Analytical methods in differential equations – Syllabus

Strum Theory. Proof of existence of eigenvalues of the Strum-Liouville problem, and properties of these eigenvalues. Properties of the Eigen functions of the Strum-Liouville Problem. The adjoint operator, and the self-adjoint operator. The Fredholm Alternative theorem. Solvability conditions. Bessd Functions. The Legendre Polynomials. Other special functions such as the gamma Function, the Beta Functions. The Hilbert-Schmidt theorem. Convergence theorems for series of Eigen functions. The Rayleigh-Ritz theorem. The Fourier Transform. The Laplace Transform. Application of all the above to PDEs (with 2 or 3 special variables, as well as time).