Lubin G. Vulkov

On the convergence of finite difference schemes for the heat equation with concentrated capacity

Summary. We investigate the convergence of difference schemes for the one-dimensional heat equation when the coefficient at the time derivative (heat capacity) is

Finite difference methods, theory and applications: 6th international conference, FDM 2014 Lozenetz, Bulgaria, June 18-23, 2014 revised selected papers

c\left( x\right) =1+K\delta \left( x-\xi \right). K = \const > 0

Applications of Steklov-Type Eigenvalue Problems to Convergence of Difference Schemes for Parabolic and Hyperbolic Equations with Dynamical Boundary Conditions

represents the magnitude of the heat capacity concentrated at the point

The Immersed Interface Method for a Nonlinear Chemical Diffusion Equation with Local Sites of Reactions

x=\xi

On the Convergence of Difference Schemes for Hyperbolic Problems with Concentrated Data

. An abstract operator method is developed for analyzing this equation. Estimates for the rate of convergence in special discrete energetic Sobolevs norms, compatible with the smoothness of the solution are obtained.

Operator's Approach to the Problems with Concentrated Factors

This volume is the Proceedings of the First Conference on Finite Difference Methods which was held at the University of Rousse, Bulgaria, 10--13 August 1997. The conference attracted more than 50 participants from 16 countries. 10 invited talks and 26 contributed talks were delivered. The volume contains 28 papers presented at the Conference. The most important and widely used methods for solution of differential equations are the finite difference methods. The purpose of the conference was to bring together scientists working in the area of the finite difference methods, and also people from the applications in physics, chemistry and other natural and engineering sciences.

Two-grid quasilinearization approach to ODEs with applications to model problems in physics and mechanics☆

Parabolic and hyperbolic equations with dynamical boundary conditions, i.e which involve first and second order time derivatives respectively, are considered. Convergence and stability of weighted difference schemes for such problems are discussed. Norms arising from Steklov-type eigenvalues problems are used, while in previously investigations, norms corresponding to Neumanns or Robins boundary conditions are used. More exact stability conditions are obtained for the difference schemes parameters.

Analysis of Immersed Interface Difference Schemes for Reaction-diffusion Problems with Singular Own Sources

A diffusion equation with nonlinear localized chemical reactions is considered in this paper. As a result of the reactions, although the equation is parabolic, the derivatives of the solution are discontinuous across the interfaces (local sites of reactions). A second-order accurate immersed interface method is constructed for the diffusion equation involving interfaces. The new method is more accurate than the standard approach and it does not require the interfaces to be grid points. Several experiments that confirm second-order accuracy are presented. The efficiency of the proposed algorithm is also demonstrated for solving blow up problems. The proposed technique could be extended for construction of efficient numerical algorithms on uniform grids for the present equations with moving interfaces [9] but more analysis is required.

Quasilinearization numerical scheme for fully nonlinear parabolic problems with applications in models of mathematical finance

Hyperbolic equations with unbounded coefficients and even generalized functions (in particular, Dirac-delta functions) occur both naturally and artificially and must be treated in numerical schemes. An abstract operator method is proposed for studying these equations. For finite difference schemes approximating several one-dimensional initial-boundary value problems convergence rate estimates in special discrete energetic Sobolevs norms, compatible with the smoothness of the solutions, are obtained.

On the blow-up of finite difference solutions to the heat-diffusion equation with semilinear dynamical boundary conditions

In this paper finite-difference schemes approximating the one-dimensional initial-boundary value problems for the heat equation with concentrated capacity are derived. An abstract operators method is developed for studying such problems. Convergence rate estimates consistent with the smoothness of the data are obtained.