Home – Home – Jornals – Numerical Analysis and Applications 2025 number 1

The problem of determining the parameters of a large system of non-autonomous differential equations consisting of subsystems connected in an arbitrary order by non-local boundary conditions is solved. Unknown parameters participate both in differential equations and in boundary conditions. The problem under study is reduced to a parametric optimal control problem with a mean-square residual criterion estimating the degree of non-fulfillment of additionally specified boundary conditions. To apply first-order numerical methods, analytical formulas for the components of the gradient of the objective functional in the space of optimized parameters are obtained. The analysis of the obtained results of computer experiments is made using a test problem as an example.

This paper aims to provide an efficient numerical method based on the second Chebyshev wavelets for solving the fractional Langevin equation. Applying this operational matrix of fractional-order integration of second Chebyshev wavelets converts the original problem into a system of algebraic equations, which could be solved by the Newton method. After analyzing the method, the error bound is estimated. Moreover, the method's efficiency through a few numerical examples is evaluated.

We consider nonlinear dynamical systems as a model of interaction of components of a gene network which regulates early stages of an embryonic stem cells state. A parametric analysis of these dynamical systems is performed in order to describe the (non)uniqueness and stability of their equilibriums. We have obtained a criterion of existence of periodic trajectories near these points and localized these oscillations on the phase portraits of dynamical systems which describe these processes. A special software for cloud numerical experiments with these systems has been elaborated.

This article explores the computational intricacies of H. Rutishauser's quotient-difference (Q-D) algorithm and C programming code, a revolutionary advancement in polynomial analysis. Our specific focus is on cubic polynomials featuring absolute, distinct non-zero real roots, emphasizing the algorithm's distinctive capability to simultaneously approximate all zeros independently of external data. Notably, it proves invaluable in diverse domains, such as determining continuous fraction representations for meromorphic functions and serving as a powerful tool in complex analysis for the direct localization of poles and zeros. To bring this innovation into practice, the article introduces a meticulously crafted C language program, complete with a comprehensive algorithm and flowchart. Supported by illustrative examples, this implementation underscores the algorithm's robustness and effectiveness across various real-world scenarios.

Rational techniques for verifying the congruence of complex matrices are discussed. An algorithm is said to be rational if it is finite and uses only arithmetical operations. An important part in verifying the congruence of nonsingular matrices play their cosquares. The verification gets complicated if there are eigenvalues of modulus 1 in the spectrum of cosquares; this is especially true if such eigenvalues are defective. In this direction, the most advanced result is the rational algorithm for matrices A and B whose cosquare is the direct sum J_m (1) ⊕ J_m (1). Here, this algorithm is extended to the case where the cosquare is the direct sum of two Jordan blocks of distinct orders. This extension is heavily dependent on additional facts concerning the solutions to the matrix equation X - J^Τ_m(1)XJ_m(1) = 0. found in the present paper.

Traditionally, to solve the hydrodynamic equations a Godunov method is used, whose main component is the solution of a Riemann problem to compute the fluxes of the conservative variables through the interfaces. Most numerical Riemann solvers are based on partial or full spectral decompositions of the Jacobian matrix with the spatial derivatives. However, when using complex hyperbolic models and various types of equations of state, even partial spectral decompositions are quite difficult to find analytically. Such hyperbolic systems include the equations of special relativistic magnetic hydrodynamics. In this paper, a numerical Riemann solver is constructed by means of a viscosity matrix on the basis of Chebyshev polynomials. This scheme does not require information about the spectral decomposition of the Jacobian matrix, while considering all types of waves in its design. To reduce the dissipation of the numerical solution, a piecewise parabolic reconstruction of the physical variables is used. The behavior of the numerical method is studied by using some classical test problems.

A process of searching on the sphere for the best (in a sense) cubature formulas that are invariant under the transformations of different dihedral rotation groups is described. The parameters of the new cubature formulas of the 6th, 10th, and 12th order of accuracy are given to 16 significant digits. A table which contains the main characteristics of all the best to date cubature formulas of the dihedral rotation group up to the 29th order of accuracy is given.

For the case of two spatial variables, a finite-difference scheme of the predictor--corrector type is constructed for solving nonlinear dispersion equations of wave hydrodynamics with a higher order of approximation of the dispersion relation. The numerical algorithm is based on splitting the original system of equations into a hyperbolic system and a scalar equation of the elliptic type. Two methods of approximating the elliptic part are considered. For each of the variants of the difference scheme, dissipation and dispersion analysis is performed, stability conditions are obtained, formulas for the phase error are analyzed, and the behavior of the harmonic attenuation coefficient is studied. A comparative analysis is carried out to identify the advantages of each of the schemes.