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Axioms
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Mathematical Analysis in Application to Solving Mathematical and Technical Problems
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Mathematical Analysis in Application to Solving Mathematical and Technical Problems

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (30 December 2023) | Viewed by 11768

Special Issue Editors

Dr. Victor Orlov

E-Mail Website
Guest Editor
Department of Higher Mathematics, Institute of Digital Technologies and Mathematical Modeling in Construction (ICTMS), National Research Moscow State University (NRU MGSU), 129337 Moscow, Russia
Interests: analytical theory of differential equations; nonlinear differential equations; differential equations with fractional derivatives; mathematical modeling; computational mathematics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Michal Feckan

E-Mail Website
Guest Editor
Department of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratislava, Bratislava, Slovakia
Interests: dynamical systems; fractional systems; functional analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The Special Issue is devoted to the application of mathematical analysis to solving mathematical problems areas of differential equations and optimal control, mathematical physics, mathematical problems of artificial intelligence, and interdisciplinary applications of mathematical theory in the study of building structures and energy in calculations. In particular, we propose the development of a method for substantiating mathematical models based on differential equations, with both ordinary and fractional derivatives in various fields of human activity, and the development of the mathematical apparatus of artificial intelligence elements to expand the capabilities of classical numerical methods. These are interesting for use in the following circumstances: when obtaining conditions for the solvability of differential equations in the analytical theory of differential equations; when proving existence and uniqueness theorems for solutions of non-linear differential equations with moving singular points, both in the domain of analyticity and in a neighborhood of moving singular points, as well as when constructing analytical approximate solutions of such equations; and when proving existence and uniqueness theorems for solutions to equations with fractional derivatives and proving convergence in numerical calculations.

The Special Issue is, naturally, open to new ideas beyond the topics listed above. We hope that this initiative will be attractive to experts in the theory of differential equations with both ordinary and fractional derivatives, as well as their theoretical and practical applications in various fields of human activity.

Researchers from interdisciplinary fields are invited to submit original research results and review articles to this issue.

Contributions may be made on a rolling basis until the deadline. Upon completion of the peer review process, submissions will be selected for publication based on their quality and relevance. 

Dr. Victor Orlov
Prof. Dr. Michal Feckan
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. Axioms is an international peer-reviewed open access monthly 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 2400 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

  • a theorem on the existence and uniqueness of a solution to a nonlinear differential equation in the domain of analyticity
  • a theorem on the existence and uniqueness of a solution to a nonlinear differential equation in the vicinity of a moving singular point
  • existence theorem for a solution to a differential equation with fractional derivatives
  • analytical approximate solution of a nonlinear differential equation in the domain of analyticity, a priori and a posteriori error estimates
  • analytical approximate solution of a nonlinear differential equation in the vicinity of a moving singular point, a priori and a posteriori error estimates
  • approximate analytical and numerical methods for solving a differential equation with fractional derivatives
  • verification of the fractional derivative index for the mathematical model of the process under study
  • mathematical substantiation of the equation with a fractional derivative as a model of the process under study
  • mathematical models of building structures
  • stability analysis of fractional differential equations
  • mathematical model of electrical networks

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Published Papers (8 papers)

