Category: Uncategorized

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).

• Obtain solvability conditions for a non-homogeneous ODE with boundary conditions.
• Obtain and use the Green function for ODEs and PDEs
• Obtain solutions to PDEs using generalized Fourier series and special functions.
• Obtain solutions to ODEs and PDEs using integral transforms special functions.

1. Single variable and multi variable calculus
2. A first course in ODEs (mandatory)
3. A first course in PDEs (recommended)
4. Complex functions

1. A four-hour lecture course which provides four academic points.
2. A weekly homework assignment will be provided.
3. There will be a final exam.
4. One of the questions at the final exam will be similar to one of the questions in the homework.
5. The lecturer will prepare formula sheets for use at the final exam. These formula sheets will be available on the course website throughout the semester.
6. A large collection of old exam will be available for students on the course website.