MTH 482/582 Class Notes

MTH 482/582 Class Notes

The book I'll rely on the most is Berg and McGregor (BM) Elementary Partial Differential Equations. I will also refer to J. D. Logan's (L) Book, Applied Partial Differential Equations.

  • Sturm Liouville Theory, Orthogonal Expansions.
  • Fourier Series.
  • PDE's, Introduction.
  • Parabolic PDE.
  • Elliptic PDE.
  • Part 1, Nonhomogeneous heat Equation.
  • Cylindrical/Polar Coordinates, the Heat and Laplace's Equations.
  • First Order Hyperbolic PDE's
  • Wave Equation, Second Order Hyperbolic PDE's.
  • Fourier Transforms.
  • Examples of Convolution.
  • PDE's on infinite and semi-infinite domains.
  • Discrete Convolution.
  • Green's Function Solution of Elliptic Problems in n-dimensions.
  • Miscelaneous Problems.
  • Mathematica CDF of the Green's Function



    Week Section Topic
    Week 1 (L) Ch 3.1-3.2, (BM) Ch 4 Sturm Liouville Theory, Orthogonal function expansions
    Week 2 (BM)6 Ch (L) Orthogonal Expansions 3.3 3.4, (BM) 4, 5, (L) 4.1 4.2 Boundary Value Problems, Fourier Series
    Week 3 (BM) 5, (L) 4.1 4.2 Boundary Value Problems, Introduction to Green's Functions.
    Week 4-6 (BM) 3, (L) 4.3 4.4 4.5 Separation of Variables, Laplace's Equation, the Heat Equation. (BM) 8, The Wave Equation.
    Week 7 Fritz Johns PDE book. Advection, The 1-Way Wave Equation.
    Week 8 (BM) 9, (L) Ch 2.1, 2.2 The Cauchy Problem for the Heat Equation and the Wave Equation
    Week 9 (BM) 9, (L) Ch 2.4, 2.6, 2.7 The Fourier and the Laplace Transform.
    Week 10 (L) Ch 2.3, 2.5 Sources and Duhamel's Principle, Green's Function Solutions to Elliptic problems.
  • MATHEMATICA TOOLS:
    Fourier Series of Simple Functions from the Wolfram Demonstrations Project by Alain Goriely





    Laplace's Equation on a Square from the Wolfram Demonstrations Project by David von Seggern





    Laplace's Equation on a Circle from the Wolfram Demonstrations Project by David von Seggern





    Heat Transfer along a Rod from the Wolfram Demonstrations Project by Stephen Wilkerson





    Heat Flow through a Pipe from the Wolfram Demonstrations Project by Enrique Zeleny





    The Vibrating String from the Wolfram Demonstrations Project by Alain Goriely





    Vibrations of a Rectangular Membrane from the Wolfram Demonstrations Project by Alain Goriely





    Solutions of the Elliptic Membrane Problem from the Wolfram Demonstrations Project by Enrique Zeleny

    2D Wave Propagation from the Wolfram Demonstrations Project by Jason Beaulieu