Math 412-501

Theory of Partial Differential Equations

Part IV (2 weeks):

  • Green's functions
  • Dirac delta function
  • Uniqueness of solutions
  • Review for the final exam

    • Haberman's book: Chapters 9 and 11 (certain sections)

    Lecture 4-1: Green's functions. Dirac delta function.
  • Haberman 9.1-9.2

    • Lecture 4-2: More on the Dirac delta function. Green's functions for ODEs.
  • Haberman 9.3

    • Lecture 4-3: Green's functions for the heat and wave equations.
  • Haberman 11.2-11.3

    • Lecture 4-4: Green's function for the wave equation.
  • Haberman 11.2

    • Lecture 4-5: Uniqueness of solutions of PDEs. The maximum principle.
  • Haberman 2.5.3

    • Lecture 4-6: Review for the final exam.
  • Haberman 1-5, 7, 8.1-8.3, 8.6, 9.1-9.3, 10, 11.2.5-11.2.6, 11.3.7, 12.3-12.5




    • Part III (4 weeks):

  • Bessel functions
  • Nonhomogeneous problems
  • Fourier transforms

    • Haberman's book: Chapters 7, 8, 10

    Lecture 3-1: Heat equation in an arbitrary domain. Spectrum of Laplace's operator.
  • Haberman 7.2, 7.4

    • Lecture 3-2: Spectral properties of the Laplacian. Bessel functions.
  • Haberman 7.4-7.7

    • Lecture 3-3: Bessel functions.
  • Haberman 7.7-7.8

    • Lecture 3-4: Applications of Bessel functions.
  • Haberman 7.7-7.8

    • Lecture 3-5: Wave equation in an arbitrary domain. Laplace's equation in a cylinder.
  • Haberman 7.7, 7.9

    • Lecture 3-6: Nonhomogeneous problems. Method of eigenfunction expansion.
  • Haberman 8.1-8.3, 8.6

    • Lecture 3-7: Poisson's equation. Complex form of Fourier series. Fourier transforms.
  • Haberman 8.6, 3.6, 10.1-10.2, 10.5

    • Lecture 3-8: Properties of Fourier transforms.
  • Haberman 10.3-10.4

    • Lecture 3-9: Convolution theorem. Applications of Fourier transforms.
  • Haberman 10.4

    • Lecture 3-10: Applications of Fourier transforms (continued).
  • Haberman 10.5-10.6

    • Lecture 3-11: Review for Exam 3.
  • Haberman 7.4-7.9, 8.1-8.3, 8.6, 10.1-10.6




    • Part II (4 weeks):

    Separation of variables:
  • General one-dimensional heat equation
  • Higher-dimensional heat and wave equations
  • Laplace's equation
  • Sturm-Liouville eigenvalue problems

    • Haberman's book: Sections 1.5, 2.5, 4.4-4.5, and Chapters 5 and 7

    Lecture 2-1: Higher-dimensional heat equation.
  • Haberman 1.5

    • Lecture 2-2: Higher-dimensional wave equation. Complex-valued functions and Laplace's equation.
  • Haberman 4.5

    • Lecture 2-3: Separation of variables for the one-dimensional wave equation. Laplace's equation in a rectangle.
  • Haberman 4.4, 2.5.1

    • Lecture 2-4: Laplace's equation in polar coordinates.
  • Haberman 2.5.2

    • Lecture 2-5: Laplace's equation in polar coordinates (continued). Heat conduction in a rectangle.
  • Haberman 2.5.2, 7.3

    • Lecture 2-6: Heat and wave equations in box-shaped regions.
  • Haberman 7.1-7.3

    • Lecture 2-7: Sturm-Liouville eigenvalue problems.
  • Haberman 5.1-5.3

    • Lecture 2-8: Sturm-Liouville eigenvalue problems (continued).
  • Haberman 5.3-5.6

    • Lecture 2-9: Sturm-Liouville eigenvalue problems (continued).
  • Haberman 5.5, 5.10

    • Lecture 2-10: Sturm-Liouville eigenvalue problems (continued). Hilbert space.
  • Haberman 5.5-5.10

    • Lecture 2-11: Review for Exam 2.
  • Haberman 1.5, 2.5, 4.4-4.5, 5.1-5.10, 7.1-7.3




    • Part I (4 weeks):

  • Introductory examples
  • Linearity and homogeneity
  • Separation of variables
  • Fourier series

    • Haberman's book: Chapters 1-4 and 12.3-12.5
    Fulling's notes: pages 1-37

    Lecture 1: Introduction. Heat equation.
  • Haberman 1.1, 1.2.1-1.2.4

    • Lecture 2: Diffusion equation. Wave equation. Boundary conditions.
  • Haberman 1.2.5, 4.1-4.2, 1.3

    • Lecture 3: Steady-state solutions of the heat equation. D'Alembert's solution of the wave equation.
  • Haberman 1.4, 12.3

    • Lecture 4: D'Alembert's solution (continued).
  • Haberman 12.3-12.5

    • Lecture 5: Linearity and homogeneity.
  • Haberman 2.2

    • Lecture 6: Separation of variables.
  • Haberman 2.1, 2.3

    • Lecture 7: Eigenvalue problems. Solution of the initial-boundary value problem for the heat equation.
  • Haberman 2.3

    • Lecture 8: Fourier's solution of the initial-boundary value problem (continued).
  • Haberman 2.4, 4.4

    • Lecture 9: Fourier series.
  • Haberman 3.1-3.3

    • Lecture 10: Fourier series (continued). Gibbs' phenomenon.
  • Haberman 3.1-3.5

    • Lecture 11: Review for Exam 1.
  • Haberman 1.1-1.4, 2.1-2.4, 3.1-3.5, 4.1-4.4, 12.3-12.5