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Vladislav V. Kravchenko

Publications and source records attributed to Vladislav V. Kravchenko.

At least 73 records · Page 4Linked to original sources
arXiv2011

Spatial pseudoanalytic functions arising from the factorization of linear second order elliptic operators

Biquaternionic Vekua-type equations arising from the factorization of linear second order elliptic operators are studied. Some concepts from classical pseudoanalytic function theory are generalized onto the considered spatial case. The derivative and antiderivative of a spatial pseudoanalytic function are introduced and their applications to the second order elliptic equations are considered.

math.CV↗
arXiv2011

Complete systems of recursive integrals and Taylor series for solutions of Sturm-Liouville equations

Consider an arbitrary complex-valued, twice continuously differentiable, nonvanishing function $ϕ$ defined on a finite segment $[a,b]\subset \mathbb{R}$. Let us introduce an infinite system of functions constructed in the following way. Each subsequent function is a primitive of the preceding one multiplied or divided by $ϕ$ alternately. The obtained system of functions is a generalization of the system of powers ${(x-x_{0}%)^{k}}_{k=0}^{\infty}$. We study its completeness as well as the completeness of its subsets in different functional spaces. This system of recursive integrals results to be closely related to so-called $L$-bases arising in the theory of transmutation operators for linear ordinary differential equations. Besides the results on the completeness of the system of recursive integrals we show a deep analogy between the expansions in terms of the recursive integrals and Taylor expansions. We prove a generalization of the Taylor theorem with the Lagrange form of the remainder term and find an explicit formula for transforming a generalized Taylor expansion of a function in terms of the recursive integrals into a usual Taylor expansion. As a direct corollary of the formula we obtain the following new result concerning solutions of the Sturm-Liouville equation. Given a regular nonvanishing complex valued solution $y_{0}$ of the equation $y^{\prime\prime}+q(x)y=0$, $x\in(a,b)$, assume that it is $n$ times differentiable at a point $x_{0}% \in\lbrack a,b]$. We present explicit formulas for calculating the first $n$ derivatives at $x_{0}$ for any solution of the equation $u^{\prime\prime}+q(x)u=λu$. That is, an explicit map transforming the Taylor expansion of $y_{0}$ into the Taylor expansion of $u$ is constructed.

math.CA↗
arXiv2010

Solution of boundary value and eigenvalue problems for second order elliptic operators in the plane using pseudoanalytic formal powers

We propose a method for solving boundary value and eigenvalue problems for the elliptic operator D=divpgrad+q in the plane using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be constructed following a simple algorithm consisting in recursive integration. This system of solutions is used for solving boundary value and spectral problems for the operator D in bounded simply connected domains. We study theoretical and numerical aspects of the method.

math.AP↗
arXiv2010

On Bers generating functions for first order systems of mathematical physics

Considering one of the fundamental notions of Bers' theory of pseudoanalytic functions the generating pair via an intertwining relation we introduce its generalization for biquaternionic equations corresponding to different first-order systems of mathematical physics with variable coefficients. We show that the knowledge of a generating set of solutions of a system allows one to obtain its different form analogous to the complex equation describing pseudoanalytic functions of the second kind and opens the way for new results and applications of pseudoanalytic function theory. As one of the examples the Maxwell system for an inhomogeneous medium is considered, and as one of the consequences of the introduced approach we find a relation between the time-dependent one-dimensional Maxwell system and hyperbolic pseudoanalytic functions and obtain an infinite system of solutions of the Maxwell system. Other considered examples are the system describing force-free magnetic fields and the Dirac system from relativistic quantum mechanics.

math-ph↗
arXiv2009

Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator

In [8] it was shown that the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper we demonstrate a fundamental property of this operator: it transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a "complicated" Vekua equation from positive formal powers and a generating sequence of "simpler" Vekua equation. Similar results is obtained regarding construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theory are considered and examples are given to illustrate the procedure.

math.CV↗
arXiv2009

A finite-sum representation for solutions for the Jacobi operator

We obtain a finite-sum representation for the general solution of the Jacobi second-order difference equation D(p(n-1)Du(n-1))+q(n)u(n)=l r(n)u(n) in terms of a nonvanishing solution corresponding to some fixed value of the spectral parameter l=l_0. Applications of this representation to some results on the boundedness of solutions are given as well as illustrating examples.

math-ph↗
arXiv2009

Conformal Mapping of Right Circular Quadrilaterals

We study conformal mappings from the unit disk to circular-arc quadrilaterals with four right angles. The problem is reduced to a Sturm-Liouville boundary value problem on a real interval, with a nonlinear boundary condition, in which the coefficient functions contain the accessory parameters t,lambda of the mapping problem. The parameter lambda is designed in such a way that for fixed t, it plays the role of an eigenvalue of the Sturm-Liouville problem. Further, for each t a particular solution (an elliptic integral) is known a priori, as well as its corresponding spectral parameter lambda. This leads to insight into the dependence of the image quadrilateral on the parameters, and permits application of a recently developed spectral parameter power series (SPPS) method for numerical solution. Rate of convergence, accuracy, and computational complexity are presented for the resulting numerical procedure, which in simplicity and efficiency compares favorably with previously known methods for this type of problem.

math.CV↗
arXiv2009

Biquaternions for analytic and numerical solution of equations of electrodynamics

