"Partial Differential Equations". Faculty of Engineering. (Ya. Yakubov)

Background: ordinary differential equations, functions of complex variables, harmonic analysis.

  1. String or wave equation. Initial and boundary value conditions (fixed and free boundary conditions). The d'Alembert method for an infinitely long string. Characteristics.
  2. Wave problems for half-infinite and finite strings.
  3. Sturm-Liouville problem.
  4. A solution of a problem for a finite string with fixed and free boundary conditions by the method of separation of variables. The uniqueness proof by the energy method. Well-posedness of a vibrating string problem.
  5. Second order linear equations with two variables: classification of the equations in the case of constant and variable coefficients, characteristics, canonical forms.
  6. Laplace and Poisson equations. Maximum principle. Well-posedness of the Dirichlet problem. Laplace equation in a rectangle. Laplace equation in a circle and Poisson formula. A non-wellposed problem - the Cauchy problem. Green formula and its using for Neumann problems. Uniqueness of a solution of the Dirichlet problem.
  7. The method of separation of variables for the one-dimensional heat equation. Maximum principle. Uniqueness for the one-dimensional heat equation. The Cauchy problem for heat equations. Green function.
  8. Non-homogeneous heat equations, Poisson equations in a circle and non-homogeneous wave equations.
  9. Free vibrations in circular membranes. Bessel equations.
  • Tikhonov, A. N. and Samarskii, N. A., "Equations of Mathematical Physics", Pergamon Press, Oxford, 1963.
  • Weinberger, H. F., "A first Course in Partial Differential Equations", Dover, New York, 1995.
  • Pinchover, Y. and Rubinstein, Y., "Mavo lemishvaot diferenzialijot xelkijot", The Technion, 2002 (in Hebrew).