Vekua Type Research Articles

Toward the theory of semi-linear Beltrami equations

We study the semi-linear Beltrami equation $\omega_{\bar{z}}-\mu(z) \omega_z=\sigma(z)q(\omega(z))$ and show that it is closely related to the corresponding semi-linear equation of the form ${\rm div} A(z)\nabla\,U(z)=G(z) Q(U(z)).$ Applying the theory of completely continuous operators by Ahlfors-Bers and Leray-Schauder, we prove existence of regular solutions both to the semi-linear Beltrami equation and to the given above semi-linear equation in the divergent form, see Theorems 1.1 and 5.2. We also derive their representation through solutions of the semi-linear Vekua type equations and generalized analytic functions with sources. Finally, we apply Theorem 5.2 for several model equations describing physical phenomena in anisotropic and inhomogeneous media.

Hilbert and Poincaré problems for semi-linear equations in rectifiable domains

The study of the boundary value problem with arbitrary measurable data originated in the dissertation of Luzin where he investigated the Dirichlet problem for harmonic functions in the unit disk. Recently, in \cite{R7}, we studied the Hilbert, Poincaré and Neumann boundary value problems with arbitrary measurable data for generalized analytic and generalized harmonic functions and provided applications to relevant problems in mathematical physics. The present paper is devoted to the study of the boundary value problem with arbitrary measurable boundary data in a domain with rectifiable boundary corresponding to semi-linear equation with suitable nonlinear source. We construct a completely continuous operator and generate nonclassical solutions to the Hilbert and Poincaré boundary value problems with arbitrary measurable data for Vekua type and Poisson equations, respectively. Based on that, we prove the existence of solutions of the Hilbert boundary value problem for the nonlinear Vekua type equation with arbitrary measurable data in a domain with rectifiable boundary. It is necessary to point out that our approach differs from the classical variational approach in PDE as it is based on the geometric interpretation of boundary values as angular (along non-tangential paths) limits. The latter makes it possible to also obtain a theorem on the boundary value problem for directional derivatives, and, in particular, of the Neumann problem with arbitrary measurable data for the Poisson equation with nonlinear sources in any Jordan domain with rectifiable boundary. As a result we arrive at applications to some problems of mathematical physics.

Infinitesimal bendings for classes of two-dimensional surfaces

ABSTRACT Infinitesimal bendings for classes of two-dimensional surfaces in R 3 are investigated. The techniques used to construct the bending fields include reduction to solvability of Bers–Vekua type equations and systems of differential equations with periodic coefficients.

Hilbert problem with measurable data for semilinear equations of the Vekua type

We prove the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the nonlinear equations of the Vekua’s type ∂z f (z ) = h (z )q(f (z )). The found solutions differ from the classical ones, because our approach is based on the notion of boundary values in the sense of angular limits along nontangential paths. The results obtained can be applied to the establishment of existence theorems for the Poincaré and Neumann boundary-value problems for the nonlinear Poisson equations of the form ΔU (z )=H (z )Q(U (z )) with arbitrary measurable boundary data with respect to the logarithmic capacity. They can be also applied to the study of some semilinear equations of mathematical physics modeling such processes as the diffusion with absorption, plasma states, stationary burning etc. in anisotropic and inhomogeneous media.

On boundary-value problems for semi-linear equations in the plane

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Hölder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

On boundary-value problems for semi-linear equations in the plane

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk $\mathbb{D}$ is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with H\"{o}lder continuous data for generalized analytic functions, i.e., continuous complex-valued functions $f(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial _{\bar{z}}f\,+\,af\,+\ b{% \overline{f}}\,=\,c\,$, where the complex-valued functions $a,b$, and $c$ are assumed to belong to the class $L^{p}$ with some $p>2$ in smooth enough domains $D$ in $\mathbb{C}$. Our last paper \cite{GNRY} contained theorems on the existence of nonclassical solutions of the Hilbert boundary-value problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions $f:D\to\mathbb{C}$ such that $\partial_{\bar z}f = g$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincar\'e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G\in L^p, p>2$, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article \cite{GNRY} and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $\partial _{\bar{z}% }f(z)=h(z)q(f(z))$. On this basis, existence theorems were also established for the Poincar\'{e} boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form $\triangle \,U(z)=H(z)Q(U(z))$ with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Use of the higher-order plate theory of I. N. Vekua type in problems of dynamics of heterogeneous plane waveguides

The dynamics of elastic plane waveguides is studied on the basis of the extended formulation of the plate theory of N th order. The plate model is based on the Lagrangian formalism of analytical dynamics combined with the dimensional reduction approach and the biorthogonal expansion of the spatial distribution of the displacement. The boundary conditions shifted from the faces onto the base plane are interpreted as constraints for the variational formulation of two-dimensional plate models. The normal wave dispersion in plates is modelled, the convergence of the approximate solutions is studied using the known exact solution for a plane layer as a reference. The proposed plate theory is used to analyse the normal wave dispersion in power graded waveguides of both symmetric and asymmetric structures, the locking phase frequencies for various power indices are computed.

