Industrial Mathematics - Alexiades
M475 - Spring 2022
Industrial Mathematics
Modeling, Analysis, and Computation of interesting scientific / technological / industrial problems
commonly known as Computational Science

Prof. Vasilios ALEXIADES   Ayres 213   974-4922
                    Office Hours: TR 1:10-2:00 and by arrangement (email me:
  • Prerequisites: Calculus (M141,142,241), ODEs (M231), and familiarity with a programming language (Matlab, Python, Julia, R, Fortran, C/C++)
  • Attendance is mandatory.
  • No textbook to buy!
  • Work and Grading:   No exams!
        8-10 Lab/Homework assignments: 40% , Project assignments: 40% , Term/Team Project: 20%
  • Do not hesitate to talk to me if you are facing difficulties.
  • Do not fall behind! Several things need to be done concurrently. Faster pace in the beginning, slower later...

  • All incidents of academic misconduct will be reported to the Student Judicial Affairs office.
  • If you need an accommodation based on the impact of a disability, please contact me privately.
       Contact the Office of Disability Services (2227 Dunford Hall, 974-6087) to coordinate reasonable accommodations for documented disabilities.

  • Computational Science : doing Science by means of computation ("in silico").
      Involves: scientific problem → math problem → computational algorithm → numerical solution → implications for original scientific problem. CSE-HPC.jpg
      It has become the 3rd pillar of Science, complementing Theory and Experiment.
  • Real scientific/technological/managerial problems canNOT be solved explicitly/exactly.
      Need to be solved numerically (approximately), so need effective approximations/algorithms and to understand effects of errors in the calculations.
  • Want algorithms to be: effective, accurate, reliable, efficient and robust !
      These aims often play against each other, so trade-offs need to be made...
  • Issues of verification, validation, uncertainty quantification are becoming increasingly important.

    The course will simulate the core aspects of Computational Science including: • modeling and computational simulation of physical phenomena
    writing reports writing proposals collaborating with colleagues on a Team/Term project • and presenting your work.

    Start thinking about a Term project topic right away!

    Topics / Content

    I.  Crystal precipitation
      - physical model leading to ODE system
      - about ODEs - well posedness of IVP
      - equilibria - root finding (Newton method) - plotting
      - analysis of the model
      - Euler scheme - computational errors 
                     - consistency-stability-convergence
                     - implementation
      - classical RK4 and other numerical schemes
    II. Air pollution: Advection and Diffusion Processes
      - the general conservation law  ut + div F = 0
                         - derivation from first principles
                         - conservation of species
                         - advective and diffusive fluxes
                         - continuity equation
                         - constitutive laws (for non-advective fluxes)
      - finite volume discretization of  ut + Fx = 0 - explicit/implicit
      - diffusion ( F = −Dux )  - parabolic PDEs - boundary conditions
                                - explicit scheme - CFL condition
                                - super-time-stepping acceleration
      - advection ( F = uV )    - explicit upwind scheme
                                - CFL condition - implementation
      - linear advection - wave propagation
                         - 1st order PDEs - method of characteristics
      - advection-diffusion ( F = uV − Dux ) 
                                - explicit scheme - CFL condition
                                - effect of small/large Peclet number
      - a few words about Lax-Wendroff and other schemes
    III. Chemical reactions via mass action kinetics
    IV. Uncertainty Quantification and parameter estimation
    Some other possible topics:
    V. Melting and Freezing
      - phase-change basics, moving boundary problems
      - Stefan Problem, exact solution, analytic approximations
      - enthalpy formulation, explicit scheme
    VI. The catalytic converter
      - diffusion-reaction model
      - control problem
      - calculus of variations - Euler-Lagrange equation
      - numerical scheme for the forward model                   
    VII. Electron beam lithography (inverse problems)
      - forward scattering (dose to exposure)
      - inverse problem (exposure to dose) - ill posed problem
      - Fourier-Poisson integral solution of diffusion equation
      - Fourier series solution of diffusion equation
      - Fourier series approximation of the inverse problem
      - Discrete Fourier Transform, FFT

    ------------ Some comments from happy students -------------
  • "Thank you for a very interesting and informative class. I looked forward to taking it and am incredibly glad I did."  
  • "You made this class very interesting, challenging, and (dare i say it) fun ... I REALLY enjoyed the final project and feel more confident in my abilities because of this class."  
  • "This class was one of the best, if not the best, of my college career. I really enjoyed it."  
  • "Extremely relevant course material, broken down in a very understandable method by instructor"  
  • "... the best math class I've had so far.... I really learned a lot and plan to use it."  
  • "Loved it. It's the best class I've ever taken"  
  • "... For someone who enjoys programming, and has a real desire to see what all this math can be used for, it has been a terrific course."