R. Chapko | Ivan Franko National University of Lviv (original) (raw)
Papers by R. Chapko
Ukrains’kyi Matematychnyi Zhurnal, 2022
UDC 519.6 We give an overview of recent methods based on semi-discretisation in time for the inve... more UDC 519.6 We give an overview of recent methods based on semi-discretisation in time for the inverse ill-posed problem of calculating the solution of evolution equations from time-like Cauchy data. Specifically, the function value and normal derivative are given on a portion of the lateral boundary of a space-time cylinder and the corresponding data is to be generated on the remaining lateral part of the cylinder for either the heat or wave equation. The semi-discretisation in time constitutes of applying the Laguerre transform or the Rothe method (finite difference approximation), and has the feature that the similar sequence of elliptic problems is obtained for both the heat and wave equation, only the values of certain parameters change. The elliptic equations are solved numerically by either a boundary integral approach involving the Nystreom method or a method of fundamental solutions (MFS). Theoretical properties are stated together with discretisation strategies in space. Sys...
Ukrainian Mathematical Journal, 2017
Computational Methods in Applied Mathematics, 2000
Computational Methods in Applied Mathematics, 2000
ABSTRACT The paper is concerned with a Cauchy problem for the Laplace equation in a bounded domai... more ABSTRACT The paper is concerned with a Cauchy problem for the Laplace equation in a bounded domain containing a cut. The aim is to reconstruct the solution on the cut from the values of the solution and its normal derivative on the external boundary of the domain. The authors propose an alternating iterative method that at each iteration solves a direct mixed problem for the Laplace operator with either Dirichlet or Neumann condition imposed on the cut. They prove that each direct problem is well-posed using a potential approach, and that the iterative procedure converges. For the numerical approximation of the mixed boundary value problems an integral equation approach with trigonometrical quadrature for the full discretization is used. Some numerical tests illustrate the feasibility of the proposed method.
Computational Methods in Applied Mathematics, 2000
Electronic Journal of Boundary Elements, 2009
Electronic Journal of Boundary Elements, 2011
DIPED - 99. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory. Proceedings of 4th International Seminar/Workshop (IEEE Cat. No.99TH8402), 1999
ABSTRACT A numerical solution of the initial boundary value problem for the heat equation has a g... more ABSTRACT A numerical solution of the initial boundary value problem for the heat equation has a great significance as regards the number of applications in engineering sciences. This problem arises also at the solution of the inverse boundary value problems in thermal tomography. The approximate solution can be found with the boundary integral equations method which may be used in the diverse variants. We combine the Cayley transform and the boundary integral equations for the numerical solution of the interior initial boundary value problem for the heat equation
PAMM, 2006
We consider the inverse Dirichlet boundary value problem for the Laplace equation that consists i... more We consider the inverse Dirichlet boundary value problem for the Laplace equation that consists in the reconstruction of the bounded inclusion in the 2D domain with infinite boundary from Cauchy data observed on it. In order to solve this problem we apply the Landweber [3] and hybrid [1] methods and investigate -mostly numerically -their goals and defects in the case of semi-infinite regions.
ABSTRACT We investigate the problem of determining the stationary temperature field on an inclusi... more ABSTRACT We investigate the problem of determining the stationary temperature field on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the temperature (or the heat flux) is known, and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in combination with Tikhonov regularization for the stable determination of the temperature and flux on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green’s functions, and properties of these equations are shown in an L 2 -setting. An effective way of discretizing these boundary integral equations based on the Nyström method and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations can be obtained with small computational effort, and the accuracy is increasing with the length of the portion of the boundary where the additionally data is given.
Mathematics and Computers in Simulation, 2014
and sharing with colleagues.
Journal of Mathematical Sciences, 2012
ABSTRACT We consider the boundary-value problem of stationary heat conduction in a three-dimensio... more ABSTRACT We consider the boundary-value problem of stationary heat conduction in a three-dimensional domain formed by a layer and a half-space with a cavity bounded by a smooth closed surface. On the contact boundary of the layer and half-space, conditions of ideal thermal contact are satisfied, and on the other boundary of the layer, a heat flow is given. Convective heat exchange with a medium of zero temperature occurs over the surface of the cavity. Using a constructed Green matrix for the corresponding layered domain, we reduce the boundary-value problem to a Fredholm integral equation of the second kind with an unknown function on the surface of the cavity. The numerical solution is performed using sincquadratures, Gauss–Legendre quadrature formulas, and the projection method with spherical basis functions. Results of numerical experiments are presented.
