arXiv subjects

Vladislav V. Kravchenko

Publications and source records attributed to Vladislav V. Kravchenko.

At least 37 records · Page 2Linked to original sources

arXiv2020

A transmutation operator method for solving the inverse quantum scattering problem

The inverse quantum scattering problem for the perturbed Bessel equation is considered. A direct and practical method for solving the problem is proposed. It allows one to reduce the inverse problem to a system of linear algebraic equations, and the potential is recovered from the first component of the solution vector of the system. The approach is based on a special form Fourier-Jacobi series representation for the transmutation operator kernel and the Gelfand-Levitan equation which serves for obtaining the system of linear algebraic equations. The convergence and stability of the method are proved as well as the existence and uniqueness of the solution of the truncated system. Numerical realization of the method is discussed. Results of numerical tests are provided revealing a remarkable accuracy and stability of the method.

math-ph↗

arXiv2020

Transmutation operators and a new representation for solutions of perturbed Bessel equations

New representations for an integral kernel of the transmutation operator and for a regular solution of the perturbed Bessel equation of the form $-u^{\prime\prime}+\left(\frac{\ell(\ell+1)}{x^{2}}+q(x)\right)u=ω^{2}u$ are obtained. The integral kernel is represented as a Fourier-Jacobi series. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to $ω$. For the coefficients of the series convenient for numerical computation recurrent integration formulas are obtained. The new representation improves the ones from arXiv:1609.06679 and arXiv:1712.01363 for large values of $ω$ and $\ell$ and for non-integer values of $\ell$. The results are based on application of several ideas from the classical transmutation (transformation) operator theory, asymptotic formulas for the solution, results connecting the decay rate of the Fourier transform with the smoothness of a function, the Paley-Wiener theorem and some results from constructive approximation theory. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.

math.CA↗

arXiv2020

A Neumann series of Bessel functions representation for solutions of the radial Dirac system

A new representation for a regular solution of the radial Dirac system of a special form is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to the spectral parameter. For the coefficients of the series convenient for numerical computation recurrent integration formulas are given. Numerical examples are presented.

math-ph↗

arXiv2019

Time-dependent one-dimensional electromagnetic wave propagation in inhomogeneous media: exact solution in terms of transmutations and Neumann series of Bessel functions

The time-dependent Maxwell system describing electromagnetic wave propagation in inhomogeneous isotropic media in the one-dimensional case reduces to a Vekua-type equation for bicomplex-valued functions of a hyperbolic variable, see arXiv:1001.0552. In arXiv:1410.4873 using this reduction a representation of a general solution of the system was obtained in terms of a couple of Darboux-associated transmutation operators arXiv:1111.4449. In arXiv:1508.02738 a Fourier-Legendre expansion of transmutation integral kernels was obtained. This expansion is used in the present work for obtaining an exact solution of the problem of the transmission of a normally incident electromagnetic time-dependent plane wave through an arbitrary inhomogeneous layer. The result can be used for efficient computation of the transmitted modulated signals. In particular, it is shown that in the classical situation of a signal represented in terms of a trigonometric Fourier series the solution of the problem can be written in the form of Neumann series of Bessel functions with exact formulas for the coefficients. The representation lends itself to numerical computation.

math-ph↗

arXiv2018

A representation of the transmutation kernels for the Schrödinger operator in terms of eigenfunctions and applications

The representations of the kernels of the transmutation operator and of its inverse relating the one-dimensional Schrödinger operator with the second derivative are obtained in terms of the eigenfunctions of a corresponding Sturm-Liouville problem. Since both series converge slowly and in general only in a certain distributional sense we find a way to improve these expansions and make them convergent uniformly and absolutely by adding and subtracting corresponding terms. A numerical illustration of the obtained results is given.

