Conservative linear equations arise in many areas of application, including continuum mechanics or high-frequency geometrical optics approximations. This kind of equation admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In this paper, we study the convergence of a class of finite-difference numerical schemes and introduce an app...

In this paper we discuss the problem of an adaptive parameter choice in onesided finite difference schemes for the numerical differentiation in case when noisy values of the function to be differentiated are available only at the given points. This problem is motivated by diabetes therapy management, where it is important to provide estimations of the future blood glucose trend from current and...

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and they determine the different scales in the solution to this problem. A Shishkin piecewise–uniform mesh is constructed, which is used, in conjunct...

The method of difference potentials was originally proposed by Ryaben’kii and can be interpreted as a generalized discrete version of the method of Calderon’s operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the ...

We give sufficient conditions under which the convergence of finite difference approximations in the space variable of possibly degenerate second order parabolic and elliptic equations can be accelerated to any given order of convergence by Richardson’s method.

We give theorems on implicit difference inequalities generated by initial-boundary value problems for parabolic functional differential equations. We apply this result for the investigation of the stability of difference schemes. Classical solutions of mixed problems are approximated in the paper by solutions of suitable implicit difference methods. The proofs of the convergence of difference m...

Various classes of stable finite difference schemes can be constructed to obtain a numerical solution. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In this study, we use a simple boundary value problem for a one-dimensional parabolic equation to discuss the selection of an approximation with respect to time. We conside...

A bstract. The three level operator finite difference schemes on nonuniform on time grids in llilbert spaces of finite dimension are consid­ ered. A priori estimates for uniform stability on the initial conditions are received under natural assumptions on operators and non uniform time grids. The results obtained here are applied to study stability o f the three levels weighted schemes of secon...

We develop a new high order accurate time-discretisation technique for initial value problems. We focus on problems that originate from a space discretisation using high order finite difference methods on summation-by-parts form with weak boundary conditions, and extend that technique to the time-domain. The new timediscretisation method is global and together with the approximation in space, i...

After a brief survey on the parametric deformable models, we develop an iterative method based on the finite difference schemes in order to obtain energy-minimizing snakes. We estimate the approximation error, the residue, and the truncature error related to the corresponding algorithm, then we discuss its convergence, consistency, and stability. Some aspects regarding the prosthetic sugical me...