Modeling of the dynamics of plane functionally graded waveguides based on the different formulations of the plate theory of I. N. Vekua type

The dynamics of elastic functionally graded waveguides with symmetric through-thickness structure is considered on the background of the new formulation of the Nth-order plate theory. The general plate model is based on the Lagrangian formalism of analytical dynamics and the dimensional reduction approach combined with the biorthogonal expansion technique applied to the spatial distribution of the displacement vector field. The use of compact basis is shown, the normal wave propagation in a plane waveguide is modeled, and the convergence is analyzed using the isotropic homogeneous plate as a test example. Finally, the same formulation is used to model the normal wave dispersion in functionally graded plates. The proposed method is a base for the investigation of the effect of uncertainties and for the solution of inverse coefficient problems for functionally graded thin-walled structures.

On the equations of the analytical dynamics of the quasi-3D plate theory of I. N. Vekua type and some their solutions

The plate theory of Nth order is constructed on the background of the Lagrangian variational formalism of analytical dynamics of continuum systems and the dimensional reduction approach of I. N. Vekua – A. A. Amosov. The plate model is defined within the configuration space, the set of field variables, and the Lagrangian density. The field variables are determined by the coefficients of the biorthogonal expansion of the spatial displacement vector field with respect to the dimensionless normal coordinate. The dynamic equations are derived as Lagrange equations of the second kind of the two-dimensional continuum system. The dynamics of the plane elastic layer is considered as an example, the normal wave propagation is described on the basis of refined plate theories of various orders, and the convergence of approximate solutions to the exact solution of the three-dimensional elastodynamics problem is analyzed for different wavenumbers.

A Cauchy kernel for the Hermitian submonogenic system

Hermitian monogenic functions are the null solutions of two complex Dirac type operators. The system of these complex Dirac operators is overdetermined and may be reduced to constraints for the Cauchy datum together with what we called the Hermitian submonogenic system (see [8], [9]). This last system is no longer overdetermined and it has properties that are similar to those of the standard Dirac operator in Euclidean space, such as a Cauchy–Kowalevski extension theorem and Vekua type solutions. In this paper, we investigate plane wave solutions of the Hermitian submonogenic system, leading to the construction of a Cauchy kernel. We also establish a Stokes type formula that, when applied to the Cauchy kernel provides an integral representation formula for Hermitian submonogenic functions.

Vekua type hierarchical models for prismatic shells with mixed conditions on face surfaces

I. Vekua constructed hierarchical models for elastic prismatic shells, in particular, plates of variable thickness, when on the face surfaces either stresses or displacements are known. In the present paper other hierarchical models for cusped, in general, elastic isotropic and anisotropic prismatic shells are constructed and analyzed, namely, when on the face surfaces (i) a normal to the projection of the prismatic shell component of a stress vector and parallel to the projection of the prismatic shell components of a displacement vector, (ii) a normal to the projection of the prismatic shell component of the displacement vector and parallel to the projection of the prismatic shell components of the stress vector are prescribed. We construct also hierarchical models, when other mixed conditions are given on face surfaces. In the zero approximations of the models under consideration peculiarities (depending on sharpening geometry of the cusped edge) of correct setting boundary conditions at edges are investigated. In concrete cases some boundary value problems are solved in an explicit form. As an example of application of the constructed Vekua-type models to composite structures, an unidirectional lamina with fibers parallel to x2-axis under shear strain is considered. Tension–compression is treated as well.