math.CA↗

arXiv2018

A right inverse operator for $\operatorname{curl}+λ$ and applications

A general solution of the equation $\operatorname{curl}\vec{w}+λ\vec {w}=\overrightarrow{g},\,λ\in\mathbb{C},\,λ\neq0$ is obtained for an arbitrary bounded domain $Ω\subset\mathbb{R}^{3}$ with a Liapunov boundary and $\overrightarrow{g}\in W^{p,\operatorname{div}}\left( Ω\right) =\left\{ \overrightarrow{u}\in L^{p}\left( Ω\right) :\,\operatorname{div}\overrightarrow{u}\in L^{p}\left( Ω\right) ,\,1<p<\infty\right\} $. The result is based on the use of classical integral operators of quaternionic analysis. Applications of the main result are considered to a Neumann boundary value problem for the equation $\operatorname{curl}\vec{w}+λ\vec {w}=\overrightarrow{g}$ as well as to the nonhomogeneous time-harmonic Maxwell system for achiral and chiral media.

math-ph↗

arXiv2018

The Bergman kernel for the Vekua equation

The paper extends some well-known results for analytic functions onto solutions of the Vekua equation $\partial _{\overline{z}}W=aW+b\overline{W}$ regarding the existence and construction of the Bergman kernel and of the corresponding Bergman projection operator.

math.AP↗

arXiv2018

A method for computation of scattering amplitudes and Green functions of whole axis problems

A method for the computation of scattering data and of the Green function for the one-dimensional Schrödinger operator $H:=-\frac{d^2}{dx^2}+q(x)$ with a decaying potential is presented. It is based on representations for the Jost solutions in the case of a compactly supported potential obtained in terms of Neumann series of Bessel functions (NSBF), an approach recently developed in arXiv:1508.02738. The representations are used for calculating a complete orthonormal system of generalized eigenfunctions of the operator $H$ which in turn allow one to compute the scattering amplitudes and the Green function of the operator $H-λ$ with $λ\in\mathbb{C}$.

math.CA↗

arXiv2018

Generalized wave polynomials and transmutations related to perturbed Bessel equations

The transmutation (transformation) operator associated with the perturbed Bessel equation is considered. It is shown that its integral kernel can be uniformly approximated by linear combinations of constructed here generalized wave polynomials, solutions of a singular hyperbolic partial differential equation arising in relation with the transmutation kernel. As a corollary of this results an approximation of the regular solution of the perturbed Bessel equation is proposed with corresponding estimates independent of the spectral parameter.

math.CA↗

arXiv2017

Pricing double barrier options on homogeneous diffusions: a Neumann series of Bessel functions representation

This paper develops a novel analytically tractable Neumann series of Bessel functions representation for pricing (and hedging) European-style double barrier knock-out options, which can be applied to the whole class of one-dimensional time-homogeneous diffusions even for the cases where the corresponding transition density is not known. The proposed numerical method is shown to be efficient and simple to implement. To illustrate the flexibility and computational power of the algorithm, we develop an extended jump to default model that is able to capture several empirical regularities commonly observed in the literature.

q-fin.CP↗

arXiv2017

Spectral parameter power series for arbitrary order linear differential equations

Let $L$ be the $n$-th order linear differential operator $Ly = ϕ_0y^{(n)} + ϕ_1y^{(n-1)} + \cdots + ϕ_ny$ with variable coefficients. A representation is given for $n$ linearly independent solutions of $Ly=λr y$ as power series in $λ$, generalizing the SPPS (spectral parameter power series) solution which has been previously developed for $n=2$. The coefficient functions in these series are obtained by recursively iterating a simple integration process, begining with a solution system for $λ=0$. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of $n$-th order initial value problems and spectral problems.

math.CA↗

arXiv2017

On construction of transmutation operators for perturbed Bessel equations

A representation for the kernel of the transmutation operator relating the perturbed Bessel equation with the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of the regular solution of the perturbed Bessel equation is given presenting a remarkable feature of uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of application to the solution of Dirichlet spectral problems is presented.

