# Makarov V. L.

### Resonant equations with classical orthogonal polynomials. II

Gavrilyuk I. P., Makarov V. L.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 455-470

UDC 517.9

We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite
and the Laguerre orthogonal polynomials, and propose an algorithm of finding their particular and general solutions in the
closed form. The algorithm is especially suitable for the computer-algebra tools, such as Maple. The resonant equations
form an essential part of various applications, e.g., of the efficient functional-discrete method for the solution of operator
equations and eigenvalue problems. These equations also appear in the context of supersymmetric Casimir operators for the di-spin algebra, as well as of the square operator equations $A^2u = f$ , e.g., of the biharmonic equation.

### Vladyslav Kyrylovych Dzyadyk (on his 100th birthday)

Dzyadyk Yu. V., Golub A. P., Kovtunets V. V., Letychevs’kyi O. A., Lukovsky I. O., Makarov V. L., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zadiraka V. K.

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 147-150

### Resonant equations with classical orthogonal polynomials. I

Gavrilyuk I. P., Makarov V. L.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 190-209

In the present paper, we study some resonant equations related to the classical orthogonal polynomials and propose an algorithm of finding their particular and general solutions in the explicit form. The algorithm is especially suitable for the computer algebra tools, such as Maple. The resonant equations form an essential part of various applications e.g. of the efficient functional-discrete method aimed at the solution of operator equations and eigenvalue problems. These equations also appear in the context of supersymmetric Casimir operators for the di-spin algebra, as well as for the square operator equations $A^2u = f$; e.g., for the biharmonic equation.

### Anatolii Mykhailovych Samoilenko (on his 80th birthday)

Antoniouk A. Vict., Berezansky Yu. M., Boichuk О. A., Gutlyanskii V. Ya., Khruslov E. Ya., Kochubei A. N., Korolyuk V. S., Kushnir R. M., Lukovsky I. O., Makarov V. L., Marchenko V. O., Nikitin A. G., Parasyuk I. O., Pastur L. A., Perestyuk N. A., Portenko N. I., Ronto M. I., Sharkovsky O. M., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 3-6

### Exact and approximate solutions of spectral problems for the Schrödinger operator on (−∞,∞) with polynomial potential

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 79-93

New exact representations for the solutions of numerous one-dimensional spectral problems for the Schr¨odinger operator with polynomial potential are obtained by using a technique based on the functional-discrete (FD) method. In cases where the ordinary FD-method is divergent, we propose to use its modification, which proved to be quite efficient. The obtained theoretical results are illustrated by numerical examples.

### On the 100th birthday of outstanding mathematician and mechanic Yurii Oleksiiovych Mytropol’s’kyi (03.01.1917 – 14.06.2008)

Berezansky Yu. M., Boichuk О. A., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Parasyuk I. O., Perestyuk N. A., Samoilenko A. M., Sharkovsky O. M.

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 132-144

### Exponentially convergent method for an abstract nonlocal problem with integral nonlinearity

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1587-1597

We consider a problem for the first-order differential equation with unbounded operator coefficient in Banach space and a nonlinear integral nonlocal condition. An exponentially convergent method for the numerical solution of this problem is proposed and justified under assumption that the indicated operator coefficient A is strongly positive and certain existence and uniqueness conditions are satisfied. This method is based on the reduction of the posed problem to an abstract Hammerstein equation, discretization of this equation by the collocation method, and its subsequent solution by the fixed-point iteration method. Each iteration of the method involves the Sinc-based numerical evaluation of the exponential operator function represented by the Dunford – Cauchy integral over the hyperbola enveloping the spectrum of A. The integral part of the nonlocal condition is approximated by using the Clenshaw – Curtis quadrature formula.

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk О. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

### Volodymyr Semenovych Korolyuk (on his 90th birthday)

Bratiichuk N. S., Gusak D. V., Kovalenko I. N., Lukovsky I. O., Makarov V. L., Samoilenko A. M., Samoilenko I. V.

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1151-1152

### Exponentially Convergent Method for the First-Order Differential Equation in a Banach Space with Integral Nonlocal Condition

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1029–1040

For the first-order differential equation with unbounded operator coefficient in a Banach space, we study the nonlocal problem with integral condition. An exponentially convergent algorithm for the numerical solution of this problem is proposed and justified under the assumption that the operator coefficient *A* is strongly positive and certain existence and uniqueness conditions are satisfied. The algorithm is based on the representations of operator functions via the Dunford–Cauchy integral along a hyperbola covering the spectrum of *A* and the quadrature formula containing a small number of resolvents. The efficiency of the proposed algorithm is illustrated by several examples.

### Yurii Stephanovych Samoilenko (on his 70th birthday)

Berezansky Yu. M., Boichuk О. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Nizhnik L. P., Samoilenko A. M., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1408-1409

### Anatolii Mykhailovych Samoilenko (on his 75th birthday)

Berezansky Yu. M., Boichuk О. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Perestyuk N. A., Portenko N. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 3 - 6

### FD-method for solving the nonlinear Klein - Gordon equation

Dragunov D. V., Makarov V. L., Sember D. A.

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1394-1415

We propose a functional-discrete method for solving the Goursat problem for the nonlinear Klein-Gordon equation. Sufficient conditions for the superexponential convergence of this method are obtained. The obtained theoretical results are illustrated by a numerical example.

