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# Home – Home – Jornals – Numerical Analysis and Applications 2022 number 2

## Numerical Analysis and Applications

### 2022 year, number 2

#### 1.Numerical algorithm for solving Prandtl equations with induced pressure in periodic case

R.K. Gaydukov
National Research University В«Higher School of Economics», Moscow, Russia
Keywords: double-deck structure, averaging, Prandtl equations with induced pressure, periodic perturbations, numerical modeling

Abstract >>
A viscous liquid flow along a semi-infinite plate with small periodic irregularities on the surface was considered for large Reynolds numbers. The flow near the plate is described by Prandtl equations with induced pressure which are non-classical PDE, because they contain a limiting term. The main goal is to construct a numerical algorithm for solving these equations with periodic boundary conditions. The results of numerical modeling of the flow are presented.

#### 2.Multiplicative model the allocation of components of the time series

V.I. Zorkal’tsev1,2, M.N. Polkovskaya3
1Limnological Institute of the Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia
2Baikal State University, Irkutsk, Russia
3Irkutsk State Agrarian University named after A.A. Yezhevsky, P. Molodezhny, Russia
Keywords: time series decomposition, axiomatic approach to method selection, multiplicative model

Abstract >>
This paper is devoted to the substantiation of a multiplicative model of time series decomposition based on the axiomatic approach. In this model, the original time series is represented as a component-by-component product of the selected components. The components are described in the form of monomials. To determine the values of variable monomials, a function is minimized that measures deviations from the unit of all components of the product of the selected components from the values of the corresponding components of the original series. In the model under consideration, all the components of the original series and the selected components are positive numbers. Four requirements are formulated for methods of selecting components. It is proved that all these requirements are met if and only if the time series decomposition is performed by a multiplicative model. As an example, we consider a model for selecting trends and seasonal fluctuations from monthly series of efficient data.

#### 3.Mathematical modeling and computational aspects of multi-criteria optimization of the conditions of the laboratory catalytic reaction

K.F. Koledina1,2, I.M. Gubaydullin1,2, S.N. Koledin2
1Institute of Petrochemistry and Catalysis of RAS, Ufa, Russia
2Ufa State Petroleum Technological University, Ufa, Russia
Keywords: multicriteria optimization, kinetic model, benzylalkyl ethers, molar ratios of initial reagents, Pareto approximation

Abstract >>
Based on the previously developed kinetic model of the catalytic reaction of the synthesis of benzylalkyl ethers, two- and three-criteria optimization of the conditions was carried out. The problem of multicriteria optimization is formulated with the definition of variable parameters: reaction temperature, proportionality coefficient of the starting reagents, reaction time; optimality criteria: yield of a target and by-products; restrictions on variable parameters. The computational aspects of multicriteria optimization by a grid algorithm (sensing) are examined. The calculated front values (optimality criteria) and Pareto sets (variable parameters) determine the exhaustive values of the reaction conditions and allow the decision maker to choose the most optimal ones. This made it possible to give technological recommendations for the industrial implementation of the synthesis of a benzyl butyl ether in the presence of a metal complex catalyst with a maximum yield of target products and a minimum content of by-products.

#### 4.Using piecewise-linear reconstruction to constructing a low-dissipation HLL method for numerical solution of hydrodynamics equation

I.M. Kulikov
Institute of Computational Mathematics and Mathematical Geophysics of Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia
Keywords: numerical modeling, computational astrophysics, HLL solver

Abstract >>
In this paper, one construction of the original «Harten-Lax-van Leer» method using a piecewise-linear reconstruction of physical variables is described. The obtained numerical method makes possible to reproduce a low-dissipation solution at discontinuities. To verify the method, we used the classical problems with an analytical solution based on various configurations of shock waves, contact discontinuities, and rarefaction waves. On the Sod-like problem, the order of accuracy of the developed numerical method was studied, it was shown that the main suppression of the order of accuracy occurs when the rarefaction wave is reproduced. The numerical method was verified by means of a three-dimensional Sedov test of a point explosion, and on the problem of a supernova Ia type explosion with two symmetric ignition points, leading to the formation of a G1.9+0.3 like remnant.

#### 5.Regularization of Fourier series with approximate coefficients for the problem of phase probability density function estimation

M.L. Maslakov1,2, V.V. Egorov1,2
1Russian Institute of Power Radiobuilding, St. Petersburg, Russia
2Saint Petersburg State University of Aerospace Instrumentation, St. Petersburg, Russia
Keywords: angle estimation, phase, phase probability distribution function, Fourier series, regularization, multiparameter regularization

Abstract >>
This paper considers the problem of phases probability distribution function estimation for phase-shift-keying signals. The modulating sequence, and, accordingly, the values of the symbols phases, as well as the statistical characteristics of this sequence are unknown. The Fourier coefficients are calculated based on a limited sample for estimation of phases probability distribution function. In this case, the obtained Fourier coefficients are regularized. Application of multiparameter regularization for increasing the estimation accuracy are considered. The numerical simulation results are presented.

#### 6.Behavior change a virus-resistant HIV-1 mathematical model

Rabiu Musa, Robert Willie, Nabendra Parumasur
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
Keywords: South Africa, HIV-1, resistance, behavior change, partial and total abstinence, demography

Abstract >>
Resistance to HIV-1 disease developed by some exposed individuals has shown a promising sign in the fight against HIV-1 infection. Behavior change has also become one of the most important protection strategies against HIV-1 pandemic, but both of them have been widely neglected by mathematical modelers. In this paper, a new virus resistance HIV-1 mathematical model incorporating behavior change is formulated and analyzed rigorously for both partial and total abstinence cases. The calculated reproduction number is used to establish the local stability of the disease-free equilibrium points using the approach of Watmough and Driessche in both cases. Using appropriate demographic and epidemiological data for South Africa in the numerical simulation, the positive effect of behavior change in the midst of HIV-1 resistance is thoroughly examined, and this strategy is absolutely effective in dealing with the threat of HIV-1. This work also provides a better result than what is obtainable in the majority of the referenced related works.

#### 7.An efficient logarithmic barrier method without line search for convex quadratic programming

Soraya Chaghoub1, Djamel Benterki2
1School of Mathematical Science Institute of Mathematics, Nanjing Normal University, Nanjing, China
2Laboratory of Fundamental and Numerical Mathematics, Setif, Algeria
Keywords: quadratic programming, linear programming, interior point methods, line search, approximate function

Abstract >>
In this work, we deal with a convex quadratic problem with inequality constraints. We use a logarithmic barrier method based on some new approximate functions. These functions have the advantage that they allow computing the displacement step easily and without consuming much time contrary to a line search method, which is time-consuming and expensive to identify the displacement step. We have developed an implementation with MATLAB and conducted numerical tests on some examples of considerable size. The obtained numerical results show the accuracy and efficiency of our approach.

#### 8.Approximation properties by some modified Szasz-Mirakjan-Kantorovich operators

Rishikesh Yadav1, Ramakanta Meher1, Vishnu Narayan Mishra2
1Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology Surat, Gujarat,India