MTH 481- 581 : MATHEMATICAL METHODS FOR ENGINEERS AND SCIENTISTS
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General information
Assignments

The numbers below refer to the section/problem numbers in the textbook (10th Edition)

The homework solutions must be written up clearly and neatly. Pay attention to the use of proper mathematical notation. Do not overdo the exposition but show enough work to justify your answers. Extra work on graphing and plotting or the use of technology is always welcome, but the hand calculations are most important (unless stated otherwise). Group discussions are permitted but no exchange of written materials is allowed; otherwise the cases will be considered as academic dishonesty.
Assignments
  1. 9/29: Introduction, overview and review of main concepts. Classification and vocabulary. Order, PDE/ODE linear/nonlinear, scalar/systems, IVP/BVP.
    Read: review Chapters 1-3 of the textbook.
    HW 1 due Friday 10/3: 3.2/7, 24, 28; 3.3/17, 35, 40; 3.6/2, 13, 15, 30.
    481 turn in: 3.2/28, 3.3/35, 3.6/13
    581 turn in: 3.2/28, 3.3/35, 3.6/13, 30
  2. 10/1: continue review. local/global existence theory for nonlinear/linear problems. Recall typical methods of solutions of equations with (constant) coefficients. Why do we need series solutions ?
  3. 10/3: Reviw sequences and series. Convergence and divergence. Criteria for convergence of series: ratio test (and other criteria).
    Read 5.1
    HW 2 due Friday 10/10: 5.1/1-16, 21-26
    481, 581 turn in: 5.1/12, 14, 19.
  4. 10/6: Power series and Taylor series. Solving ODEs with power series.
    Power Series Notes by Prof. Showalter. See also Web Study Guide to Sequences and Series.
  5. 10/8: Solving ODEs with power series. (First oder example).
  6. 10/10: Second order DE; solution near an ordinary point.
    Read 5.2, 5.3
    HW 3 due Friday 10/17: 5.2/1,2,4,15,17,21; 5.3/1,4,5-8,17,20,21; 5.4/1,3,5,8,17,21,22,26
    481 turn in: 5.2/4; 5.3/4;
    581 turn in: 5.2/4,21; 5.3/4;
    HW 3b due Monday 10/20: 5.4/1,22 (both 481/581)
    Start making your collection of equations: Airy, Bessel, Chebyshev, Euler, Hermite, Legendre. You should find out what they represent, and what their solutiosn are like. Why ? (TBD)
  7. 10/13: Nonanalytic function which is infinitely smooth. What is an ordinary point. Example of solving an ODE with variable coefficients.
  8. 10/15: Continue examples of solving ODEs near ordinary points.
  9. 10/17: Solving Euler equations, and more generally near (regular) singular points.
  10. 10/20 continue solving ODEs near regular singular points.
  11. 10/22: no class today.
    MIDTERM Friday, 10/24. Open books, open notes.
    Scope of exam: material from sections 5.1-5.4.
  12. 10/27: start systems of ODEs. Review of linear algebra. Read 7.2-7.3.
    HW (do not turn in): 7.2/4,8-10,15,21,22,25; 7.3/5,8,11,13-15,16,19,25.
    Problems 7.3/31-34 are very important and contain very useful information.
  13. 10/29: eigenvalues, eigenvectors and diagonalization of matrices.
  14. 10/31: continue eigenvalues/eigenvectors.
  15. 11/3: [Read 7.4] general theory of solving linear systems of ODEs.
    HW4 due Nov. 7:
    (a) Prepare a concise summary of equations of each of the following types: Airy, Bessel, Euler, Legendre. (State the equation and discuss the parameters, if any. Discuss the methods of solution and behavior near singular points. Discuss general solution.) [At most one two-sided sheet of paper is allowed so the word "concise" is critical, since you have 1/2 of a page for each equation] Use the textbook as the source of information; if other sources are used, cite them properly.
    (b) Turn in: 7.4/6.
    Do not turn in: 7.4/4-5,7
  16. 11/5/14: Solving homogeneous linear systems with constant coefficients. Diagonalization and Ansatz. [Read 7.5]
  17. 11/7/14: Examples for nondefective matrices with real roots. Plotting phase portraits.
  18. 11/10/14: Continue examples, real roots, nondefective matrix with eigenvalues other than e_1,e_2. Show how to use PPLANE. Asymptotic behavior of solutions. Origin as equilibrium point (attractor, repeller, saddle).
    HW 5 due Nov. 14: 481 and 581 turn in 7.5/5,7,9,12,25; 7.6/9,13. 581 also turn in 7.5/20.
    Do not turn in: 7.5/1-6, 24-27, 31. 7.6/1,3,14,15.
  19. 11/12/14: Planar systems with complex roots.
  20. 11/14/14: Subspaces and spanning sets of vectors. Basis, dimension.
    ODE systems with defective matrices. [Read 7.8]
  21. 11/17/14: Fundamental matrices. [Read 7.7]
  22. 11/19/14: Wrap up solving problems with constant coefficients. How to solve nonhomogeneous problems [Read 7.9].
    HW 6 due Nov. 26 (Wednesday): 7.7/12. (Extra: 7/7/14-15); 7.8/2; 7.9/1.
    [451: choose two, 551: do all four] of 9.1/1,4,10,11. For these problems solve parts a-c. (you can use computer for part d but this is not mandatory)
    Do not turn in: 7.7/1,3,5;7.8/1-5,7,15. 9.1/1-12. 9.1/13, 14, 16
    Read 9.1/prepare concise summary of phase portraits. Record Table 9.1.1.
    Extra: analyze behavior of a planar system with matrix [1 1; a 1] stated in class depending on the parameter a.
  23. 11/21/14: Recap of different phase plane portraits and qualitative behavior of solutions. Stability, asymptotic stability, unstability [Read 9.1] Trace-determinant plot (Pbms 9.1/20-21). Applications: springs and mass system. (Read 7.1)
  24. 11/24/14: Other applications from chemistry and biology (predator-prey model). [9.4 and 9.5]
  25. 11/26/14: Linearization of systems. Equilibria. [Read 9.2 and 9.3]
    HW 7 due Dec. 5 (Friday): All do (i) and (ii) or (i) and (iii). 581 students also do 9.4/17
    (i) 9.3/5-6(a-c) (part d optional).
    (ii) Use systems from 9.2/6-7 (a-d)
    (iii) use systems from 9.2/6-7 to determine equilibria and their type, AND solve 9.3/27-28.

    Do not turn in: 9.3/1-4, 7-11.
  26. 12/1/14: continue examples. Basin of attraction. Separatrices.
  27. 12/3/14: a system with periodic orbits. Methods of analysis of nonlinear systems other than linearization. Bifurcations.
  28. 12/5/14: Review.
    Final Problem Set: 481 and 581 do: 7.6/9; 9.3/13; 9.7/16abcd; 9.7/17a. (Due Monday, Dec. 9 by 4:00pm, in my office).