Elementary Partial Differential Equations

Most of the topics of this exam are covered in the following courses: Mathematics in Modern Science I (Mate 6675) and Elementary Partial Differential Equations (Mate 6677). Topics:

  • Theory of Fourier series and mathematical validation of the method of separation of variables: Convergence, term-by-term differentiation and integration of Fourier series, applications of the latter to boundary value problems for heat, wave and Laplace equations.
  • Fist order quasi linear equations: Linear transport equation, Burgers equation and method of characteristics.
  • Elementary distribution theory: Test functions and convergence, distributions represented by locally integrable functions and other important distributions, operations on distributions, fundamental solutions of differential operators.
  • Heat (Diffusion) equation: Fundamental solution, initial value problem and uniqueness, Duhamel’s principle, maximum principle.
  • Laplace equation: Harmonic functions, Green’s formula, mean value theorem, maximum principle, Dirichlet and Neumann boundary value problems and uniqueness property, Green’s functions.
  • Wave equation: Initial value problem and uniqueness, Duhamel’s principle, d’Alembert formula: domain of dependence, region of influence.


Bibliography:

  • Introduction to Partial Differential Equations and Hilbert Space Methods, Gustafson, John Wiley and Sons, 1997.
  • Applied Partial Differential Equations with Fourier series and Boundary Value Problems, R. Haberman, Pearson, 2013.
  • Partial Differential Equations in Action from Modeling to Theory, S. Salsa, Springer, 2008.
  • Introduction to Partial Differential equations, G. Folland, Princeton University Press, 1995.
  • Partial Differential Equations, Fritz John, Springer Verlag, 1982.
  • Partial Differential Equations, J. Rauch, Springer Verlag, 1991.
  • Partial Differential Equations, Lawrence C. Evans, AMS, 2000.
  • Partial Differential Equations: An Introduction to Partial Differential Equations, W. A. Strauss, 2007