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August 2010 - August 2012
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Publications (55)
УДК 517.5 Знайдено необхідні та достатні умови існування продовження через гладку криву для градієнтів функцій, які визначені та є бігармонічними функціями у відповідних областях, що межують з даною кривою. Навіть більше, знайдене продовження визначає градієнт бігармонічної функції в області, яка є об'єднанням зазначених областей та кривої.
New representations of solutions of Lamé system with real coefficients via monogenic functions in the biharmonic algebra are found.
УДК 517.54, 517.95 Розглядається кусково-неперервна бігармонічна задача у куті і відповідна їй крайова задача типу задачі Шварца для моногенних функцій у комутативній бігармонічній алгебрі. Вказані задачі редуковано до системи інтегральних рівнянь на додатній півпрямій. Показано, що на кожному відрізку цієї півпрямої множина розв'язків системи збіг...
We prove that any commutative and associative algebra 𝔹* of the second rank with unit over the field of complex numbers ℂ contains bases {e1, e2} for which 𝔹*-valued “analytic” functions Φ(xe1 + ye2), where x and y are real variables, satisfy a homogeneous partial differentional equation of the fourth order with complex coefficients whose character...
The statement that any two-dimensional algebra 𝔹* of the second rank with unity over the field of complex numbers contains such a basis {e1; e2} that 𝔹*-valued “analytic” functions Φ(xe1 + ye2) (x, y are real variables) satisfy such a fourth-order homogeneous partial differential equation with complex coefficients that its characteristic equation h...
The statement that any two-dimensional algebra $\mathbb{B}_{\ast}$ of the second rank with unity over the field of complex numbers contains such a basis $\{e_{1},e_{2}\}$ that $\mathbb{B}_{\ast}$-valued ''analytic'' functions $\Phi(xe_{1}+ye_{2})$ ($x$, $y$ are real variables) satisfy such a fourth-order homogeneous partial differential equation wi...
УДК 517.9 Доведено, що кожна комутативна й асоціативна алгебра другого рангу з одиницею над полем комплексних чисел містить такі базиси що -значні „аналітичні'' функції ( – дійсні змінні) задовольняють однорідне рівняння з частинними похідними четвертого порядку та комплексними коефіцієнтами, характеристичне рівняння якого має один подвійний корінь...
Among all two-dimensional algebras of
the second rank with unity $e$ over the field of complex numbers $\mathbb{C}$, we found
a semi-simple
algebra $\mathbb{B}_{0}:= \{c_1 e+c_2 \omega: c_k\in\mathbb{C}, k=1,2\},\,
\omega^2=e$, containing basеs $\{e_1,e_2\}$,
such that
$\mathbb{B}_{0}$-valued ``analytic'' functions $\Phi(xe_1+ye_2)$ ($x$, $y$ are...
A commutative algebra \( \mathbbm{B} \) over the complex field with a basis {e1, e2} satisfying the conditions \( {\left({e}_1^2+{e}_2^2\right)}^2=0,{e}_1^2+{e}_2^2\ne 0 \) is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type Φ (xe1+...
Among all two-dimensional commutative associative algebras of the second rank with unity over the field of complex numbers we want to find all pairs \((\mathbb {B}_{\ast }, \{e_1,e_2\})\), where \(\mathbb {B}_{\ast }\) is an algebra and {e
1, e
2} are its bases such that \(e_1^4+ 2p e_1^2 e_2^2 + e_2^4 = 0\) for every fixed p, − 1 < p < 1. This pro...
A commutative algebra $\mathbb{B}$ over the complex field with a basis $% \{e_{1},e_{2}\}$ satisfying the conditions $(e_{1}^{2}+e_{2}^{2})^{2}=0$, $% e_{1}^{2}+e_{2}^{2}\neq 0$ is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type $\...
УДК 517.5, 539.3 Серед двовимірних алгебр другого рангу з одиницею над полем комплексних чисел знайдено напівпросту алгебру що містить базиси такі, що -значні ,,аналітичні'' функції ( - дійсні змінні) задовольняють однорідне рівняння з частинними похідними четвертого порядку, яке має лише прості ненульові характеристики.Наведено повний опис множини...
We develop a hypercomplex method of solving of boundary value problems for biharmonic functions. This method is based on a relation between biharmonic functions and monogenic functions taking values in a commutative algebra associated with the biharmonic equation. We consider Schwarz-type boundary value problems for monogenic functions that have re...
We consider a class of plane orthotropic deformations of the form εx = σx + a12σy, γxy = 2(p − a12)Txy, εy = a12σx + σy, where σx, Txy, σy and\( {\upvarepsilon}_x\frac{\upgamma_{xy}}{2},{\upvarepsilon}_Y \) are components of the stress tensor and the deformation tensor, respectively, real parameters p and a12 satisfy the inequalities: -1 < p < 1, -...
