Fall 2016

Prof Yitzchak Rubinstein

יום ב 11:00 - 09:00 בגוטמן [32] חדר 210
יום ה 14:00 - 12:00 בבנין 90 (מקיף ז’) [90] חדר 227
  1. Linear first order partial differential equations; characteristics - particles trajectories in a continuum; the Cauchy problem, propagation of singularities; complete integral and general solution.

  2. System of two linear first order partial differential equations; classification; normal and canonical form; solution of the Cauchy problem for a hyperbolic system.

  3. Classification of second order partial differential equations with a linear main part; canonical form; characteristics; propagation of singularities; Cauchy-Kovalevskaya theorem; physical phenomena leading to equations of various types.

  4. One dimensional wave equation - example of a hyperbolic equation; initial and boundary conditions; Cauchy and boundary value problems; propagating waves method; D’Alambert’s formula; boundary value problems on a semi-axis and a segment; propagation of singularities; separation of variables; non-homogeneous problems; Duhamel’s principle.

  5. One dimensional heat equation - example of a parabolic equation; typical problems - the Cauchy and boundary value problem; moments; solutions of heat equation on the axis, similarity variable and solution, fundamental solution and it’s properties, solution of the Cauchy problem; boundary value problems on a semi-axis, on a segment, separation of variables; non-homogeneous problems, Duhamel’s principle; Green’s functions; maximum principle and comparison theorems.

  6. The Laplace’s equation - example of an elliptic equation; harmonic, sub- and super- harmonic functions and their properties, mean value theorem, maximum principle, Hopf’s lemma; Hadamard’s example and typical boundary value problems for elliptic equations; comparison theorems for linear and quasi-linear elliptic equations; fundamental solutions an their physical meaning; Green’s functions, method of images, inversion; separation of variables.