Malanchuk O. The differential-symbol method of solving the two-point problems for partial differential equations

The dissertation is devoted to investigation of solvability of the problems with local two-point in time conditions which contain differential polynomials, for partial differential equations in the spaces of entire functions. The kernel of the problem for partial differential equation of second order in time variable in which local two-point conditions are given and generally infinite order in spatial variables is investigated. Sufficient conditions of existence of the nontrivial solutions in the classes of entire functions, and also necessary and sufficient conditions of existence of nontrivial quasipolynomial solutions are established. The differential-symbol method of constructing the solutions is proposed. The conditions of existence only trivial solution of the problem are found. The conditions of existence and uniqueness of the solution of the problem for homogeneous partial differential equation of second order in time and generally infinite order in spatial variables with nonhomogeneous local two-point conditions, and also the problem for nonhomogeneous partial differential equation of second order in time and generally infinite order in spatial variables with homogeneous local two-point conditions in the classes of entire functions, continuously differentiable functions and in classes of quasipolynomials are established. The differential-symbol method of constructing the solutions of the problem is proposed. The problem with local two-point conditions for partial differential equation of second order in time variable and generally infinite order in spatial variables when the characteristic determinant of the problem identically equals to zero is investigated by differential-symbol method. The conditions of existence and nonexistence solution of the problem are found. In the case of existence of solutions of the problem the method of their construction is proposed.