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The numerical approach for study of the 3D-nonlinear problem for the spherically-nonsymmetric polaron is considered. An expansions in spherical harmonics are used for the approximation of the solution. The iterative Newton's scheme with an additional parametrization of the initial equation for the solving of the nonlinear problem is proposed. The results of numerical modelling are discussed. The comparison...
The iterative schemes based on the Continuous analogue of the Newton's method (CANM) have been applied to the solving a number of the QCD problems. A mathematical statement of boundary problems is formulated for the QCD-inspired potential quarkonium model on the basis of Schwinger-Dyson and Bethe-Salpeter equations. The explicit form of these equations depends on a chosen effective potential...
A parallel implementation of the SOR iterative method is presented for the solution of block banded linear systems. The algorithm is based on the block reordering of the coefficient matrix used by the domain decomposition methods. It is proved that the obtained iteration matrix maintains the same spectral properties of the corresponding sequential method and also the same optimal parameter of relaxation...
The aim of this paper is to obtain the conditions of applicability of FLAC3 finite elements in the plane. Some approximate properties and applications are discussed.
Iterative schemes based on continuous analogue of the Newton's method have been successfully used in computational modeling of various nonlinear physical problems. The paper concerns the utilization of this method to solve magnetostatic field equations. The extended convergence domain of the method makes possible to construct fast, reliable, robust iteration scheme for magnetic field calculations...
An iterative algorithm for numerical simulation of coupled heat-mass transfer and chemical reaction around flat interface in two-phase steady laminar flow is presented. The mathematical model is based on the boundary layer approximation of 2D Navier-Stokes equations and corresponding convection-diffusion equations for heat and concentration in both phases. The obtained system of nonlinear partial...
The bulk synchronous parallel (BSP) model promises scalable and portable software for a wide range of applications. A BSP computer consists of several processors, each with private memory, and a communication network that delivers access to remote memory in uniform time. Numerical linear algebra computations can benefit from the BSP model, both in terms of simplicity and efficiency. Dense LU...
In this paper we consider initial value problems for heat equation with discontinuous heat flow and concentrated heat capacity in interior points or at the boundary. Convergence of the Crank-Nicolson scheme is analyzed via the concept of elliptic projection. Namely, second order convergence is proved for the corresponding elliptic problems in special norms. Then, splitting the error of the heat problem...
Large ozone concentrations have harmful effects on forests and crops when these exceed some critical levels. It is believed that the damages in USA due to high ozone concentrations exceed several billions dollars. Therefore it is worthwhile to investigate different actions that could be applied in the attempts to reduce the harmful effects. One needs reliable mathematical models in such studies. Reliable...
Many numerical methods for the approximation of ordinary differential equations (ODEs) are obtained by using Linear Multistep Formulae (LMF). Such methods, however, in their usual implementation suffer of heavy theoretical limitations, summarized by the two well known Dahlquist barriers. For this reason, Runge-Kutta schemes have become more popular than LMF, in the last twenty years. This situation...
In this paper, we study an ill-posed, nonlinear inverse problem in heat conduction and hydrology applications. In [2], the problem is linearized to give a linear integral equation, which is then solved by the Tikhonov method with the identity as the regularization operator. We prove in this paper that the resulting equation is well-condition and has clustered spectrum. Hence if the conjugate gradient...
For target α of the Nth-degree polynomial P (z), ¦δ*/δ¦≡ ¦(z* −α)/(z −α)¦=O [σδ¦q−1] < 1 if q > 1 and ¦σδ¦ ≪ 1, regardless of ¦δ¦ itself. Even if α is not a zero but the centroid of a cluster, the recomputed multiplicity estimate m (z) could lead to a component zero. In global iterations, popular methods proved inadequate, yet for symmetric clusters the CLAM formula z*=...
A numerical method is developed for a time dependent reaction diffusion two dimensional problem. This method is deduced by combining an alternating direction technique and the central finite difference scheme on some special piecewise uniform meshes. We prove that this method is uniformly convergent with respect to the diffusion parameter ε, achieving order 1 in both spatial and time variables. The...
A class of higher order methods is investigated which can be viewed as implicit Taylor series methods based on Hermite quadratures. Improved automatic differentiation techniques for the claculation of the Taylor-coefficients and their Jacobians are used. A new rational predictor is used which can allow for larger step sizes on stiff problems.
The T algebra, related to the discrete sine transform, is an efficient tool for approximating Toeplitz matrices arising in image processing. We present two applications concerning the computation of singular values and the preconditioning of least squares problems.
We consider here a stochastic discrete minimax control problem with infinite horizon. We prove the existence of solution, we characterize it and we present iterative methods to compute it numerically.
An algorithm with result verification for linear systems involving uncertainties in the input data is realized for Maple. We give an overview on the collection of procedures designed as a package and report on numerical experiments which confirm the feasibility of implementing such algorithms in computer algebra systems. The package will be made available under the name Velisy for the Maple share...
A common Monte Carlo approach for linear algebra problems is presented. The considered problems are inverting a matrix B, solving systems of linear algebraic equations of the form Bu=b and calculating eigenvalues of symmetric matrices. Several algorithms using the same Markov chains with different random variables are described. The presented algorithms contain iterations with a resolvent matrix...
The strengthened Sobolev spaces are naturally connected, e.g., with such important (two or three-dimensional) problems of mathematical physics as those in theory of plates and shells with stiffeners or in the capillary hydrodynamics involving the surface tension. These nonstandard Hilbert spaces allow also to set variational and operator problems on composed manifolds of different dimensionality....
Using the notion of a block P-matrix, introduced previously by the authors, a characterization of the nonsingularity (Schur stability, resp.) of all convex combinations of three nonsingular (Schur stable, resp.) real matrices is derived.
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