# Gavrilyuk I. P.

### 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.

### 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.

### 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.

### Estimates for the convergence of the penalty method for second-order variational elliptic inequalities

Gavrilyuk I. P., Sazhenyuk V. S., Voitsekhovskii S. A.

Ukr. Mat. Zh. - 1987. - 39, № 2. - pp. 245–250

### Exact and truncated schemes of any order of accuracy for a class of one-dimensional variational inequalities

Ukr. Mat. Zh. - 1986. - 38, № 5. - pp. 563–568