Course information

Fourier Series: Trigonometric series, Convergence theorems, special type Fourier expansions, generalized Fourier series, orthogonal coordinate systems, complete systems, Bessel inequality.  Boundary Value Problems: Linear boundary value problems, eigenfunctions and eigenvalues,, Sturm-Liouville problems, non-homigeneous problems.  Introduction to Partial Differential Equations: Fundamental notions, Classification of second order semi-linear differential equations.  Laplace equation: The Dirichlet and Newmann boundary value problems. Compatibility condition. The separation of variables technique in Cartesian, polar, cylindrical and spherical coordinates. The non-homogeneous problem. Helmholtz equation.  Heat equation: Initial-Boundary value problems for bounded domains, the no-homogeneous diffusion problem.  Wave equation: Initial- Boundary value problems, the infinite length string, D’Alembert solution, the circular drum problem.  Integral transformations: Fourier transform, Sine and Cosine Fourier transforms, Hankel transform, application of integral transforms to the solution of initial-boundary value problems. Use of computational software for the study of problems arising in partial differential equations.