Vitalii S. Shpakivskyi’s scientific contributions

We prove that a locally bounded and differentiable in the sense of Gateaux function given in a finite-dimensional commutative Banach algebra over the complex field is also differentiable in the sense of Lorch.

In this paper, an analog of the conformable fractional derivative is defined in an arbitrary finite-dimensional commutative associative algebra. Functions taking values in the indicated algebras and having derivatives in the sense of a conformable fractional derivative are called φ-monogenic. A relation between the concepts of φ-monogenic and monogenic functions in such algebras has been established. Two new definitions have been proposed for the fractional derivative of the functions with values in finite-dimensional commutative associative algebras.

In this chapter, for monogenic functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that is obtained in the previous chapter and yields analogs of Cauchy–Riemann conditions and the continuity of Gâteaux derivatives of all orders for monogenic functions. In such a way, analogs of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved. As a result, different equivalent definitions for the mentioned monogenic functions are established.

A spatial potential solenoidal vector field symmetric with respect to the axis Ox is described in a meridian plane xOr in terms of the axial-symmetric potential φ and Stokes’ flow function ψ that satisfy the system of equations r∂φ(x, r)∕∂x = ∂ψ(x, r)∕∂r, r∂φ(x, r)∕∂r = −∂ψ(x, r)∕∂x degenerating on the axis Ox. One of the ways of solving boundary-value problems for axial-symmetric potentials and Stokes’ flow functions is based on the integral representations for the function φ and ψ. In this chapter, we obtain integral representations for an axial-symmetric potential and Stokes’ flow function in an arbitrary simply connected domain symmetric with respect to the axis Ox in the meridian plane via the holomorphic functions of a complex variable.

We develop a functionally analytic method for effective solving boundary problems in a meridian plane of a spatial axial-symmetric potential field. This method is based on integral expressions for the axial-symmetric potential and Stokes’ flow function in an arbitrary simply connected domain symmetric with respect to an axis that are presented in Chap. 18. In such a way, in this chapter, we develop a method of transition from the Dirichlet boundary problem for axial-symmetric potential in a bounded domain to the Cauchy singular integral equation on the real axis. In the case that is important for applications where the boundary is a smooth curve satisfying certain additional requirements, the mentioned singular integral equation is reduced to the Fredholm integral equation of the second kind. In the case where the boundary is a circle, the solution of the Dirichlet boundary problem is explicitly obtained.

In finite-dimensional commutative associative algebra, the concept of σ-monogenic function is introduced. Necessary and sufficient conditions for σ-monogeneity have been established. In some low-dimensional algebras, with a special choice of σ, the representation of σ-monogenic functions is obtained using holomorphic functions of a complex variable. We proposed the application of σ-monogenic functions with values in two-dimensional biharmonic algebra to representation of solutions of two-dimensional biharmonic equation.