"Partial Differential Equations". Faculty of Engineering. (Ya. Yakubov)
Background: ordinary differential equations, functions of complex variables, harmonic analysis.
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).
- String or wave equation. Initial and boundary value conditions (fixed and free
boundary conditions). The d'Alembert method
for an infinitely long string. Characteristics.
- Wave problems for half-infinite and finite strings.
- Sturm-Liouville problem.
- 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
- Second order linear equations with two variables: classification of the equations
in the case of constant and variable coefficients, characteristics, canonical forms.
- Laplace and Poisson equations. Maximum principle. Well-posedness
of the Dirichlet problem. Laplace equation in a rectangle. Laplace equation in a circle and Poisson
A non-wellposed problem - the Cauchy
problem. Green formula and its using for Neumann problems. Uniqueness of a solution
of the Dirichlet problem.
- 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.
- Non-homogeneous heat equations, Poisson equations in a circle and
non-homogeneous wave equations.
- Free vibrations in circular membranes. Bessel