MTH 623 : Partial Differential Equations - Spring 2022
Links
General information
General information
Instructor: Malgorzata Peszynska, Professor of Mathematics (Contact information including office hours on instructor's department website)
Class: Lecture: MWF 13:00-13:50pm, STAG 113.
Course information, syllabus and assignments: see CANVAS.
Schedule: [textbook] {assignments}

    == Week 1
  • Class overview.
  • More on the overview. Example of a weak solution.
  • Revisit weak and distributional solutions to conservation laws. Shock speed calculation two ways.

    • == Week 2
  • 4/4: convex conjugate functions (Legendre transformation). Worksheet on weak solutions to linear conservation law. sketching solutions of conservation laws with convex and concave flux functions.
  • 4/6: Lax-Oleinik formulas for calculating solutions to conservation laws.
  • 4/8: L-O continued. Cole-Hopf transformation.

    • == Week 3
  • 4/11: Entropy conditions: (I) Lax; (II) Oleinik (chord); and (III) with entropy function and flux.
  • 4/13: Example of (III) entropy conditions. Recall integral formulation of conservation laws. The shape of rarefaction fan.
  • 4/15: A slew of conservation laws: flood model, adsorption models, Buckley-Leverett, traffic flow. Worksheet on the qualitative behavior.

    • == Week 4
  • 4/18: Wrap up conservation laws. Worksheet on traffic flow and on signoid flux functions.
  • 4/20: How to get the velocity. (Flow model). Connection between pressure and density. Examples: Euler equation. The "porous medium equation".
  • 4/22: The actual porous medium models.

    • == Week 5
  • 4/25: Single phase slightly compressible low in porous media: linear or nonlinear depending oin primary unknonwns.
  • 4/27: Parabolic PDE with dominant diffusion or advection or reaction. How to determine the dominant dynamics. Ex.: linear transport, or gravity driven flow slightly compressible flow.

    • Multiphase flow equations. For incompressible and immiscible case, make additional assumptions and simplify to Richards equation or Buckley-Leverett problem.
  • 4/29: Continuum mechanics models starting with indicial notation (Einstein summation notation).

    • == Week 6
  • 5/2: Strain in solid mechanics. Different steps of linearizaiton for small deformations. Lagrangian and Green's finite deformation tensor. Lagrangian infinitesimal strain tensor.
  • 5/4: Stress and its connection to strain: compression and shear. Reducing stress-strain tensor (81 components) to two using Lame Coefficientrs or Young's modulus/ Poisson's ratio.
  • 5/6: Material derivative. Lagrangian vs Eulerian description of motion. Exercises.

    • == Week 7
  • 5/9: Continue calculations in Lagrangian or Eulerian frame.
  • 5/11: Fluids models.
  • 5/13: no in-class meeting. See online content.

    • == Week 8
  • 5/15: Fluids: models starting with most general Newtonian fluid models: Navier-Stokes equations for compressible and viscous fluids. Special approximations of such fluids. Exact soluitions, e.g., Hagen-Poiseuille flow.
  • 5/17: Wrap up fluids and solids. Start asymptotic methods (overview); algebraic examples
  • 5/20: Class cancelled due to SIAM PNW.

    • == Week 9
  • 5/23: Asymptotics: singular and regular perturbations in differential equations. Boundary later and inner/outer solutions.
  • 5/25: Asymptotic expansions for differential equations with periodic coefficients.
  • 5/27: Applications of homogenization for problems with periodic data: elasticity, fluids, heat conduction. Intrduction to systems of conservation laws.

    • == Week 10
  • 5/30: No class (Memorial Day).
  • 6/1: Solving a linear system of conservation laws. Hugoniot locus. Recall what is difference for nonlinear conservation laws.
  • 6/3: Wrap-up/review.