Journal of Integral Equations and Applications, 2007
ABSTRACT The paper discribes a new method to solve an inverse boundary value problem connected wi... more ABSTRACT The paper discribes a new method to solve an inverse boundary value problem connected with the Dirichlet problem Δu=0 in D, u=0 on Γ; u=f on Λ (D∈ℝ 2 is a doubly connected domain with the boundary ∂D consisting of two disjoint closed C 2 curves Γ and Λ such that Γ is contained in the interior of Λ). The inverse problem is, given the Dirichlet data f=u| Λ on Λ and the Neumann data g=∂u ∂ν| Λ , determine the shape of the interior boundary Γ· The suggested method consists of the following two steps. In the first step we assume a priori that we can place an auxiliary closed curve C in the interior of Γ and represent u as a single-layer potential with the Green’s function for the interior of Λ as a kernel and with an unknown density on C. We obtain an ill-posed linear operator equation for the unknown density. We can solve the obtained equation of the first kind and define an approximation for the solution u, for example, via Tikhonov regularization. In the second step the unknown boundary curve is found by determining a curve on which the boundary condition u=0 is satisfied in some least squares sense. This method may be interpreted as a hybrid of a decomposition method suggested by A. Kirsch and R. Kress and a regularized Newton method for solving a nonlinear operator equation. Numerical examples demonstrate the feasibility of the present method.
Journal of Integral Equations and Applications, 1999
We present a method for the numerical solution of first order nonstationary problems with a pseud... more We present a method for the numerical solution of first order nonstationary problems with a pseudodifferential operator coefficient on a manifold. Using the Cayley transform we get an explicit representation of the exact solution and reduce the problem to a sequence of stationary equations which then are transformed into hypersingular integral equations of the second kind. For the numerical solution of these integral equations we use a collocation procedure based on appropriate trigonometric interpolation quadratures. Using the numerical solution of the integral equations and the explicit representation of the exact soution, we get a fully discrete approximation with respect to time and to spatial discretization parameters. In the case of a circle, the analysis of convergence and error estimates are given which show automatic dependence of the error order on the smoothness of the exact solution (the spectral property with respect to both the time and the spatial discretization parameters).
Journal of Integral Equations and Applications, 2008
Ukrains’kyi Matematychnyi Zhurnal, 2022
UDC 519.6 We give an overview of recent methods based on semi-discretisation in time for the inve... more UDC 519.6 We give an overview of recent methods based on semi-discretisation in time for the inverse ill-posed problem of calculating the solution of evolution equations from time-like Cauchy data. Specifically, the function value and normal derivative are given on a portion of the lateral boundary of a space-time cylinder and the corresponding data is to be generated on the remaining lateral part of the cylinder for either the heat or wave equation. The semi-discretisation in time constitutes of applying the Laguerre transform or the Rothe method (finite difference approximation), and has the feature that the similar sequence of elliptic problems is obtained for both the heat and wave equation, only the values of certain parameters change. The elliptic equations are solved numerically by either a boundary integral approach involving the Nystreom method or a method of fundamental solutions (MFS). Theoretical properties are stated together with discretisation strategies in space. Sys...
Ukrainian Mathematical Journal, 2017
Computational Methods in Applied Mathematics, 2000
Computational Methods in Applied Mathematics, 2000
ABSTRACT The paper is concerned with a Cauchy problem for the Laplace equation in a bounded domai... more ABSTRACT The paper is concerned with a Cauchy problem for the Laplace equation in a bounded domain containing a cut. The aim is to reconstruct the solution on the cut from the values of the solution and its normal derivative on the external boundary of the domain. The authors propose an alternating iterative method that at each iteration solves a direct mixed problem for the Laplace operator with either Dirichlet or Neumann condition imposed on the cut. They prove that each direct problem is well-posed using a potential approach, and that the iterative procedure converges. For the numerical approximation of the mixed boundary value problems an integral equation approach with trigonometrical quadrature for the full discretization is used. Some numerical tests illustrate the feasibility of the proposed method.
