Nytrebych Z. Differential-symbol method for solving problems for partial differential equations

The present dissertation deals with the differential-symbol method (DSM) for solving Cauchy problem, multipoint problems, non-local, integral boundary value problems as well as Dirichlet type problem for partial differential equations with constant and time-dependent coefficients. Firstly, we specify the conditions of univalent solvability of the problems, then we construct their solutions in the classes of entier functions with some restrictions on their order as well as in infinite order Sobolev spaces. The solutions of those problems are represented as actions of differential expressions onto certain entire or meromorphic functions of parameters. The parameters are assumed to be zeros after action of differential expressions. We investigate the null sets of some problems. We establish the interrelation between the representations of solutions of the same problem that have been obtained by means of the DSM, the operator method and the Fourier transform method. We obtain formulas for the limit passage from the solution of multipoint problem to the one of Cauchy problem, from the solution of multipoint problem to the one of Dirichlet type problem in a layer, from the solution of multipoint problem to the one of multipoint problem with smaller number of nodes. DSM is transferred to the case of problems for ordinary differential equations as well as for the evaluation of entier functions of matrix argument.