Among all two-dimensional commutative algebras of the second rank a totally of all their biharmonic bases $\{e_1,e_2\}$, satisfying conditions $\left(e_1^2+ e_2^2\right)^{2} = 0$, $e_1^2 + e_2^2 \ne 0$, is found in an explicit form. A set of "analytic" (monogenic) functions satisfying the biharmonic equation and defined in the real planes generated...
We consider a class of plane orthotropic deformations of the form \(\varepsilon_{x} = \sigma_x + a_{12} \sigma_y\), \(\gamma_{xy} = 2 \left(p-a_{12}\right) \tau_{xy}\), \(\varepsilon_{y}= a_{12}\sigma_x+\sigma_y\), where \(\sigma_x\), \(\tau_{xy}\), \(\sigma_y\) and \(\varepsilon_{x}\), \(\frac{\gamma_{xy}}{2}\), \(\varepsilon_{y}\) are components...
The author’s name should read S. V. Gryshchuk.
For an algebra B0≔c1e+c2ω:ck∈ℂk=12, e2 = ω2 = e, eω = ωe = ω, over the field of complex numbers ℂ, we consider arbitrary bases (e, e2) such that e+2pe22+e24=0 for any fixed p > 1. We study B0-valued “analytic” functionsΦxe+ye2=U1xye+U2xyie+U3xye2+U4xyie2such that their real-valued components Uk,k=1,4¯, satisfy the equation for the stress function u...
A solution of the elliptic type PDE of the 4th order, being a reduction of the Eqs. of stress function corresponding to any case of plane anisotropy which is not equal to isotropy (proved by S.\,G.~Mikhlin), is described in terms of hypercomplex `analytic' functions with values in two-dimensional semisimple algebra over the field of complex numbers...
We consider Schwartz-type boundary value problems for monogenic functions in a commutative algebra \(\mathbb {B}\) over the field of complex numbers with the bases {e1, e2} satisfying the conditions \((e_1^2+e_2^2)^2=0\), \(e_1^2+e_2^2\ne 0\). The algebra \(\mathbb {B}\) is associated with the biharmonic equation, and considered problems have relat...
We consider a commutative algebra B over the field of complex numbers with a basis {e 1 , e 2 } satisfying the conditions (e12+e22)2=0, e12+e22≠0. We consider a Schwarz-type boundary value problem for “analytic” B-valued functions in a simply connected domain. This problem is associated with BVPs for biharmonic functions. Using a hypercomplex analo...
Among all two-dimensional commutative and assosiative algebras of the second rank with the unity \(e\) over the field of complex numbers \(\mathbb{C}\) we find a semi-simple algebra \(\mathbb{B}_{0} := \{c_1 e+c_2 \omega: c_k\in\mathbb{C}, k=1,2\}\), \(\omega^2=e\), containing a basis \((e_1,e_2)\), such that \( e_1^4 + 2p e_1^2 e_2^2 + e_2^4 = 0 \...
Among all two-dimensional algebras of the second rank with unity e over the field of complex numbers ℂ, we find a semisimple algebra 𝔹0: ={c1e + c2ω : ck ∈ ℂ, k = 1, 2}, ω² = e, containing bases (e1, e2) such that \( {e}_1^4+2p{e}_1^2{e}_2^2+{e}_2^4=0 \) for any fixed p > 1. A domain {(e1, e2)} is described in the explicit form. We construct 𝔹0-val...
We consider a commutative algebra 𝔹 over the field of complex numbers with a basis {e1, e2} satisfying the conditions
(e12+e22)2=0,e12+e22≠0. $ (e_{1}^{2}+e_{2}^{2})^{2}=0, e_{1}^{2}+e_{2}^{2}\neq 0. $ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic 𝔹-valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i...
A commutative algebra $\mathbb{B}$ over the field of complex numbers with the bases $\{e_1,e_2\}$ satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$, is considered. The algebra $\mathbb{B}$ is associated with the biharmonic equation. Consider a Schwartz-type boundary value problem on finding a monogenic function of the type $\Phi(xe_...
A commutative algebra $\mathbb{B}$ over the field of complex numbers with the bases $\{e_1,e_2\}$ satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$, is considered. The algebra $\mathbb{B}$ is associated with the biharmonic equation. Consider a Schwartz-type boundary value problem on finding a monogenic function of the type $\Phi(xe_...
Consider the commutative algebra $\mathbb{B}$ over the field of complex
numbers with the bases $\{e_1,e_2\}$ such that $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$.