Computational Methods in Applied Mathematics, 2000
Electronic Journal of Boundary Elements, 2009
Electronic Journal of Boundary Elements, 2011
DIPED - 99. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory. Proceedings of 4th International Seminar/Workshop (IEEE Cat. No.99TH8402), 1999
ABSTRACT A numerical solution of the initial boundary value problem for the heat equation has a g... more ABSTRACT A numerical solution of the initial boundary value problem for the heat equation has a great significance as regards the number of applications in engineering sciences. This problem arises also at the solution of the inverse boundary value problems in thermal tomography. The approximate solution can be found with the boundary integral equations method which may be used in the diverse variants. We combine the Cayley transform and the boundary integral equations for the numerical solution of the interior initial boundary value problem for the heat equation
PAMM, 2006
We consider the inverse Dirichlet boundary value problem for the Laplace equation that consists i... more We consider the inverse Dirichlet boundary value problem for the Laplace equation that consists in the reconstruction of the bounded inclusion in the 2D domain with infinite boundary from Cauchy data observed on it. In order to solve this problem we apply the Landweber [3] and hybrid [1] methods and investigate -mostly numerically -their goals and defects in the case of semi-infinite regions.
ABSTRACT We investigate the problem of determining the stationary temperature field on an inclusi... more ABSTRACT We investigate the problem of determining the stationary temperature field on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the temperature (or the heat flux) is known, and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in combination with Tikhonov regularization for the stable determination of the temperature and flux on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green’s functions, and properties of these equations are shown in an L 2 -setting. An effective way of discretizing these boundary integral equations based on the Nyström method and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations can be obtained with small computational effort, and the accuracy is increasing with the length of the portion of the boundary where the additionally data is given.
Mathematics and Computers in Simulation, 2014
and sharing with colleagues.
Journal of Mathematical Sciences, 2012
ABSTRACT We consider the boundary-value problem of stationary heat conduction in a three-dimensio... more ABSTRACT We consider the boundary-value problem of stationary heat conduction in a three-dimensional domain formed by a layer and a half-space with a cavity bounded by a smooth closed surface. On the contact boundary of the layer and half-space, conditions of ideal thermal contact are satisfied, and on the other boundary of the layer, a heat flow is given. Convective heat exchange with a medium of zero temperature occurs over the surface of the cavity. Using a constructed Green matrix for the corresponding layered domain, we reduce the boundary-value problem to a Fredholm integral equation of the second kind with an unknown function on the surface of the cavity. The numerical solution is performed using sincquadratures, Gauss–Legendre quadrature formulas, and the projection method with spherical basis functions. Results of numerical experiments are presented.
Journal of Integral Equations and Applications, 2007
ABSTRACT The paper discribes a new method to solve an inverse boundary value problem connected wi... more ABSTRACT The paper discribes a new method to solve an inverse boundary value problem connected with the Dirichlet problem Δu=0 in D, u=0 on Γ; u=f on Λ (D∈ℝ 2 is a doubly connected domain with the boundary ∂D consisting of two disjoint closed C 2 curves Γ and Λ such that Γ is contained in the interior of Λ). The inverse problem is, given the Dirichlet data f=u| Λ on Λ and the Neumann data g=∂u ∂ν| Λ , determine the shape of the interior boundary Γ· The suggested method consists of the following two steps. In the first step we assume a priori that we can place an auxiliary closed curve C in the interior of Γ and represent u as a single-layer potential with the Green’s function for the interior of Λ as a kernel and with an unknown density on C. We obtain an ill-posed linear operator equation for the unknown density. We can solve the obtained equation of the first kind and define an approximation for the solution u, for example, via Tikhonov regularization. In the second step the unknown boundary curve is found by determining a curve on which the boundary condition u=0 is satisfied in some least squares sense. This method may be interpreted as a hybrid of a decomposition method suggested by A. Kirsch and R. Kress and a regularized Newton method for solving a nonlinear operator equation. Numerical examples demonstrate the feasibility of the present method.
Journal of Integral Equations and Applications, 1999
We present a method for the numerical solution of first order nonstationary problems with a pseud... more We present a method for the numerical solution of first order nonstationary problems with a pseudodifferential operator coefficient on a manifold. Using the Cayley transform we get an explicit representation of the exact solution and reduce the problem to a sequence of stationary equations which then are transformed into hypersingular integral equations of the second kind. For the numerical solution of these integral equations we use a collocation procedure based on appropriate trigonometric interpolation quadratures. Using the numerical solution of the integral equations and the explicit representation of the exact soution, we get a fully discrete approximation with respect to time and to spatial discretization parameters. In the case of a circle, the analysis of convergence and error estimates are given which show automatic dependence of the error order on the smoothness of the exact solution (the spectral property with respect to both the time and the spatial discretization parameters).
Journal of Integral Equations and Applications, 2008