ENGR597, Session 12

Nonlinear Systems I. Perturbation Methods and Methods of Weighted Residuals

Fall, 2001

Selected mathematical algorithms for solving commonly encountered nonlinear engineering, physical, chemical and biological systems in research will be introduced to graduate and undergraduate students. These topics will be analytical in nature and will be of interests to students of a broad spectrum of background. The discussion will be accompanied with representative examples which are solved mostly in Mathcad, Matlab and Mathematica environments. At the end of the semester, students are expected to have learned

A. Linear vs. Nonlinear Systems
     1. Sources of Nonlinearity
     2. Mathematical and Physical (Process) Limitations in Dealing Nonlinear Systems
     3. Review of Linear Algorithms for Solving ODE and PDE

B. Perturbation Methods
     1. Straight Forward (or the Regular) Asymptotic Expansions
     2. Singular Perturbation Methods
       a. Matched Asymptotic Expansions (MMAE)
       b. Method of Composite Expansion (MCE)
       c. Method of Strained Parameter
       d. PLK Methods - Method of Strained Coordinates
       e. Renormalization Methods
       g. Method of Averaging

C. Method of Weighted Residuals
     1. Orthogonal Set of Functions
     2. Jacobi and Legendre Polynomials
     3. Interpolation and Integration by Quadratures
     4. Methods of Weighted Residuals
      a. Collocation Method
      b. Galerkin Method
      c. Subdomain Method
      d. Moment Method
      e. Least-Square method
      f. Orthogonal Collocation Method

D. Nonlinear Dynamic Systems
     1. Phase Plans, Flows and Bifurcations of One-Dimensional Systems
     2. Oscillations, Limit Cycles, Poincare Maps and Hopf Bifurcations of Two-Dimensional Systems
     3. Strange Attractors and Liapunov Exponents of Higher Dimension (>2) Systems
     4. Fractals and Chaos

E. Fundamentals of Partial Differential Equations
1. Classification
2. Solution of Quasilinear First-Order PDE by Characteristic Curves

II. INSTRUCTOR AND MEETING TIME

Instructor: Professor Wei-Yin Chen
Office: Room 140, Anderson Hall
Telephone: 915-5651
E-mail: cmchengs@olemiss.edu
Classes: 4:00 pm - 5:15 pm, Mondays and Wednesdays
Office Hours: 2:00 - 3:30 pm Tuesdays and Thursdays, or by appointment

III. TEXTBOOKS:

Rice, R., and D.D. Do, "Applied Mathematics and Modeling for Chemical Engineers," Wiley, New York (1994).
Chen, W.Y., "Supplementary Volume for Advanced Engineering Mathematics," University of Mississippi (2001).

General References

Kreyszig, E., "Advanced Engineering Mathematics," 8-th Edition, Wiley (1999).
Wylie, C.R., and L.C. Barrett, "Advanced Engineering Mathematics," 6-th ed., McGraw-Hill, New York (1995).

Advanced General References

Varma, A., and M. Morbidelli, "Mathematical Methods in Chemical Engineering," Oxford University Press (1997).
Denn, M.M., "Process Modeling," Longman, New York (1986).

Classical Approaches of Solving Linear Partial Differential Equations

Carslaw, H.S., and J.C. Jaeger, "Conduction of Heat in Solids," Oxford Univ. Press, London, Great Britain (1959).
Crank, J, "The Mathematics of Diffusion," 2nd Ed., Clarendon Press, Oxford, Great Britain (1975).

Perturbation Methods

*Nayfeh, A., "Perturbation Methods," Wiley (1973).
Nayfeh, A. "Introduction to Perturbation Techniques," Wiley (1981).
Nayfeh, A. "Problems in Perturbation," Wiley (1985).
Van Dyke, M., "Perturbation Methods in Fluid Mechanics," Annotated ed., Parabolic Press (1975).
Hinch, E.J., "Perturbation Methods," Cambridge (1991).
Kevorkian, J., and J.D. Cole, "Perturbation Methods in Applied Mathematics," Springer-Verlag (1981).
Bender, C.M., and S.A. Orszag, "Advanced mathematical Methods for Scientists and Engineers," McGraw-Hill, New York (1978); Springer-Verlag, New York (1999).
Holmes, M., "Introduction to Purturbation Methods," Springer-Verlag, New York(1995).

Weighted Residuals

*Finlayson, B.A., "The Method of Weighted Residuals and Variational Principles," Academic Press (1972).
*Villadsen, J., and M.L. Michelsen, "Solution of Differential Equation Models by Polynomial Approximations," Prentice Hall (1978).
Finlayson, B.A., "Nonlinear Analysis in Chemical Engineering," McGraw-Hill, New York, (1980).

Historical Review and Applications of Weighted Residuals:

Finlayson, B.A., and L.E. Scriven, "The Method of Weighted Residuals - A Review," Applied Mechanics Reviews, 19(9), 735-748 (1966).
Perlmutter, D.D., "Stability of Chemical Reactors," Prentice-Hall, New Jersey (1972).
Yu, W.C., M.M. Denn, and J. Wei, "Radial Effects in Moving Bed Coal Gasifier," Chem. Eng. Sci., 38(9), 1467-1481 (1983).

Nonolinear Dynamic Systems
Short Introductions

Varma, A., and M. Morbidelli, "Mathematical Methods in Chemical Engineering," Chapter 2, Oxford University Press (1997).
Bequette, B.W., "Process Dynamics - Modeling, Analysis and Simulation," Chapters 13 through 17, Prentice Hall, New Jersey (1998).
Baker, G.L., and J.P. Gollub, "Chaotic Dynamics, An Introduction," Cambridge University Press, New York (1990).

