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Mathematics
Special Issues
Construction and Research of Mathematical Models
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Construction and Research of Mathematical Models

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (15 November 2021) | Viewed by 6471

Special Issue Editors

Prof. Dr. Dmitry Sergeevich Kulyabov

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Guest Editor
1. Department of Applied Probability and Informatics, Peoples' Friendship University of Russia (RUDN University), Moscow 117198, Russia
2. Joint Institute for Nuclear Research, 6 Joliot-Curie st, Dubna, Moscow 141980, Russia
Interests: special relativity; optics; differential geometry; general relativity; electrodynamics; mathematical modeling
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Leonid Sevastianov

E-Mail Website
Guest Editor
1. Department of Applied Probability and Informatics, Peoples' Friendship University of Russia (RUDN University), Moscow 117198, Russia
2. Joint Institute for Nuclear Research, 6 Joliot-Curie st, Dubna, Moscow 141980, Russia
Interests: mathematical modeling; computational physics; waveguide and integrated optics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Vladimir Korenkov

E-Mail Website
Guest Editor
Scientific Director of Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research, 6 Joliot-Curie St., 141980 Dubna, Moscow Region, Russia
Interests: computing and networking; grid technologies and cloud calculations; parallel and distributed computations; visualization and multimedia systems; distributed data storages; big data; programme engineering
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematical modelling is a powerful scientific method. This method is the main method for studying physical laws. From physics, the method of mathematical modelling has spread to other areas of science. But the language of mathematics is only one of the languages for implementing models. With the development of computers, new types of modelling have appeared: simulation modelling, surrogate modelling. Furthermore, to describe the same phenomenon, it is possible to use not one model, but a whole ensemble of models. This kind of approach, when different realizations of models and ensembles of models are applied, we call the multi-model approach.

This Special Issue focuses on the use of current advances in mathematical modelling in aspects of research and application of the multi-model approach. Besides, we consider various combined aspects of model investigation, for example, a combination of analytical and numerical research methods. We invite you to contribute and submit your latest research work.

Prof. Dr. Dmitry Kulyabov
Prof. Leonid Sevastianov
Dr. Vladimir Vasilievich Korenkov
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Mathematical modelling
  • Multi-model approach
  • Analytical-numerical methods
  • Surrogate modelling

Published Papers (3 papers)

