analytic methods and their applications in engineering and physical systems.  The topics include linear partial differential equations, separation of variables, Green's functions, Fourier transforms, complex functions and Laplace transforms.  Both formulation of physical problems and analysis of mathematical solutions are emphasized. 
The materials are mainly based on that in the course taught by Professor Chiang C. Mei at MIT (1.131: Mathematical Analysis and Modeling in Engineering).  "1.131" was one of my favorite courses which I have taken when I was a graduate student at MIT.    

Prerequisites:

Mulitivariable Calculus plus an undergraduate course of Applied Mathematics or Engineering Mathematics on ordinary differential equation

Textbook:

 Chiang C. Mei, "Mathematical Analysis in Engineering", Cambridge University Press, 1995 ( Frontmatter, Google Book)

References:

Grading: Based on the grades of one midterm examination (50%), one final examination (50%)

 Syllabus:
This syllabus is a good faith estimate -- it is subject to change.
Week
Topics
Reading
 1   Formulation of Physical Problems
  • Transverse vibration of a taut string (differential approach)
  • Longitudinal vibration of an elastic rod (differential approach)

    • 1.1
    1.2
     2
  • Traffic flow on a freeway (integral approach)
  • Seepage flow through a porous medium soil (integral approach)
  • Diffusion in a stationary medium (integral approach)
    		• 1.3
    1.4
    1.5
     3
    		  One-dimensional Wave Equation
  • Waves on an infinite-long string due to initial disturbance --- d'Alembert's solution
  • Method of characteristics
  • Domain of Dependence
  • Range of influence
         Finite Domains and Separation of Variables
  • Method of separation of variables
    		• 3.1
     
     
     

    4.1
    4
  • One-dimensional diffusion --- heat equation with Neumann and Dirichlet conditions 
  • Eigenfunctions and base vectors --- Orthognality
  • Heat equation with Robbins boundary conditions --- Partially insulated slab
    		• 4.2
    4.3
    4.4
    5
  • Sturm-Liouville problems
    		• 4.5
    6
  • Inhomogeneous wave equation --- steady forcing
  • Inhomogeneous wave equation --- transient forcing
    		• 4.6
    4.7
    7  Unbounded Domains and Fourier Transform
  • Fourier transform

    • 7.1 
    8
  • One-dimensional diffusion along on an infinite-long rod
  • One-dimensional diffusion due to localized source --- Delta function
  • One-dimensional diffusion with discontinuous initial condition --- Heaviside step function
    		• 7.2.1
    7.2.2
    7.2.3
    9  Midterm examination  
    10
  • Forced waves along a one-dimensional, infinite-long string
  • Fourier sine and cosine transforms
    		• 7.3
    7.6
    11
  • Diffusion along a semi-infinite domain
     Complex Variables
  • Complex numbers and functions
    		• 7.7

    9.1  9.2
    12
  • Branch point, branch cut and Riemann surface
  • Analytical functions -- Cauchy-Riemann condition
  • Cauchy's theorem
    		• 9.3
    9.4
    9.8
    13
  • Cauchy's integral formula
  • Isolated singularities --- branch point, poles and essential singularity
  • Evaluation of real integral by complex integration
  • Jordan's lemma
    		• 9.9
    9.11
    9.12
    9.13
    14
  • Forced harmonic waves and radiation conditions --- application of contour integration
    		• 9.21
    15  Laplace Transform and Initial Value Problem
  • Laplace transform
  • Derivatives and convolution theorem

    • 10.1
    10.2
    16
  • Initial value problem --- coupled pendula
  • Initial value problem --- string-oscillator system
    		• 10.3
    10.5
    17  Final examination  


    Notes for Review of Undergraduate Applied Mathematics:
    The following notes were prepared for the undergraduate courses -- Applied Mathematics I and II, which were taught by me at the former Department of Oceanography at Taiwan Ocean University (1994 to 1997) and the Department of Earth Sciences at National Central University (2002 and 2003).  They have not been updated since 1997.  The textbook used is: Advanced Engineering Mathematics by Erwin Kreyszig; John Wiley & Sons; 8th edition, 1998. 

    Useful materials and links:

    Java Applets: (You need Java software for these webpages. Get it free from java.sun.com.)