# Gryshchuk S. V.

### Commutative сomplex algebras of the second rank with unity and some cases of the plane orthotropy. II

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1382-1389

For an algebra $B_0 = \{ c_1e + c_2\omega : c_k \in C, k = 1, 2\} , e_2 = \omega 2 = e, e\omega = \omega e = \omega$, over the field of complex numbers $C$, we сonsider arbitrary bases $(e, e_2)$, such that$e + 2pe^2_2 + e^4_2 = 0$ for any fixed $p > 1$. We study $B_0$ -valued “analytic” functions $\Phi (xe+ye_2) = U_1(x, y)e + U_2(x, y)ie + U_3(x, y)e_2 + U_4(x, y)ie_2$ such that their real-valued components $U_k, k = 1, 4$, satisfy the equation for the stress function $u$ in the case of orthotropic plane deformations $$\biggl( frac{\partial^4}{\partial x^4} + 2p\frac{\partial^4}{\partial x^2 \partial y^2} + \frac{\partial^4}{\partial y^4}\biggr)u(x, y) = 0,$$ here, $x$ and $y$ are real variables. All functions $\Phi$ for which $U_1 \equiv u$ are described in the case of a simply connected domain. Particular solutions of the equilibrium system of equations in displacements are found in the form of linear combinations of the components $U_k , k = 1, 4$, of the function $\Phi$ for some plane orthotropic media.

### Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. I

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1058-1071

Among all two-dimensional algebras of the second rank with unity $e$ over the field of complex numbers $C$, we find a semisimple algebra $B_0 = \{ c_1e + c_2\omega: c_k \in C, k = 1, 2\} , \omega^2 = e$, containing bases $(e_1, e_2)$, such that $e^4_1 + 2pe^2_1e^2_2 + e^4_2 = 0$ for every fixed $p > 1$. A domain $\{ (e1, e2)\}$ is described in the explicit form. We construct $B_0$ -valued “analytic” functions $\Phi$ such that their real-valued components satisfy the equation for the stress function $u$ in the case of orthotropic plane deformations $$\biggl(\frac{\partial^4}{\partial x^4} + 2p \frac{\partial^4}{\partial x^2 \partial y^2} + \frac{\partial^4}{\partial y^4}\biggr) u(x, y) = 0,$$ where $x, y$ are real variables.

### Monogenic functions in a biharmonic algebra

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1587-1596

We present a constructive description of monogenic functions that take values in a commutative biharmonic algebra by using analytic functions of complex variables. We establish an isomorphism between algebras of monogenic functions defined in different biharmonic planes. It is proved that every biharmonic function in a bounded simply connected domain is the first component of a certain monogenic function defined in the corresponding domain of a biharmonic plane.

### Integral representations of generalized axially symmetric potentials in a simply connected domain

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 160-177

We obtain integral representations of generalized axially symmetric potentials via analytic functions of a complex variable that are defined in an arbitrary simply connected bounded domain symmetric with respect to the real axis. We prove that these integral representations establish a one-to-one correspondence between analytic functions of a complex variable that take real values on the real axis and generalized axially symmetric potentials of certain classes.