We give an overview of recent advances in analysis of equations of electrodynamics with the aid of biquaternionic technique. We discuss both models with constant and variable coefficients, integral representations of solutions, a numerical method based on biquaternionic fundamental solutions for solving standard electromagnetic scattering problems, relations between different operators of mathematical physics including the Schrodinger, the Maxwell system, the conductivity equation and others leading to a deeper understanding of physics and mathematical properties of the equations.

math-ph↗
arXiv2008

Spectral parameter power series for Sturm-Liouville problems

We consider a recently discovered representation for the general solution of the Sturm-Liouville equation as a spectral parameter power series (SPPS). The coefficients of the power series are given in terms of a particular solution of the Sturm-Liouville equation with the zero spectral parameter. We show that, among other possible applications, this provides a new and efficient numerical method for solving initial value and boundary value problems. Moreover, due to its convenient form the representation lends itself to numerical solution of spectral Sturm-Liouville problems, effectively by calculation of the roots of a polynomial. We discuss examples of the numerical implementation of the SPPS method and show it to be equally applicable to a wide class of singular Sturm-Liouville problems as well as to problems with spectral parameter dependent boundary conditions.

math-ph↗
arXiv2008

Zakharov-Shabat system and hyperbolic pseudoanalytic function theory

In [1] a hyperbolic analogue of pseudoanalytic function theory was developed. In the present contribution we show that one of the central objects of the inverse problem method the Zakharov-Shabat system is closely related to a hyperbolic Vekua equation for which among other results a generating sequence and hence a complete system of formal powers can be constructed explicitly.

math-ph↗
arXiv2008

Solution of the equation d/dx(pdu/dx)+qu=cu by a solution of the equation d/dx(pdu/dx)+qu=0

We give a simple solution of the equation d/dx(pdu/dx)+qu=cu whenever a nontrivial solution of d/dx(pdu/dx)+qu=0 is known. The method developed for obtaining this result is based on the theory of pseudoanalytic functions and their relationship with solutions of the stationary two-dimensional Schrodinger equation. The final result, that is the formula for the general solution of the equation d/dx(pdu/dx)+qu=cu has a simple and easily verifiable form.

math-ph↗
arXiv2007

On a complex differential Riccati equation

We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical "one-dimensional" results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral theorem.

math.AP↗
arXiv2007

On Beltrami fields with nonconstant proportionality factor on the plane

We consider the equation rotB+aB=0 (1) in the plane with a being a real-valued function and show that it can be reduced to a Vekua equation of a special form. In the case when a depends on one Cartesian variable a complete system of exact solutions of the Vekua equation and hence of equation (1) is constructed based on L. Bers' theory of formal powers.

math-ph↗
arXiv2006

Recent developments in applied pseudoanalytic function theory

We present recently obtained results in the theory of pseudoanalytic functions and its applications to elliptic second-order equations. The operator (divpgrad+q) with p and q being real valued functions is factorized with the aid of Vekua type operators of a special form and as a consequence the elliptic equation (divpgrad+q)u=0, (1) reduces to a homogeneous Vekua equation describing generalized analytic (or pseudoanalytic) functions. As a tool for solving the Vekua equation we use the theory of Taylor and Laurent series in formal powers for pseudoanalytic functions developed by L. Bers. The series possess many important properties of the usual analytic power series. Their applications until recently were limited mainly because of the impossibility of their explicit construction in a general situation. We obtain an algorithm which in a really broad range of practical applications allows us to construct the formal powers and hence the pseudoanalytic Taylor series in explicit form precisely for the Vekua equation related to equation (1). In other words, in a bounded domain this gives us a complete (in C-norm) system of exact solutions of (1).

math.CV↗
arXiv2006

On a factorization of second order elliptic operators and applications

We show that given a nonvanishing particular solution of the equation (divpgrad+q)u=0 (1) the corresponding differential operator can be factorized into a product of two first order operators. The factorization allows us to reduce the equation (1) to a first order equation which in a two-dimensional case is the Vekua equation of a special form. Under quite general conditions on the coefficients p and q we obtain an algorithm which allows us to construct in explicit form the positive formal powers (solutions of the Vekua equation generalizing the usual powers of the variable z). This result means that under quite general conditions one can construct an infinite system of exact solutions of (1) explicitly, and moreover, at least when p and q are real valued this system will be complete in ker(divpgrad+q) in the sense that any solution of (1) in a simply connected domain can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of . Finally we give a similar factorization of the operator (divpgrad+q) in a multidimensional case and obtain a natural generalization of the Vekua equation which is related to second order operators in a similar way as its two-dimensional prototype does.

math.AP↗
arXiv2005

New applications of pseudoanalytic function theory to the Dirac equation

In the present work we establish a simple relation between the Dirac equation with a scalar and an electromagnetic potentials in a two-dimensional case and a pair of decoupled Vekua equations. In general these Vekua equations are bicomplex. However we show that the whole theory of pseudoanalytic functions without modifications can be applied to these equations under a certain not restrictive condition. As an example we formulate the similarity principle which is the central reason why a pseudoanalytic function and as a consequence a spinor field depending on two space variables share many of the properties of analytic functions. One of the surprising consequences of the established relation with pseudoanalytic functions consists in the following result. Consider the Dirac equation with a scalar potential depending on one variable with fixed energy and mass. In general this equation cannot be solved explicitly even if one looks for wave functions of one variable. Nevertheless for such Dirac equation we obtain an algorithmically simple procedure for constructing in explicit form a complete system of exact solutions (depending on two variables). These solutions generalize the system of powers of z in complex analysis and are called formal powers. With their aid any regular solution of the Dirac equation can be represented by its Taylor series in formal powers.

math-ph↗