## Numerical solution of one-dimensional Focker-Plank-Kolmogorov equation

The paper considers the algorithm of numerical solution of the Focker-Plank-Kolmogorov equation for the probability density of a solution of a stochastic differential equation. Its solution is approximated by cubic splines on the time-dependent moving grid.

## Tensor preconditioners in decomposition methods

One of the most effective approach in solution of mesh and finite-element SLAEs *Au = f*, arrising in approximation of two-dimensional (or multi-dimensional) problems is the decomposition method. The essense of the method consists in special choice of easy-invertible linear transformation *H* and in a successive realization of iterating process
*u...*

## Explicit-implicit domain decomposition methods based on splitting for solving parabolic equations

In the article we propose and study a new noniterative domain decomposition algorithm without overlapping subdomains and with the use of splitting procedure in one of subdomains for solution multidimensional boundary value parabolic problems.

## On one domain decomposition method with nonmatching grids for solving parabolic equations

In the article we propose and study a noniterative domain decomposition algorithm without overlapping subdomains and with the use of the penalty functionals on the interface between subdomains. Such type of algorithms were consideredearlier. In all these works the error estimate for optimal penalty parameter is *O = √τ*. In this...

## On second order finite-volume approximations for 3D mixed boundary value problems

The main topic of the paper is to present a way of constructing the second order finite-volume approximations on nonuniform grids to solve 3D mixed boundary value problems for diffusion equation with piecewise constant coefficients. For obtaining the difference equations, a linear combination of the balance relations for the normal...

## A strongly S-stable method for solving stiff systems of ordinary differential equations

A strongly *S*-stable (by A. Prothero and A. Robinson) one-step noniterated method is presented. Results of numerical calculation showing the advantage of the proposed method in comparison with a similar *L*-stable method are given.

## On optimal choice of spline-smoothing parameter

In this paper we consider an abstract spline smoothing problem in Hilbert space and express Newton’s iteration formula for an optimal choice of the smoothing parameter α in terms of the residual operator
*Rαz = z – Aσα*.

## Numerical solution of Volterra integral equations of the second kind by implicit Runge-Kutta method

An effective multistep algorithm for numerical solution of Volterra integral equations of the second kind, based on the implicit Runge-Kutta (RK) method, is constructed. The choice of the Gauss scheme for the implicit RK method allows to obtain algorithm, having a higher approximation order for one-step method and maintaining the...

## The specific basis of trigonometric functions in the problem of approximate solution of integral equations with the kernel of the kind K(x-t)

In this paper we will deal with the approximate solution of Fredholm's and Volterra's equations with the kernel of the kind *K*(*x-t*). We shall use the known algorithm for the search of the approximate solution in the form of a linear combination of preassigned basic functions*φ...*

## Optimization of energy functional for variational splines

The aim of this paper is to suggest optimization procedure for the refinement of energy functional in variational spline approximation problem. Our approach is based on a separation of measurement data between two sets. First is the set of basic measurements (nodes of spline), second is the set of control...