Analytical methods in differential equations – Syllabus


Sturm’s theory. Proof of the 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, and 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).

At the end of the course the student will be able to

At the end of the course, the students will be able to

  • Obtain the solvability condition for an in-homogeneous linear ODE with boundary conditions.
  • Obtain and use Green’s function for ODEs and PDEs.
  • Obtain solutions to PDEs using generalized Fourier series, using special functions, and being acquainted with their properties.
  • Obtain solutions for ODEs and PDEs using integral transforms, using special functions.