Partial Differential Equations

Third year course, Amsterdam University College, Fall 2016.

This is a course for third year students in the Science major in AUC that I have developed in 2013 and also taught in 2014 and 2016. There are two lectures per week and four written tests.

Outline

  1. Introduction
    1. Examples
    2. Speed of light from Maxwell's equations
  2. ODEs
    1. First order
      • Basic concepts
      • First order differential equations
      • Geometrical interpretation as slope fields
      • Existence and uniqueness for Cauchy problem
      • Separable equations
      • Exact equations
      • Integrating factor
      • Linear equations
    2. Second order
      • Equations reducible to first order
      • Linear equations
      • Existence and uniqueness
      • Principle of superposition
      • Fundamental sets of solutions and Wronskian
      • General solution of homogeneous equation and inhomogeneous equation
      • Linear second order equations with constant coefficients
  3. First order PDEs
    1. Basic definitions
    2. Linear operators
    3. Homogeneous and inhomogeneous linear equations
    4. First order linear PDEs with constant and variable coefficients
    5. Characteristic curves
    6. Derivation of simple transport equation
  4. The wave equation
    1. General solution to the wave equation on the line
    2. Initial value problem and d'Alembert formula
    3. Vibrating string
    4. Causality, dependence and influence domains, characteristic lines.
    5. Waves in 3D and Huygens principle
    6. Energy conservation
    7. Boundary value problems
    8. Dirichlet and Neumann coundary conditions
    9. Even and odd functions
    10. Semi-infinite vibrating string
    11. Reflection of waves
    12. Wave equation on the interval
    13. Separated solutions
    14. Dirichlet and Neumann boundary conditions on the interval
    15. Well-posed problems
    16. Stability of initial value problem
  5. Fourier series
    1. Eigenvalue problems
    2. Sine and cosine Fourier series
    3. Sine and cosine Fourier series
    4. Orthogonality and coefficient formulas
    5. Solution of the wave equation initial value problems with Dirichlet, Neumann boundary conditions.
    6. Periodic boundary conditions
    7. Full Fourier series
    8. Symmetric boundary conditions and orthogonality
    9. Pointwise, uniform and mean-square convergence
    10. Convergence theorems
  6. Heat equation
    1. Derivation of heat equation
    2. Weak and strong maximum principle
    3. Proof of weak maximum principle
    4. Uniqueness and stability using maximum principle
    5. Energy method
    6. Uniqueness and stability using energy method
    7. Green’s function
    8. Poisson formula for heat equation on the line
    9. Invariance properties
    10. The heat equation on the half-line
    11. Separation of variables for the heat equation on the interval
    12. Smoothing property of the heat equation
  7. Fourier transform
    1. Heuristic derivation of Fourier transform inversion formula
    2. Definition of Fourier and inverse Fourier transforms
    3. Examples of Fourier transforms
    4. Fourier transform of the gaussian function
    5. Properties of the Fourier transform
    6. Fourier transform of the derivative
    7. Convolution and its Fourier transform
    8. Solution formula for the heat equation in terms of Fourier transform
    9. Inversion theorems
    10. Fourier transform as a unitary transformation on L^2(R)
    11. Cauchy-Schwarz and triangle inequalities
    12. Parseval's equality
  8. Laplace equation
    1. Laplace and Poisson equations
    2. Complex analytic functions, Cauchy-Riemann equations
    3. Maximum principle, proof of the weak case
    4. Uniqueness and stability of Dirichlet problem
    5. Separation of variables on a rectangle
    6. Laplacian operator in polar coordinates
    7. Separation of variables on a circle
    8. Poisson formula
    9. Mean value property
    10. Neumann b.v.p.
    11. Proof of strong maximum principle
  9. Laplace transform
    1. Definition of Laplace transform
    2. Main properties of Laplace transform
  10. Second order linear PDEs
    1. Comparison of heat, wave and Laplace equations
    2. Classification of second order linear PDEs in two variables