О ВАРИАЦИОННЫХ УРАВНЕНИЯХ РАСШИРЕННОЙ ТЕОРИИ N-ГО ПОРЯДКА УПРУГИХ ОБОЛОЧЕК И ИХ ПРИЛОЖЕНИИ К НЕКОТОРЫМ ЗАДАЧАМ ДИНАМИКИ

One of topical problems of thick shells theory consists in an accurate modelling of high-frequency shell vibrations and wave propagation, as well as in the application of developed mathematical models to engineering problems of structural dynamics. Many observed processes of both periodic and non-steady shell dynamics cannot be adequately simulated on the groundwork of most traditional hypothesis-based shell theories. Therefore the engineering practice requires new shell models considering higher degrees of freedom besides the displacement and rotation vectors and taking into account the interference of tangential and transversal motions. Here a variant of the N’th order shell model of I.N. Vekua type is analyzed. The proposed model improves the A. Amosov’s shell theory; it is based on the Lagrange’s formalism of analytical mechanics extended to continuum systems, on the dimensional reduction approach, and on the biorthogonal expansion technique. The shell is finally represented by a material surface framed by a set of generalized coordinates (or field variables) and a scalar generator function – the surficial density of the Lagrangian. The dynamic equations for the shell are formulated as generalized Lagrange equations of the second kind. A propagation of normal waves in a plane elastic layer is considered, the corresponding homogeneous problem of N’th order plate theory is formulated, and its solution analysis is extended. In particular, the investigation of the second longitudinal propagating mode and its modelling by approximated plate theories are expanded. The main feature of the second mode is the “inverse wave” effect, i.e. the opposite signs of the phase and group velocities at small wavenumbers. It was shown earlier that the accurate dispersion curves for both phase frequency and group velocity can be obtained on the basis of the fourth- and fifth-order theories. Using the known solution of the spectral problem of N’th order plate theory the eigenfunctions are obtained. The second propagating mode evaluation at the smallest wavenumbers is modelled, the forms of the second propagating wave are investigated, and the approximation of the waveforms at some wavenumbers by plate theories of various orders is estimated. The accuracy of the approximation of the “inverse wave” by some applied theories of various orders based on the orthogonal expansions of displacement vector is discussed. © PNRPU

Solutions to a class of polynomially generalized Bers–Vekua equations using Clifford analysis

In this paper a class of polynomially generalized Vekua–type equations and of polynomially generalized Bers–Vekua equations with variable coefficients defined in a domain of Euclidean space are discussed. Using the methods of Clifford analysis, first the Fischer–type decomposition theorems for null solutions to these equations are obtained. Then we give, under some conditions, the solutions to the polynomially generalized Bers–Vekua equation with variable coefficients. Finally, we present the structure of the solutions to the inhomogeneous polynomially generalized Bers–Vekua equation.

Representation of solutions of a singular Cauchy–Riemann equation in the plane

This article deals with the Bers–Vekua type equations with a singular point. Series representations of solutions and Green functions are constructed. A generalized Cauchy formula and its consequences are obtained.

On the Solution of Discrete Vekua Equations

We consider finite difference equations of Vekua type. Main goal of the paper is to prove a representation formula for the solution of homogeneous equations in the form of a product with one factor being a discrete holomorphic function.

Discrete Vekua equations with constant coefficients in the complex and quaternionic case

A discrete version of complex Vekua type equations is considered. We construct a repre\-sentation formula for the solution of the homogeneous complex equation and investigate the inhomogeneous equation. Similar to the continuous case this representation formula is a product of two functions. We also factorize the solution of a homogeneous Vekua type equation in the quaternionic case. In the complex plane we analyse both factors in detail and study the relation between the non-holomorphic factors in the discrete and continuous case.

Linear Generalized Analytic Functions

Complex-valued continuous functions are approximated using solutions of complex partial differential equations of Vekua type.

On the Representation of Generalized Analytic Vectors

This paper is concerned with the representation of generalized analytic vectors which are defined as solutions of certain elliptic systems of linear first-order partial differential equations in two complex variables. For the explicite representation of solutions we give integral operators of Vekua type as well as differential operators of Bauer type which map holomorphic vectors onto the set of solutions. For the existence of the differential operators we give a necessary and sufficient condition. We discuss properties of these operators and show how to construct the kernels of the integral operators using the differential operators. Finally we present examples for systems of differential equations for which all the solutions can be represented explicitely.

Boundary Value Problems for Partial Differential Equations in the Plane Revisited

AbstractBoth boundary value problems the DIRICHLET and the RIEMANN‐HILBERT problems, were solved by the author in the SOBOLEV space W1,p(D), 2 < p < ∞, for the elliptic differential equation in [8], [11] and [12]. In fact in most of the literature so far it is the case of p > 2 only which has been studied extensively. The results obtained so far relied on the fact that the VEKUA‐type operator TD maps the LEBESGUE space Lp(D) Into the HÖLDER space (D̄) for α= (p–2)/p. By modifying both VEKUA type singular integral operators TD and πD it is shown here that we can then solve both boundary value problems in W1,p(D) for 1 < p < ∞. While enlarging the set of admissible SOBOLEV spaces, the set of partial differential equations that can then be solved is reduced through stricter conditions on the LIPSCHITZ constants.

Optimal control with semilinear first order complex differential equations of I. N. vekua's type

Optimal control with semilinear first order complex differential equations of I. N. vekua's type