### Oleksandr Ivanovych Stepanets’ (on the 70 th anniversary of his birthday)

Gorbachuk M. L., Lukovsky I. O., Makarov V. L., Motornyi V. P., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Sharko V. V., Zaderei P. V.

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 579-581

### Approximation of Urysohn operator with operator polynomials of Stancu type

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 318-343

We study a one-parameter family of positive polynomial operators of one and two variables that approximate the Urysohn operator. In the case of two variables, the integration domain is a "rectangular isosceles triangle". As a special case, Bernstein-type polynomials are obtained. The Stancu asymptotic formulas for remainders are refined.

### On continual interpolation nodes for operators in linear topological spaces

Kashpur O. F., Khlobystov V. V., Makarov V. L.

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 494–503

We establish conditions for the existence of continual nodes for interpolation polynomials of the integral type. This result is generalized to the case of multivariable operators. Some examples of these interpolants are analyzed.

### Fifty years devoted to science (on the 70th birthday of Anatolii Mykhailovych Samoilenko)

Berezansky Yu. M., Dorogovtsev A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Perestyuk N. A., Rebenko A. L., Ronto A. M., Ronto M. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 3–7

### Application of the FD-method to the solution of the Sturm-Liouville problem with coefficients of special form

Klymenko Ya. V., Makarov V. L.

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1140–1147

We use the functional-discrete method for the solution of the Strum-Liouville problem with coefficients of a special form and obtain the estimates of accuracy. The numerical experiment is performed by using the Maple-10 software package.

### On the 90th birthday of Yurii Alekseevich Mitropol’skii

Berezansky Yu. M., Gorbachuk M. L., Korolyuk V. S., Koshlyakov V. N., Lukovsky I. O., Makarov V. L., Perestyuk N. A., Samoilenko A. M., Samoilenko Yu. I., Sharko V. V., Sharkovsky O. M., Stepanets O. I., Tamrazov P. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 147–151

### FD-method for an eigenvalue problem with nonlinear potential

Gavrilyuk I. P., Klymenko A. V., Makarov V. L., Rossokhata N. O.

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 14–28

Using the functional discrete approach and Adomian polynomials, we propose a numerical algorithm for an eigenvalue problem with a potential that consists of a nonlinear autonomous part and a linear part depending on an independent variable. We prove that the rate of convergence of the algorithm is exponential and improves as the order number of an eigenvalue increases. We investigate the mutual influence of the piecewise-constant approximation of the linear part of the potential and the nonlinearity on the rate of convergence of the method. Theoretical results are confirmed by numerical data.

### Integral Newton-Type Polynomials with Continual Nodes

Kashpur O. F., Khlobystov V. V., Makarov V. L.

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 779-789

We construct an integral Newton-type interpolation polynomial with a continual set of nodes. This interpolant is unique and preserves an operator polynomial of the corresponding degree.

### Interpolational Integral Continued Fractions

Khlobystov V. V., Makarov V. L., Mykhal'chuk B. R.

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 479-488

For nonlinear functionals defined on the space of piecewise-continuous functions, we construct an interpolational integral continued fraction on continual piecewise-continuous nodes and establish conditions for the existence and uniqueness of interpolants of this type.

### Oleksandr Ivanovych Stepanets' (on his 60-th birthday)

Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Romanyuk A. S., Romanyuk V. S., Rukasov V. I., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zaderei P. V.

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 579-580

### Structural Analysis of One Class of Dynamical Systems

Koshlyakov V. N., Makarov V. L.

Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1089-1096

We develop the method of structural transformations of dynamical systems (proposed earlier by Koshlyakov) for systems containing nonconservative positional structures. The method under consideration is based on structural transformations that enable one to eliminate nonconservative positional terms from the original system without changing its stability properties.

### International Conference “INAMTAP’96 (Computer Science, Computational and Applied Mathematics: Theory, Application, and Prospects)”

Lukovsky I. O., Makarov V. L., Prokhur M. Z.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 611–612

### Characteristics of the sum of a multidimensional series

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 886–892

### General structure of interpolational functional polynomials

Khlobystov V. V., Makarov V. L.

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1361–1368

### Estimate of the convergence of the method of straight lines for equations of parabolic type

Burkovskaya V. L., Makarov V. L.

Ukr. Mat. Zh. - 1986. - 38, № 2. - pp. 169–173

### Estimation of convergence rate of different solutions to generalized solution of dirichlet problem for helmholtz equation of class *W* ^{1} _{2}(Ω) in convex domain

Makarov V. L., Shablii T. G., Voitsekhovskii S. A.

Ukr. Mat. Zh. - 1986. - 38, № 1. - pp. 98–101

### Straight-line method for a quasilinear equation of parabolic type with nonclassical boundary condition

Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 42 – 48

### Convergence of the interpolation process for an *x*²-analytic function

Ukr. Mat. Zh. - 1983. - 35, № 2. - pp. 151—155

### Algorithmic aspects of the method of summary representations

Lyashko I. I., Makarov V. L., Shmanenko T. L.

Ukr. Mat. Zh. - 1980. - 32, № 4. - pp. 515–522

### Interpolation of hermite type in a class of x?-harmonic functions of given smoothness

Burkovskaya V. L., Klunnik A. A., Makarov V. L.

Ukr. Mat. Zh. - 1979. - 31, № 3. - pp. 317–320

### Bounds on the zeros of a certain determinant whose elements are Bessel functions of the first and second kinds

Ukr. Mat. Zh. - 1972. - 24, № 2. - pp. 247—253