math.CA↗

arXiv2017

Transmutation operators: construction and applications

Recent results on the construction and applications of the transmutation (transformation) operators are discussed. Three new representations for solutions of the one-dimensional Schrödinger equation are considered. Due to the fact that they are obtained with the aid of the transmutation operator all the representations possess an important for practice feature. The accuracy of the approximate solution is independent of the real part of the spectral parameter. This makes the representations especially useful in problems requiring computation of large sets of eigendata with a nondeteriorating accuracy. Applications of the exact representations for the transmutation operators to partial differential equations are discussed as well. In particular, it is shown how the methods based on complete families of solutions can be extended onto equations with variable coefficients.

math.CA↗

arXiv2017

Analytic approximation of solutions of parabolic partial differential equations with variable coefficients

A complete family of solutions for the one-dimensional reaction-diffusion equation \[ u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t) \] with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem for the considered equation with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper the Dirichlet boundary conditions are considered however the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.

math.AP↗

arXiv2017

Solution of parabolic free boundary problems using transmuted heat polynomials

A numerical method for free boundary problems for the equation \[ u_{xx}-q(x)u=u_t \] is proposed. The method is based on recent results from transmutation operators theory allowing one to construct efficiently a complete system of solutions for this equation generalizing the system of heat polynomials. The corresponding implementation algorithm is presented.

math.AP↗

arXiv2017

Asymptotics with respect to the spectral parameter and Neumann series of Bessel functions for solutions of the one-dimensional Schrödinger equation

A representation for a solution $u(ω,x)$ of the equation $-u"+q(x)u=ω^2 u$, satisfying the initial conditions $u(ω,0)=1$, $u'(ω,0)=iω$ is derived in the form \[ u(ω,x)=e^{iωx}\left( 1+\frac{u_1(x)}ω+ \frac{u_2(x)}{ω^2}\right) +\frac{e^{-iωx}u_3(x)}{ω^2}-\frac{1}{ω^2}\sum_{n=0}^{\infty} i^{n}α_n(x)j_n(ωx), \] where $u_m(x)$, $m=1,2,3$ are given in a closed form, $j_n$ stands for a spherical Bessel function of order $n$ and the coefficients $α_n$ are calculated by a recurrent integration procedure. The following estimate is proved $\vert u(ω,x) -u_N(ω,x)\vert \leq \frac{1}{\vert ω\vert^2}\varepsilon_N(x)\sqrt{\frac{\sinh(2\mathop{\rm Im}ω\,x)}{\mathop{\rm Im}ω}}$ for any $ω\in\mathbb{C}\backslash \{0\}$, where $u_N(ω,x)$ is an approximate solution given by truncating the series in the representation for $u(ω,x)$ and $\varepsilon_N(x)$ is a nonnegative function tending to zero for all $x$ belonging to a finite interval of interest. In particular, for $ω\in\mathbb{R}\backslash \{0\}$ the estimate has the form $\vert u(ω,x)-u_N(ω,x)\vert \leq \frac{1}{\vertω\vert^2}\varepsilon_N(x)$. A numerical illustration of application of the new representation for computing the solution $u(ω,x)$ on large sets of values of the spectral parameter $ω$ with an accuracy nondeteriorating (and even improving) when $ω\rightarrow \pm \infty$ is given.

math.CA↗

arXiv2017

A Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations

A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm-Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter $ω$ the difference between the exact solution and the approximate one (the truncated NSBF) depends on $N$ (the truncation parameter) and the coefficients of the equation and does not depend on $ω$. A similar result is valid when $ω\in\mathbb{C}$ belongs to a strip $|Imω|<C$. This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of $ω$. Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm-Liouville equation is developed and illustrated on a test problem.

math.CA↗

arXiv2017

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

A new representation of solutions to the equation $-y"+q(x)y=ω^2 y$ is obtained. For every $x$ the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter $ω$. Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to $ω$ which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.

math.CA↗