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Research

29 pages, 939 KiB  
Article
Sharp Power Mean Bounds for Two Seiffert-like Means
by Zhenhang Yang and Jing Zhang
Axioms 2023, 12(10), 910; https://doi.org/10.3390/axioms12100910 - 25 Sep 2023
Viewed by 1018
Abstract
The mean is a subject of extensive study among scholars, and the pursuit of optimal power mean bounds is a highly active field. This article begins with a concise overview of recent advancements in this area, focusing specifically on Seiffert-like means. We establish [...] Read more.
The mean is a subject of extensive study among scholars, and the pursuit of optimal power mean bounds is a highly active field. This article begins with a concise overview of recent advancements in this area, focusing specifically on Seiffert-like means. We establish sharp power mean bounds for two Seiffert-like means, including the introduction and establishment of the best asymmetric mean bounds for symmetric means. Additionally, we explore the practical applications of these findings by extending several intriguing chains of inequalities that involve more than ten means. This comprehensive analysis provides a deeper understanding of the relationships and properties of these means. Full article
(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)
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26 pages, 341 KiB  
Article
On the Geometry of the Riemannian Curvature Tensor of Nearly Trans-Sasakian Manifolds
by Aligadzhi R. Rustanov
Axioms 2023, 12(9), 837; https://doi.org/10.3390/axioms12090837 - 29 Aug 2023
Viewed by 934
Abstract
This paper presents the results of fundamental research into the geometry of the Riemannian curvature tensor of nearly trans-Sasakian manifolds. The components of the Riemannian curvature tensor on the space of the associated G-structure are counted, and the components of the Ricci tensor [...] Read more.
This paper presents the results of fundamental research into the geometry of the Riemannian curvature tensor of nearly trans-Sasakian manifolds. The components of the Riemannian curvature tensor on the space of the associated G-structure are counted, and the components of the Ricci tensor are calculated. Some identities are obtained that are satisfied by the Riemannian curvature tensors and the Ricci tensor. A number of properties are proved that characterize nearly trans-Sasakian manifolds with a closed contact form. The structure of nearly trans-Sasakian manifolds with a closed contact form is obtained. Several classes are singled out in terms of second-order differential-geometric invariants, and their local structure is obtained. The k-nullity distribution of a nearly trans-Sasakian manifold is studied. Full article
(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)
13 pages, 314 KiB  
Article
Nonlinear Differential Equations of Flow Motion Considering Resistance Forces
by Sergej Evtushenko, Victor Kokhanenko and Olga Burtseva
Axioms 2023, 12(9), 836; https://doi.org/10.3390/axioms12090836 - 29 Aug 2023
Viewed by 782
Abstract
For a stationary potential 2D planar open high-velocity water flow of the ideal liquid, we propose a closed system of nonlinear equations considering the resistance forces to the flow from the channel bottom. Tangential stresses on jet interfaces are ignored. The resistance force [...] Read more.
For a stationary potential 2D planar open high-velocity water flow of the ideal liquid, we propose a closed system of nonlinear equations considering the resistance forces to the flow from the channel bottom. Tangential stresses on jet interfaces are ignored. The resistance force components are expressed in terms of velocity components. In this case, the flow equations can be solved through the method of characteristics, and the surface forces are reduced to equivalent volumetric forces. The system of non-linear equations is solved in the velocity hodograph plane; further, the transition to the physical plane takes place. Since the value of the hydrodynamic pressure decreases downstream of the flow, the friction forces to the flow in the first approximation can be considered by using the integral laws of resistance. At that, the form of the equations of motion in the plane of the velocity hodograph does not change. This fact is proved in the article. An example of calculating the water flow is provided. The kinecity, ordinates, and velocities of the flow along its extreme line are calculated without considering resistance forces. Validation of the model in the real flow is performed. Acceptable accuracy relative to experimental data is obtained. Full article
(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)
10 pages, 1255 KiB  
Article
Sturm-Liouville Problem with Mixed Boundary Conditions for a Differential Equation with a Fractional Derivative and Its Application in Viscoelasticity Models
by Ludmila Kiryanova and Tatiana Matseevich
Axioms 2023, 12(8), 779; https://doi.org/10.3390/axioms12080779 - 11 Aug 2023
Cited by 1 | Viewed by 1211
Abstract
In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem of a second-order differential equation containing a fractional derivative operator. The fractional differentiation operator was considered according to two definitions: Gerasimov-Caputo and Riemann-Liouville-Visualizations of the system [...] Read more.
In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem of a second-order differential equation containing a fractional derivative operator. The fractional differentiation operator was considered according to two definitions: Gerasimov-Caputo and Riemann-Liouville-Visualizations of the system of eigenfunctions, the biorthogonal system, and the distribution of eigenvalues on the real axis were presented. The numerical behavior of eigenvalues was studied depending on the order of the fractional derivative for both definitions of the fractional derivative. Full article
(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)
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14 pages, 282 KiB  
Article
