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Registration Number

0215U003273, 0112U000374 , R & D reports

Title

Monogenic and hyperholomorphic functions in finite-dimensional algebras and their application to the boundary problems of mathematical physics

Head

Gerus Oleg Fedorovych, Кандидат фізико-математичних наук

Registration Date

22-01-2015

Organization

Zhytomyr State University named after Ivan Franko

Description

We proved an analog of the integral Cauchy theorem for hyperholmorphic functions acting from a three-dimensional domain with nonsmooth boundary into any finite-dimensional commutative associative algebra. We obtained a constructive description for monogenic functions acting into a finite-dimensional algebra with unit and the radical of the maximal dimention and we proved analoges of the integral Cauchy theorem, the integral Cauchy formula, Morera theorem and Loran expansion. We state a correct set up of the Schwarz-type boundary value problem for monogenic functions of biharmonic variable acting into a biharmonic algebra associated with the main biharmonic problem. We constracted a method of reduction of Schwarz-type boundary value problem for monogenic functions in a simply connected domain to the corresponding boundary value problem for monogenic functions in a unit disk of the biharmonic plane. We find decisive formulas for solution of the Schwarz-type problem in certain type domais. We proved the analytic property for eigenvalues and eigenfunctions of one-point perturbation of a simply connected domain in a finite-dimentional space for the Sveklov problem under the assumption on eigenvalue to be a small varianced on the simply eigenvalue. For monogenic functions acting into a finite-dimensional semisimple commutative algebra we proved analoges of the Cauchy integral theorem for the curvelinar integral, Morera theorem, the Cauchy integral formula, Ostrogradski-Gauss formula and the Cauchy integral theorem for the surface integral. We proved an upper estimate for the continuity module of boundary values of the quaternionic Cauchy-type integral.

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Грищук Сергій Вікторович
Плакса Сергій Анатолійович
Пухтаєвич Роман Петрович
Шпаківський Віталій Станіславович