Lviv National University. Studies in Faculty of Applied Mathematics and Mechanics 1980-1985. Dipl. (equivalent to M.S.) in Applied Mathematics
RESEARCH INTERESTS
Numerical solution of linear and non-linear evolution inverse problems, integral equation approach
66
Scopus Publications
Scopus Publications
Numerical solution of the Richards' equation in a two-dimensional bounded homogeneous soil Ihor Borachok, Roman Chapko, Leonidas Mindrinos Journal of Hydraulic Research, 2026 In this work, we present a numerical solution to the linear Richards' equation in a two-dimensional bounded soil domain, modelling unsaturated flow through a homogeneous rectangular medium under various infiltration scenarios. The proposed method employs a two-step approach: time discretization is performed using a finite difference scheme, followed by spatial discretization using the method of fundamental solutions (MFS), a meshless technique known for its simplicity and accuracy. The resulting scheme is easy to implement and yields reliable estimates of soil moisture dynamics. The effectiveness and accuracy of the method are demonstrated through several test cases, including comparisons with available analytical solutions.
A FUNDAMENTAL SEQUENCES METHOD FOR AN INVERSE BOUNDARY VALUE PROBLEM FOR THE HEAT EQUATION IN DOUBLE-CONNECTED DOMAINS Ihor Borachok, Roman Chapko Inverse Problems and Imaging, 2025 The application of the fundamental sequences method for reconstructing the inner part of the boundary of a double-connected domain from the overdetermined Cauchy data of the solution of the heat conduction equation on the outer part of the boundary is considered. The nonlinear ill-posed problem is numerically solved by the regularized Newton's method, at each step of which direct problems for the heat equation are solved. Using Rothe's method, each direct problem is reduced to a sequence of elliptic Dirichlet problems for the inhomogeneous modified Helmholtz equation. Which, in turn, is fully discretized by the fundamental sequences method. The results of numerical examples in both two- and three-dimensional domains confirm the accuracy of the proposed method with negligible computational effort.
On the numerical solution of a parabolic Fredholm integro-differential equation by the RBF method Ihor Borachok, Roman Chapko, Oksana Palianytsia Results in Applied Mathematics, 2025 This paper presents the numerical solution of an initial boundary value problem for a parabolic Fredholm integro-differential equation (FIDE) in bounded 2D and 3D spatial domains. To reduce the dimensionality of the problem, we employ the Laguerre transformation and Rothe’s method, with both first- and second-order time discretization approximations. As a result, the time-dependent problem is transformed into a recurrent sequence of boundary value problems for elliptic FIDEs. The radial basis function (RBF) method is then applied, where each stationary solution is approximated as a linear combination of radial basis functions centered at specific points, along with polynomial basis functions. The placement of these center points is outlined for both two-dimensional and three-dimensional regions. Collocation at center points generates a sequence of linear systems with integral coefficients. To compute these coefficients numerically, parameterization is performed, and Gauss–Legendre and trapezoidal quadratures are used. The shape parameter of the RBFs is optimized through a real-coded genetic algorithm. Numerical results in both two-dimensional and three-dimensional domains confirm the effectiveness and applicability of the proposed approaches.
Numerical solution of the vertical infiltration problem in bounded profiles , I. Borachok, R. Chapko, , L. Mindrinos, and Mathematical Modeling and Computing, 2025 There is presented a numerical solution of the one-dimensional infiltration problem in bounded profiles. The soil is assumed to have constant water diffusivity and linear dependence between the hydraulic conductivity and the water content. Then, the vertical infiltration problem is modeled as an initial boundary value problem for a diffusion equation. We combine the finite difference scheme for the time variable with the fundamental sequence method for the spatial variable. The derived numerical scheme is applied to both flooding and rainfall scenarios. The convergence of the numerical approximated solution to the analytical one justifies the applicability of the method.
On the numerical solution of the Laplace equation with complete and incomplete cauchy data using integral equations CMES Computer Modeling in Engineering and Sciences, 2014
On the numerical solution of the laplace equation with complete and incomplete cauchy data using integral equations CMES Computer Modeling in Engineering and Sciences, 2014
A direct integral equation method for a cauchy problem for the laplace equation in 3-dimensional semi-infinite domains CMES Computer Modeling in Engineering and Sciences, 2012
Ivan Gavrilyuk — 60 R. Chapko, M. Hermann, B. Jovanovich, V. V.Khlobystov, M. Kutniv, R. Lazarov, I. Lukovskyj, V. Makarov, P. Matus, A. Timokha, V. Trotsenko, V. Vasylyk Computational Methods in Applied Mathematics, 2008
Professor V. L.Makarov – 65 R. Chapko, I. Gavriljuk, B. Jovanovich, M. Hermann, R. Lazarov, I. Lukovsky, P. Matus, A. Timokha, P. Vabishchevich Computational Methods in Applied Mathematics, 2006
