Syllabus
1. Laplace transform of functions Definition and basic properties. Inversion theorems, application to intial promblems in ODEs.
2. Special functions Gamma and beta funcions, Bessel functions. Gauss integration, hypergeometrical series.
3. Theory of distributions Distributions, tempered distributions, (Dirac, vp and Pf distributions). Distributional calculus (multiplication by a smooth function, tensor product, convolution, differentiation, linear transformation). Convergence of distributions, distributions with parameter, Fourier and Laplace transform of distributions and its applications: derivative, convolution, tensor product. Convolution equations, fundamental solution. Fourier transform of periodical functions and distributions, Fourier series of periodical distributions.
4. Applications of theory of distributions Laplace-Poisson equation:uniqueness, existence, Liouville theorem. Theorem of three potentials. Dirichlet problem and its solution. Use of conformal mappings to obtain solution in two dimensional domain. Heat equation: fundamental solutions, solutions with data. Heat waves, cooling of the ball. The wave equation: fundamental solutions, solutions with data.
Annotation
This one-semestral course is a continuation of the basic two year course on analysis and linear algebra for physicists.