*Strogatz, S.H., "Nonlinear Dynamics and Chaos," Addison-Wesley, Reading, MA (1994).
Nayfeh, A.H., and B. Balachandran, "Applied Nonlinear Dynamics," Wiley, New York (1995).
Alligood, K.T., T.D. Sauer, and J.A. Yorke, "Chaos - An Introduction to Dynamical Systems," Springer-Verlag, New York (1997).
Kaplan, D., and L. Glass, "Undersatnding Nonlinear Dynamics," Springer-Verlag, New York (1995).
Ott, E., "Chaos in Dynamical Systems," Cambridge Univ. Press, New York (1993).
Thompson, J.M., and H.B. Stewart, "Nonlinear Dynamics and Chaos - Geometrical Methods for Engineers and Scientists," Wiley, New York (1986).
Hilborn, R.C., "Chaos and Nonlinear Dynamics," Oxford Univ Press, New York (1994).
Drazin, P.G., "Nonlinear Systems," Cambridge Univ. Press, Cambridge, United Kingdom (1992).

Verhulst, F., "Nonlinear Differential Equations and Dynamical Systems," 2nd ed., Springer-Verlag, Berlin, Gremany (1991).
*Jordan, J.W., and P. Smith, "Nonlinear Ordinary Differential Equations," 3rd ed., Oxford Univ. Press, Oxford, England (1999).
*Hale, J., and H. Kocak, "Dynamics and Bifurcations," Springer-Verlag, New York (1991).
*Hubbard, J., and B. West, "Differential Equations: A Dynamical Systems Approach, Ordinary Differential Equations" Springer-Verlag, New York (1991).
*Hubbard, J., and B. West, "Differential Equations: A Dynamical Systems Approach, Higher-Dimensional Systems" Springer-Verlag, New York (1995).
Guckenheimer, J., and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," 5-th Printing, Springer-Verlag, New York (1996).
*Hirsch, M., and S. Smale, "Differential Equations, Dynamical Systems and Linear Algebra," Academic Press, San Diego, CA (1997).
Iooss, G., and S. Smale, "Elementary Stability and Bifurcation Theory," 2nd ed., Springer-Verlag, New York (1990).
Grimshaw, R., "Nonlinear Ordinary Differential Equations," CRC Press, Boca Raton, FL (1993).
Perko, L., "Differential Equations and Dynamical Systems," 3rd ed., Springer-Verlag, New York (2000).
Hoppensteadt, F.C., "Analysis and Simulation of Chaotic Systems," 2nd ed., Springer-Verlag, New York (2000).

Alhumaizi, K., and R. Aris, "Surveying A Dynamical System: A Study of the Gray-Scott Reaction in A Two-Phase Reactor," Longman House, Harlow, Essex, England (1995).
Aris, R., "Mathematical Modeling, A Chemical Engineer's Perspective," Academic Press, San Diego (1999).
Elnashaie, S.S.E.H., and S.S. Elshishini, "Dynamic Modelling, Bifurcation and Chaotic Behaviour of Gas-Solid Catalytic Reactors," Gordon and Breach, Amsterdam, Netherlands (1996).
Epstein, I.R., and J.A. Pojman, "An introduction to Nonlinear Chemical Dynamics," Oxford University Press, New York (1998).
Scott, S.K., "Chemical Chaos," Clarendon Press, Oxford, England (1991).
Perlmutter, D.D., "Stability of Chemical Reactors," Prentice-Hall, New Jersey (1972).
Ottino, J.M., "The Kinematics of Mixing: Stretching, Chaos and Transport," Cambridge Univ Press, Cambridge, United Kingdom (1989).
Henson, M.A., and D.E. Seborg, (editors) "Nonlinear Process Control," Prentice Hall, Upper Saddle River, New Jersey (1997).

Moon, F.C., "Chaotic and Fractal Dynamics - An Introduction for Applied Scientists and Engineers," Wiley, New York (1992).

Sastry, S., "Nonlinear Systems - Analysis, Stability and Control," Springer-Verlag, New York (1999)

Turcotte, D.L., "Fractals and Chaos in Geology and Geophysis," Cambridge Univ. Press, Cambridge, Great Britain (1992).

Murray, J.D., "Mathematical Biology," 2nd ed., Spriner-Verlag, New York (1993).

Homework will be assigned every week.

One term project will also be required for each graduate student; complex and advanced engineering problems solvable by one of the algorithms discussed in the course, e.g., perturbation method, weighted residuals, and nonolinear dynamics, will be slected, analyzed, solved and presented. These topics will have to be chosen collectively with the instructor by the first six weeks of the semester, and the completed solutions will be presented to the class at the end of the semester. Reproduction of a published work in literature is acceptable, but new topics are encouraged.

V. Grading:

Grades will be given based on homework assignments and term projects.

Total grade will be given according to a scale similar to the following:
80 < A < 100
70 < B < 79
60 < C < 69
50 < D < 59
50 < F

Because the level of difficulties of the materials, attendance is very important for the course. In accordance with Departmental policy, students missing more than 9 class periods are subject to direct grade penalties. Penalties may be assessed without regard to the student's performance.

Students are expected to keep up with the material as it is presented and submit assignments on time; most students find this difficult without regular class attendance.

Except for excessive absenteeism, student attendance is not directly factored into grades.

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