Sergii M. Torba

Vladislav V. Kravchenko, Luis J. Navarro, Sergii M. Torba

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.

Raul Castillo Perez, Vladislav V. Kravchenko, Sergii M. Torba

A spectral parameter power series (SPPS) representation for regular solutions of singular Bessel type Sturm-Liouville equations with complex coefficients is obtained as well as an SPPS representation for the (entire) characteristic function of the corresponding spectral problem on a finite interval. It is proved that the set of zeros of the characteristic function coincides with the set of all eigenvalues of the Sturm-Liouville problem. Based on the SPPS representation a new mapping property of the transmutation operator for the considered perturbed Bessel operator is obtained, and a new numerical method for solving corresponding spectral problems is developed. The range of applicability of the method includes complex coefficients, complex spectrum and equations in which the spectral parameter stands at a first order linear differential operator. On a set of known test problems we show that the developed numerical method based on the SPPS representation is highly competitive in comparison to the best available solvers such as SLEIGN2, MATSLISE and some other codes and give an example of an exactly solvable test problem admitting complex eigenvalues to which the mentioned solvers are not applicable meanwhile the SPPS method delivers excellent numerical results.

Vladislav V. Kravchenko, Sergii M. Torba

We give an overview of recent developments in Sturm-Liouville theory concerning operators of transmutation (transformation) and spectral parameter power series (SPPS). The possibility to write down the dispersion (characteristic) equations corresponding to a variety of spectral problems related to Sturm-Liouville equations in an analytic form is an attractive feature of the SPPS method. It is based on a computation of certain systems of recursive integrals. Considered as families of functions these systems are complete in the $L_{2}$-space and result to be the images of the nonnegative integer powers of the independent variable under the action of a corresponding transmutation operator. This recently revealed property of the Delsarte transmutations opens the way to apply the transmutation operator even when its integral kernel is unknown and gives the possibility to obtain further interesting properties concerning the Darboux transformed Schrödinger operators. We introduce the systems of recursive integrals and the SPPS approach, explain some of its applications to spectral problems with numerical illustrations, give the definition and basic properties of transmutation operators, introduce a parametrized family of transmutation operators, study their mapping properties and construct the transmutation operators for Darboux transformed Schrödinger operators.

Vladislav V. Kravchenko, Sergii M. Torba, Raúl Castillo-Pérez

A new representation for a regular solution of the perturbed Bessel equation of the form $Lu=-u"+\left( \frac{l(l+1)}{x^2}+q(x)\right)u=ω^2u$ is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to $ω$. For the coefficients of the series explicit direct formulas are obtained in terms of the systems of recursive integrals arising in the spectral parameter power series (SPPS) method, as well as convenient for numerical computation recurrent integration formulas. The result is based on application of several ideas from the classical transmutation (transformation) operator theory, recently discovered mapping properties of the transmutation operators involved and a Fourier-Legendre series expansion of the transmutation kernel. For convergence rate estimates, asymptotic formulas, a Paley-Wiener theorem and some results from constructive approximation theory were used. 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.

Vladislav V. Kravchenko, Sergii M. Torba

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.

Raul Castillo Perez, Vladislav V. Kravchenko, Sergii M. Torba

Spectral parameter power series (SPPS) method is a recently introduced technique for solving linear differential equations and related spectral problems. In the present work we develop an approach based on the SPPS for analysis of graded-index optical fibers. The characteristic equation of the eigenvalue problem for calculation of guided modes is obtained in an analytical form in terms of SPPS. Truncation of the series and consideration in this way of the approximate characteristic equation gives us a simple and efficient numerical method for solving the problem. Comparison with the results obtained by other available techniques reveals clear advantages of the SPPS approach, in particular, with regards to accuracy. Based on the solution of the eigenvalue problem, parameters describing the dispersion are analyzed as well.

Vladislav V. Kravchenko, Sergii M. Torba, Jessica Yu. Santana-Bejarano

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.

Kira V. Khmelnytskaya, Vladislav V. Kravchenko, Sergii M. Torba

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.

Vladislav V. Kravchenko, Sergii M. Torba

We give a brief overview of recent developments in Sturm-Liouville theory concerning operators of transmutation (transformation) and spectral parameter power series (SPPS) and propose a new method for numerical solution of corresponding spectral problems.

Vladislav V. Kravchenko, R. Michael Porter, Sergii M. Torba

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.