Geometry of Harmonic Nearly Trans-Sasakian Manifolds
by Aligadzhi R. Rustanov
Axioms 2023, 12(8), 744; https://doi.org/10.3390/axioms12080744 - 28 Jul 2023
Cited by 1 | Viewed by 852
Abstract
This paper considers a class of nearly trans-Sasakian manifolds. The local structure of nearly trans-Sasakian structures with a closed contact form and a closed Lee form is obtained. It is proved that the class of nearly trans-Sasakian manifolds with a closed contact form [...] Read more.
This paper considers a class of nearly trans-Sasakian manifolds. The local structure of nearly trans-Sasakian structures with a closed contact form and a closed Lee form is obtained. It is proved that the class of nearly trans-Sasakian manifolds with a closed contact form and a closed Lee form coincides with the class of almost contact metric manifolds with a closed contact form locally conformal to the closely cosymplectic manifolds. A wide class of harmonic nearly trans-Sasakian manifolds has been identified (i.e., nearly trans-Sasakian manifolds with a harmonic contact form) and an exhaustive description of the manifolds of this class is obtained. Also, examples of harmonic nearly trans-Sasakian manifolds are given. Full article
(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)
13 pages, 666 KiB  
Article
Numerical Analysis of New Hybrid Algorithms for Solving Nonlinear Equations
by Miguel Vivas-Cortez, Naseem Zulfiqar Ali, Awais Gul Khan and Muhammad Uzair Awan
Axioms 2023, 12(7), 684; https://doi.org/10.3390/axioms12070684 - 12 Jul 2023
Cited by 2 | Viewed by 2313
Abstract
In this paper, we propose two new hybrid methods for solving nonlinear equations, utilizing the advantages of classical methods (bisection, trisection, and modified false position), i.e., bisection-modified false position (Bi-MFP) and trisection-modified false position (Tri-MFP). We implemented the proposed algorithms for several benchmark [...] Read more.
In this paper, we propose two new hybrid methods for solving nonlinear equations, utilizing the advantages of classical methods (bisection, trisection, and modified false position), i.e., bisection-modified false position (Bi-MFP) and trisection-modified false position (Tri-MFP). We implemented the proposed algorithms for several benchmark problems. We discuss the graphical analysis of these problems with respect to the number of iterations and the average CPU time. Full article
(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)
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17 pages, 5780 KiB  
Article
Mathematical Analysis of the Vibratory Pile Driving Rate
by Armen Z. Ter-Martirosyan, Alexander N. Shebunyaev and Vitalii V. Sidorov
Axioms 2023, 12(7), 629; https://doi.org/10.3390/axioms12070629 - 26 Jun 2023
Cited by 2 | Viewed by 2075
Abstract
Vibratory piling technology does not require analytical tools to predict displacement rates and arising forces. The authors consider the problem of vibratory driving of a pile into a homogeneous unsaturated sandy massif under the action of static and dynamic loads. The purpose of [...] Read more.
Vibratory piling technology does not require analytical tools to predict displacement rates and arising forces. The authors consider the problem of vibratory driving of a pile into a homogeneous unsaturated sandy massif under the action of static and dynamic loads. The purpose of this study is to develop a new analytical solution to the problem of the vibratory pile driving rate in a homogeneous sand base taking vibration creep into account. The solution is provided for the quasi-dynamic problem statement (inertial terms in equations of motion are neglected): the sand medium develops viscous properties due to vibration under the action of the dynamic component of the load, and a pile is driven into the viscous sand base due to the static component of the vertical load. The obtained mathematical model converges with the results of laboratory flume and field experiments performed by other researchers earlier, where the pile vibratory embedding rate increased along with an increase in static loading, the amplitude of dynamic load, and vibration frequency. It can be used to predict the pile or sheet pile driving rate into the unsaturated sand base under the action of vibration, and also to evaluate the necessary parameters of pile driving to obtain the required value of the pile embedding rate. Full article
(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)
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9 pages, 276 KiB  
Article
Dependence of the Analytical Approximate Solution to the Van der Pol Equation on the Perturbation of a Moving Singular Point in the Complex Domain
by Victor Orlov
Axioms 2023, 12(5), 465; https://doi.org/10.3390/axioms12050465 - 12 May 2023
Cited by 3 | Viewed by 1128
Abstract
This paper considers a theoretical substantiation of the influence of a perturbation of a moving singular point on the analytical approximate solution to the Van der Pol equation obtained earlier by the author. A priori estimates of the error of the analytical approximate [...] Read more.
This paper considers a theoretical substantiation of the influence of a perturbation of a moving singular point on the analytical approximate solution to the Van der Pol equation obtained earlier by the author. A priori estimates of the error of the analytical approximate solution are obtained, which allows the solving of the inverse problem of the theory of error: what should the structure of the analytical approximate solution be in order to obtain a result with a given accuracy? Thanks to a new approach for obtaining a priori evaluations of errors, based on elements of differential calculus, the domain, used to obtain an analytical approximate solution, was substantially expanded. A variant of optimizing a priori estimates using a posteriori estimates is illustrated. The results of a numerical experiment are also presented. Full article
(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)
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