MTH 621-3: DIFFERENTIAL AND INTEGRAL EQUATIONS - Spring 2009
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General information
Assignments
MATLAB utilities
Assignments and schedule
  1. 3/30: Introduction and overview, review of method of characteristics
  2. 4/1: Review of travelling wave solution; examples of travelling waves.
  3. 4/3: Models of physical phenomena in the form of conservation laws: traffic flow; chromatography; glacier flow; two-phase flow (fractional flow form); conservation of mass, momentum, and energy in a pipe; Burgers' equation. Vanishing viscosity solutions.
  4. 4/6: Different notions of solutions: classical, integral, and weak solutions; connection between these.
    worksheet1: you can use download PDF or LATEX and type in solution in worksheet. (Remember to download amsnumbers.texas well if you want to process the file with my macros.
  5. 4/8: solving Burgers' equation via MOC, TW. Finding breaking time.
  6. 4/10: finding speed of discontinuity via Rankine-Hugoniot condition derived for integral and weak solutions (informal and formal arguments).
    worksheet2 parts A-D.
  7. 4/10-13: derivation of multiple (I) solutions to Riemann problem via equal area rule.
    Use worksheet2 part E for Cole-Hopf transformation for Burgers equation. Details of taking the limit of viscosity solutions.
    Construction of minimizer of a function parametrized with x,t, and initial condition.
  8. 4/15: Lax-Oleinik solution for general conservation law with convex flux function.
  9. 4/17: entropy solutions and entropy conditions (Lax, Oleinik). Entropy function and flux. Apparent non-uniqueness of weak solutions.
  10. 4/20: plot solutions to convex and non-convex flux functions in characteristics plane. Start worksheet3. You can use grid.pdf for plotting.
  11. 4/22: SYSTEMS of conservation laws. Linear acoustics: compute sound speed.
  12. 4/24: Acoustics in terms of variables [p,u]. Solving a hyperbolic system via diagonalization.
  13. 4/27: Acoustics for heterogenous media. Reflected and transmitted wave. (See animations).
  14. 4/29-5/1: Riemann problem for a linear hyperbolic system. Hugoniot locus and phase plane. Finding intermediate states and reflection/transmission coefficients for heterogeneous media. (Use MATLAB tools to visualize solutions).
  15. 5/4-6 Understanding nonlinear hyperbolic systems in 1D: simple wave, Riemann invariants for gas dynamics. Use hodograph transformation to transform a nonlinear system to a linear one.
    Worksheet4 is now available.
  16. 5/8-11 Developing general equations of motion in 2D and 3D for fluids and solids. Stress tensor, and constitutive equations.
  17. 5/13: attend Colloquium talk by Prof. K.-H. Hoffmann.
  18. 5/15: develop equations for Stokesian and Newtonian fluids as well as linear elasticity theory.
  19. 5/18-20: study wave equation in 2D, 3D (method of spherical means).
  20. 5/22-29: stationary deformation and flow in heterogeneous media.
  21. 5/25-27 (NO CLASS: Memorial Day break)
  22. 6/1: review. WORKSHEETS ARE DUE.
  23. 6/3: individual meetings to discuss worksheet solutions.
  24. 6/5: how to treat coupled systems of flow and transport. Example: pressure and saturation equations for multiphase flow equations; Richards' equation.
Class calendar is