Raúl Castillo-Pérez, Vladislav V. Kravchenko, Sergii M. Torba

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 $λ\in\mathbb{C}$.

Nelson Gutiérrez Jiménez, Sergii M. Torba

A representation in the form of spectral parameter power series (SPPS) is given for a general solution of a one dimension Dirac system containing arbitrary matrix coefficient at the spectral parameter, \[ B \frac{dY}{dx} + P(x)Y = λR(x)Y,\] where $Y=(y_1,y_2)^T$ is the unknown vector-function, $λ$ is the spectral parameter, $B = \begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$, and $P$ is a symmetric $2\times 2$ matrix, $R$ is an arbitrary $2\times 2$ matrix whose entries are integrable complex-valued functions. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a non-vanishing solution for one particular $λ= λ_0$. The existence of such solution is shown. For a general linear system of two first order differential equations \[ P(x)\frac{dY}{dx}+Q(x)Y = λR(x)Y,\ x\in [a,b], \] where $P$, $Q$, $R$ are $2\times 2$ matrices whose entries are integrable complex-valued functions, $P$ being invertible for every $x$, a transformation reducing it to a type considered above is shown. The general scheme of application of the SPPS representation to the solution of initial value and spectral problems as well as numerical illustrations are provided.

Vladislav V. Kravchenko, Josafath A. Otero, Sergii M. Torba

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.

Vladislav V. Kravchenko, Sergii M. Torba, Kira V. Khmelnytskaya

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.

Igor V. Kravchenko, Vladislav V. Kravchenko, Sergii M. Torba

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.

Vladislav V. Kravchenko, Sergii M. Torba

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.

Vladislav V. Kravchenko, Sergii M. Torba

In arXiv:1306.2914 a method for approximate solution of Sturm-Liouville equations and related spectral problems was presented based on the construction of the Delsarte transmutation operators. The problem of numerical approximation of solutions was reduced to approximation of a primitive of the potential by a finite linear combination of certain specially constructed functions obtained from the generalized wave polynomials introduced in arXiv:1208.5984 and arXiv:1208.6166. The method allows one to compute both lower and higher eigendata with an extreme accuracy. Since the solution of the approximation problem is the main step in the application of the method, the properties of the system of functions involved are of primary interest. In arXiv:1306.2914 two basic properties were established: the completeness in appropriate functional spaces and the linear independence. In this paper we present a considerably more complete study of the systems of functions. We establish their relation with another linear differential second-order equation, find out certain operations (in a sense, generalized derivatives and antiderivatives) which allow us to generate the next such function from a previous one. We obtain the uniqueness of the coefficients of expansions in terms of such functions and a corresponding generalized Taylor theorem. We construct the invertible integral operators transforming powers of the independent variable into the functions under consideration and establish their commutation relations with differential operators. We present some error bounds for the solution of the approximation problem depending on the smoothness of the potential and show that these error bounds are close to optimal in order. Also, we provide a rigorous justification of the alternative formulation of the proposed method allowing one to make use of the known initial values of the solutions at an endpoint.

Vladislav V. Kravchenko, Samy Morelos, Sergii M. Torba

A method for solving spectral problems for the Sturm-Liouville equation $(pv^{\prime})^{\prime}-qv+λrv=0$ based on the approximation of the Delsarte transmutation operators combined with the Liouville transformation is presented. The problem of numerical approximation of solutions and of eigendata is reduced to approximation of a pair of functions depending on the coefficients $p$, $q$ and $r$ by a finite linear combination of certain specially constructed functions related to generalized wave polynomials introduced in arXiv:1208.5984, arXiv:1208.6166. The method allows one to compute both lower and higher eigendata with an extreme accuracy. Several necessary results concerning the action of the Liouville transformation on formal powers arising in the method of spectral parameter power series are obtained as well as the transmutation operator for the Sturm-Liouville operator $\frac{1}{r}\left(\frac{d}{dx}p\frac{d}{dx}-q\right)$.

Kira V. Khmelnytskaya, Vladislav V. Kravchenko, Sergii M. Torba

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). Using this relation we solve the problem of the transmission through an inhomogeneous layer of a normally incident electromagnetic time-dependent plane wave. The solution is written in terms of a pair of Darboux-associated transmutation operators (see arXiv:1111.4449), and combined with the recent results on their construction (see arXiv:1208.6166, arXiv:1306.2914) can be used for efficient computation of the transmitted modulated signals. We develop the corresponding numerical method and illustrate its performance with examples.