Let $D$ be a domain in $xOy$, $D_{\zeta}:=\{xe_1+ye_2:(x,y) \in D\}\subset
\mathbb{B}$. We say that $\mathbb{B}$-valued function $\Phi \colon D_{\zeta}
\longrightarrow \mathbb{B}$, $\Phi(\z...
Considered a boundary value problem (BVP) for monogenic functions of bi\-har\-mo\-nic
variable taking values in a two-dimensional commutative Banach algebra. This BVP is associated with
the main biharmonic problem for biharmonic functions of two real variables. Developing a
reduction's scheme for this BVP for monogenic functions to BVP in a disk by...
Одержано вирази для розв’язків системи рівнянь рівноваги Ляме у зміщеннях через компоненти гіперкомплексних моногенних функцій бігармонічної змінної. Знайдено опис усіх моногенних функцій, що мають однією з дійсних компонент дану бігармонічну функцію, асоційовану з розв’язком задачі Фламана для ізотропної півплощини.
We consider a commutative algebra B over the field of complex
numbers with a basis {e_1, e_2} satisfying the conditions (e_1^{2}+e_2^{2})^2=0, e_1^2+e_2^2 ><0. Let D be a bounded domain in the Cartesian plane xOy and Dζ = {xe_1 +ye_2 : (x, y) ∈ D}. Components of every monogenic function Φ(xe_1 + ye_2) =U_{1}(x, y) e_1+U_{2}(x, y) ie_1+U_{3}(x, y) e...
We consider a commutative algebra $\mathbb{B}$ over the field of complex
numbers with a basis $\{e_1,e_2\}$ satisfying the conditions
$(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$. Let $D$ be a bounded domain in the
Cartesian plane $xOy$ and $D_{\zeta}=\{xe_1+ye_2 : (x,y)\in D\}$. Components of
every monogenic function $\Phi(xe_1+ye_2)=U_{1}(x,y)\,e_1+U_...
Let be a bounded open domain of . Let denote the outward unit normal of . We assume that the Steklov problem Δu = 0 in and on has a simple eigenvalue of . Then we consider an annular domain obtained by removing from a small-cavity size of ε > 0, and we show that under proper assumptions there exists a real valued and real analytic function defined...
A boundary value problem (BVP) for monogenic functions of biharmonic variable taking values in a two-dimensional commutative Banach algebra is considered. This BVP is associated with the main biharmonic problem for biharmonic functions of two real variables. A reduction scheme for this BVP for monogenic functions to BVP in a disk by using of expans...
For 2D bounded composite material geometrically composed by a disk of variable radius r and an outer ring it is determined in an analytic form the x-component of the effective conductivity tensor. Namely, it is shown that the x-component is a sum of geometrical progression with respect to powers of r 2 for all sufficiently small r.
We consider monogenic functions given in a biharmonic plane and taking values in a commutative algebra associated with the biharmonic equation. For the mentioned functions, we establish basic properties analogous to properties of holomorphic functions of the complex variable: the Cauchy integral theorem and integral formula, the Morera theorem, the...
Let Io be a bounded open domain of Rn. Let
νIo denote the outward unit normal to ∂Io.
We assume that the Steklov problem Δu = 0 in Io,
∂u∂νIo = λu on ∂Io has a
simple eigenvalue ~λ. Then we consider an annular domain
A(ɛ) obtained by removing from Io a small cavity of
size ɛ > 0, and we show that under proper assumptions there
exists a real valued...
We consider a two-dimensional commutative algebra B over the field of complex
numbers. The algebra B is associated with the biharmonic equation. For
monogenic functions with values in B, we consider a Schwartz-type boundary
value problem (associated with the main biharmonic problem) for a half-plane
and for a disk of the biharmonic plane. We obtain...
Let ${\mathbb{I}}^{o}$ be a bounded open domain of
${\mathbb{R}}^{n}$. Let $ \nu_{ {\mathbb{I}}^{o} }$ denote the outward unit normal to $\partial{\mathbb{I}}^{o}$.
We assume that the Steklov problem $ \Delta u=0 $ in ${\mathbb{I}}^{o}$,
$\frac{\partial u}{\partial \nu_{ {\mathbb{I}}^{o} } }=\lambda u$ on $\partial {\mathbb{I}}^{o}$ has a simple ei...
The idea of an algebraic-analytic approach to equations of mathematical physics means to find commutative Banach algebras such that monogenic functions defined on them form an algebra and have components satisfying previously given equations with partial derivatives. We obtain constructive descriptions of monogenic functions taking values in commut...
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 doma...
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 compl...
For investigation of elliptic type equations degenerating on an axis we develop a method analogous to the method of analytic function in the complex plane. We have obtained expressions of generalized axial-symmetric potentials via components of analytic functions taking values in a commutative associative Banach algebra.
