The course provides an introduction to the theory of distributions with applications to solutions of partial differential equations with constant coefficients, further the Fredholm theorems are discussed for the case of a continuous kernel on a compact set as well as Sturm-Liouville operators on bounded intervals, and applications of the separation of variables method to the solution of some boundary value problems and mixed problems.
1. Definition of spaces of distributions and basic operations, periodic distributions, tensor product and convolution. 2. Tempered distributions and the Fourier transformation. 3. The generalized Laplace transformation. 4. The Fredholm theorems for integral operators with continuous kernels on a compact set. 5. Elliptic operators, Sturm-Liouville operators on a bounded interval, the Green function. 6. Solutions of a boundary value problem for the Laplace equation on a symmetric domain. 7. Solutions of a mixed problem by the separation of variables method.
Distribution, fundamental solution, the Fourier transformation, the Laplace transformation, the heat equation, the wave equation, integral equation, Sturm-Liouville operator, boundary-value problem, mixed problem.