Band 3, Heft 2

Variational inequalitiy of the second kind in the Banach or Hilbert space is considered. A ”semi-implicit” iterative method for its solution is studied.

In this paper, a resolvent estimate is proved for discretisations of second order ordinary differential operators subject to Dirichlet boundary conditions on a finite or infinite interval. As a discretisation method, a fully discrete Galerkin method using continuous splines of order r > 2 on a locally quasi-uniform grid is considered. As a byproduct, an a priori estimate for the discretised differential operator is obtained.

This paper analyzes immersed interface difference schemes for onedimensional reaction-diffusion equations with singular own linear and nonlinear sources. Error bounds in the infinity norm based on the maximum principle are derived. Sharper bounds and a more detailed structure of the error are obtained using the asymptotic error expansion analysis method. Numerical examples confirm the theoretical results.

We consider rather a general class of equilibrium problems in a real Banach space, which involve nonsmooth convex functions. We apply the D-gap function approach to these problems and show that, under certain additional assumptions, they can be converted into a problem of finding a stationary point of a differentiable function. Based on this property, we suggest a descent type algorithm to find a solution to the initial problem. An example of applications to nonlinearly constrained equilibrium problems is also given.

This work discusses issues on the design and analysis of three-point difference schemes for the nonlinear boundary-value problem. A new effective implementation of an exact three-point difference schemes on irregular grids in terms of homogeneous three-point difference schemes of a high-accuracy order is proposed.

In the present paper, a priori estimates of the stability in the sense of the initial data of the difference schemes approximating quasilinear parabolic equations and nonlinear transfer equation have been obtained. The basic point is connected with the necessity of estimating all derivatives entering into the nonlinear part of the difference equations. These estimates have been proved without any assumptions about the properties of the differential equations and depend only on the behavior of the initial and boundary conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for the finite instant of time t 6 t0 connected with the fact that the solution of the Riccati equation becomes infinite. is already associated with the behavior of the second derivative of the initial function and coincides with the time of the exact solution destruction (heat localization in the peaking regime). A close relation between the stability and convergence of the difference scheme solution is given. Thus, not only a priori estimates for stability have been established, but it is also shown that the obtained conditions permit exact determination of the time of destruction of the solution of the initial boundary value problem for the original nonlinear differential equation in partial derivatives. In the present paper, concrete examples confirming the theoretical conclusions are given.

Chebyshev polynomials of the first and second kind are used to derive approximate solutions of the Cauchy-type singular integral equations.