Syllabus
Sturm's theory. Proof of existence of the eigenvalues of the Sturm-Liouville problem, and the properties of these eigenvalues. The properties of the eigenfunctions of the regular Sturm-Liouville problem.
The adjoint operator and the self-adjoint operator. Fredholm's Alternative theorem. The solvability conditions.
The Bessel Functions. The Legendre Polynomials. Several other special functions: the Gama functions, Beta function, Error function, the Elliptic integrals and Elliptic functions, other special functions if the students are interested.
Green's function.
The Hilbert-Schmidt theorem. Convergence theorems for series expansions in eigenfunctions. The Rayleigh-Ritz
theorem.
Fourier transform. Laplace transform.
Application of all the above to solving PDEs (in two or three spatial coordinates, as well as with time dependence).
Before the first lecture, please review your knowledge of elementary ODEs.
- Teacher: לידיה פרס-הרי