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18 pages, 854 KiB  
Article
Synthesis and Computer Study of Population Dynamics Controlled Models Using Methods of Numerical Optimization, Stochastization and Machine Learning
by Anastasia V. Demidova, Olga V. Druzhinina, Olga N. Masina and Alexey A. Petrov
Mathematics 2021, 9(24), 3303; https://doi.org/10.3390/math9243303 - 18 Dec 2021
Cited by 4 | Viewed by 2198
Abstract
The problems of synthesis and analysis of multidimensional controlled models of population dynamics are of both theoretical and applied interest. The need to solve numerical optimization problems for such a class of models is associated with the expansion of ecosystem control requirements. The [...] Read more.
The problems of synthesis and analysis of multidimensional controlled models of population dynamics are of both theoretical and applied interest. The need to solve numerical optimization problems for such a class of models is associated with the expansion of ecosystem control requirements. The need to solve the problem of stochastization is associated with the emergence of new problems in the study of ecological systems properties under the influence of random factors. The aim of the work is to develop a new approach to studying the properties of population dynamics systems using methods of numerical optimization, stochastization and machine learning. The synthesis problems of nonlinear three-dimensional models of interconnected species number dynamics, taking into account trophic chains and competition in prey populations, are studied. Theorems on the asymptotic stability of equilibrium states are proved. A qualitative and numerical study of the models is carried out. Using computational experiments, the results of an analytical stability and permanent coexistence study are verified. The search for equilibrium states belonging to the stability and permanent coexistence region is made using the developed intelligent algorithm and evolutionary calculations. The transition is made from the model specified by the vector ordinary differential equation to the corresponding stochastic model. A comparative analysis of deterministic and stochastic models with competition and trophic chains is carried out. New effects are revealed that are characteristic of three-dimensional models, taking into account the competition in populations of prey. The formulation of the optimal control problem for a model with competition and trophic chains is proposed. To find optimal trajectories, new generalized algorithms for numerical optimization are developed. A methods for the synthesis of controllers based on the use of artificial neural networks and machine learning are developed. The results on the search for optimal trajectories and generation of control functions are presented.The obtained results can be used in modeling problems of ecological, demographic, socio-economic and chemical kinetics systems. Full article
(This article belongs to the Special Issue Construction and Research of Mathematical Models)
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12 pages, 319 KiB  
Article
On the Quadratization of the Integrals for the Many-Body Problem
by Yu Ying, Ali Baddour, Vladimir P. Gerdt, Mikhail Malykh and Leonid Sevastianov
Mathematics 2021, 9(24), 3208; https://doi.org/10.3390/math9243208 - 11 Dec 2021
Cited by 5 | Viewed by 1893
Abstract
A new approach to the construction of difference schemes of any order for the many-body problem that preserves all its algebraic integrals is proposed herein. We introduced additional variables, namely distances and reciprocal distances between bodies, and wrote down a system of differential [...] Read more.
A new approach to the construction of difference schemes of any order for the many-body problem that preserves all its algebraic integrals is proposed herein. We introduced additional variables, namely distances and reciprocal distances between bodies, and wrote down a system of differential equations with respect to the coordinates, velocities, and the additional variables. In this case, the system lost its Hamiltonian form, but all the classical integrals of motion of the many-body problem under consideration, as well as new integrals describing the relationship between the coordinates of the bodies and the additional variables are described by linear or quadratic polynomials in these new variables. Therefore, any symplectic Runge–Kutta scheme preserves these integrals exactly. The evidence for the proposed approach is given. To illustrate the theory, the results of numerical experiments for the three-body problem on a plane are presented with the choice of initial data corresponding to the motion of the bodies along a figure of eight (choreographic test). Full article
(This article belongs to the Special Issue Construction and Research of Mathematical Models)
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16 pages, 1623 KiB  
Article
Bayesian Estimation of Adsorption and Desorption Parameters for Pore Scale Transport
by Vasiliy V. Grigoriev and Petr N. Vabishchevich
Mathematics 2021, 9(16), 1974; https://doi.org/10.3390/math9161974 - 18 Aug 2021
Cited by 3 | Viewed by 1478
Abstract
Stochastic parameter estimation and inversion have become increasingly popular in recent years. Nowadays, it is computationally reasonable and regular to solve complex inverse problems within the Bayesian framework. Applications of Bayesian inferences for inverse problems require investigation of the posterior distribution, which usually [...] Read more.
Stochastic parameter estimation and inversion have become increasingly popular in recent years. Nowadays, it is computationally reasonable and regular to solve complex inverse problems within the Bayesian framework. Applications of Bayesian inferences for inverse problems require investigation of the posterior distribution, which usually has a complex landscape and is highly dimensional. In these cases, Markov chain Monte Carlo methods (MCMC) are often used. This paper discusses a Bayesian approach for identifying adsorption and desorption rates in combination with a pore-scale reactive flow. Markov chain Monte Carlo sampling is used to estimate adsorption and desorption rates. The reactive transport in porous media is governed by incompressible Stokes equations, coupled with convection–diffusion equation for species’ transport. Adsorption and desorption are accounted via Robin boundary conditions. The Henry isotherm is considered for describing the reaction terms. The measured concentration at the outlet boundary is provided as additional information for the identification procedure. Metropolis–Hastings and Adaptive Metropolis algorithms are implemented. Credible intervals have been plotted from sampled posterior distributions for both algorithms. The impact of the noise in the measurements and influence of several measurements for Bayesian identification procedure is studied. Sample analysis using the autocorrelation function and acceptance rate is performed to estimate mixing of the Markov chain. As result, we conclude that MCMC sampling algorithm within the Bayesian framework is good enough to determine an admissible set of parameters via credible intervals. Full article
(This article belongs to the Special Issue Construction and Research of Mathematical Models)
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