Partial Differential Equations
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

Linear first order partial differential equations; characteristics  particles trajectories in a continuum; the Cauchy problem, propagation of singularities; complete integral and general solution.

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

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

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 semiaxis and a segment; propagation of singularities; separation of variables; nonhomogeneous problems; Duhamel’s principle.

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 semiaxis, on a segment, separation of variables; nonhomogeneous problems, Duhamel’s principle; Green’s functions; maximum principle and comparison theorems.

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 quasilinear elliptic equations; fundamental solutions an their physical meaning; Green’s functions, method of images, inversion; separation of variables.