Home – Home – Jornals – Numerical Analysis and Applications 2022 number 3

For a model of circadian oscillator represented in the form of 6-dimensional nonlinear dynamical system, conditions of uniqueness of an equilibrium point, and conditions of existence of a periodic trajectory (cycle) are established. One client-server application is elaborated in order to fulfill numerical experiments with this model on a cloud server, and to visualize results of these experiments.

The numerical algorithm without saturation for wave equation is considered. It is supposed that Laplace's operator has the discrete, valid range, and the corresponding matrix of the discrete operator Laplace has the complete set of eigenvectors. The technique speaks the example of the one-dimensional equation, but during statement is shown that the dimension is insignificant here.

This paper is devoted to constructing quadrature formulas for singular and hypersingular integrals evaluation. For evaluating the integrals with the weights (1- t )^γ₁(1+ t )^γ₂, γ₁, γ₂ > -1, defined on [-1,1], we have constructed quadrature formulas uniformly converging on [-1,1] to the original integral with the weights (1- t )^γ₁(1+ t )^γ₂, γ₁, γ₂ ≥ -1/2, and converging to the original integral for -1< t <1 with the weights (1- t )^γ₁(1+ t )^γ₂, γ₁, γ₂ > -1. In the latter case a sequence of quadrature formulas converges to evaluating integral uniformly on [-1+δ,1-δ], where δ > 0 is arbitrarily small. We propose a method for construction and error estimate of quadrature formulas for evaluating hypersingular integrals based on transformation of quadrature formulas for evaluation of singular integrals. We also propose a method of the error estimate for quadrature formulas for singular integrals evaluation based on the approximation theory methods. The results obtained were extended to hypersigular integrals.

The goal of this work is to highlight the advantages of using NonStandard Finite Differences (NSFD) numerical schemes for the resolution of Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) of which some properties of the exact solution are a-priori known, such as positivity. The main reference considered is Mickens' work [14], in which the author derives NSFD schemes for ODEs and PDEs that describe real phenomena, and therefore widely used in applications. We rigorously demonstrate that NSFD methods can have a higher order of convergence than the related classical ones, deriving also the conditions that guarantee the stability of the analyzed schemes. Furthermore, we carry out in-depth numerical tests comparing the classical methods with the NSFD ones proposed by Mickens, evaluating when the latter are decidedly advantageous.

In this paper, optimum differential schemes for the solution of the Maxwell equations with the use of the Laquerre spectral transformation are considered. Additional parameters are introduced into the differential scheme of equations for harmonics. Numerical values of these parameters are obtained by minimization of an error of differential approximation of the Helmholtz equation. The optimum values of parameters thus obtained are used when constructing differential schemes - optimum differential schemes. Two versions of optimum differential schemes are considered. It is shown that the use of optimum differential schemes leads to an increase in the accuracy of the solution of the equations. A simple modification of the differential scheme gives an increase in the efficiency of the algorithm.

This paper deals with Monte Carlo simulation of optical phenomena which appear in the lidar sensing of atmospheric clouds and water media. By numerical experiments we study peculiarities of the laser pulse propagation when the light forms expanding ring structures at the expense of multiple scattering.

In this manuscript, a new exponentially fitted operator strategy for solving a singularly perturbed parabolic partial differential equation with a right boundary layer is considered. We discretize the time variable using the implicit Euler approach and approximate the equation into first order delay differential equation with a small deviating argument using a Taylor series expansion. The two-point Gaussian quadrature formula and linear interpolation are implemented to obtain a tridiagonal system of equations. The tridiagonal system of equations is solved using the Thomas algorithm. Three numerical examples are considered to illustrate the efficiency of the present method and compared with the methods produced by different authors. Convergence of the method is analyzed. The absolute maximum error and rate of convergence are obtained for the model examples. The result shows that the present method is more accurate and ε-uniformly convergent for all ε ≤ h.

In this paper, we, first, using the reduction method in the narrow sense (the simple reduction method), have generalized the classical Gauss-Jordan method for solving finite systems of linear algebraic equations to inhomogeneous infinite systems. The generalization is based on a new theory of solutions to inhomogeneous infinite systems, proposed by us, which gives an exact analytical solution in the form of a series. Second, we have shown that the application of reduction in the narrow sense in the case of homogeneous systems gives only a trivial solution, therefore, in order to generalize the Gauss-Jordan method for solving infinite homogeneous systems, we used the reduction method in the wide sense. A numerical comparison is given that shows acceptable accuracy.