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

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 32923

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Special Issue Editors

Dr. Victor Orlov

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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

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Guest Editor

Department of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratislava, Bratislava, Slovakia
Interests: dynamical systems; fractional systems; functional analysis
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Special Issue Information

Dear Colleagues.

This issue is devoted to sections of mathematical disciplines that apply the basics of mathematical analysis to prove results in such areas as differential equations and optimal control, mathematical physics, mathematical problems of artificial intelligence, and interdisciplinary applications of mathematical theory—in particular when obtaining conditions for the solvability of differential equations in the analytical theory of differential equations; when proving the existence and uniqueness theorems of solutions of nonlinear differential equations with movable singular points both in the area of analyticity and in the vicinity of movable singular points, as well as when constructing analytical approximate solutions of such equations; when proving the existence and uniqueness theorems of solutions for equations with fractional derivatives and proving convergence during numerical calculations; when developing methods for substantiating mathematical models based on differential equations of both ordinary and fractional derivatives in various fields of human activity; when developing a mathematical apparatus of artificial intelligence elements to expand the capabilities of classical numerical methods; and when applying mathematical theory in the study of building structures and calculations.

It should be noted that the Special Issue is open to further ideas beyond the aforementioned topics.

We hope that this initiative will be attractive to experts in the field of the theory of differential equations with both ordinary and fractional derivatives, as well as their theoretical and practical applications. We recommend that you submit your current research for inclusion in a Special Issue.

Researchers working in this interdisciplinary field can present their original research results, as well as explanatory and review articles.

Contributions may be submitted on an ongoing basis before the deadline. After the review process, the materials will be selected for publication based on their quality and relevance.

Prof. Dr. Viktor N. Orlov
Prof. Dr. Michal Feckan
Guest Editors

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Keywords

the theorem of existence and uniqueness of the solution of a nonlinear differential equation in the field of analyticity
the theorem of existence and uniqueness of the solution of a nonlinear differential equation in the vicinity of a moving singular point
the theorem of the existence of a solution to a differential equation with fractional derivatives
analytical approximate solution of a nonlinear differential equation in the field 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 solutions of a differential equation with fractional derivatives
verification of the fractional derivative indicator for the mathematical model of the process under study
mathematical justification for an equation with a fractional derivative as a model of the process under study
mathematical models of building structures
stability analysis of fractional differential equations

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Published Papers (17 papers)

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Research

19 pages, 1153 KiB

Open AccessArticle

Newton-like Normal S-iteration under Weak Conditions

by Manoj K. Singh, Ioannis K. Argyros and Arvind K. Singh

Axioms 2023, 12(3), 283; https://doi.org/10.3390/axioms12030283 - 8 Mar 2023

Viewed by 1193

Abstract

In the present paper, we introduced a quadratically convergent Newton-like normal S-iteration method free from the second derivative for the solution of nonlinear equations permitting

f^{'} (x) = 0

at some points in the neighborhood of the root. Our proposed [...] Read more.

In the present paper, we introduced a quadratically convergent Newton-like normal S-iteration method free from the second derivative for the solution of nonlinear equations permitting

f^{'} (x) = 0

at some points in the neighborhood of the root. Our proposed method works well when the Newton method fails and performs even better than some higher-order converging methods. Numerical results verified that the Newton-like normal S-iteration method converges faster than Fang et al.’s method. We studied different aspects of the normal S-iteration method regarding the faster convergence to the root. Lastly, the dynamic results support the numerical results and explain the convergence, divergence, and stability of the proposed method. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

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22 pages, 362 KiB

Open AccessArticle

On the Structure of Coisometric Extensions

by Dan Popovici

Axioms 2023, 12(2), 202; https://doi.org/10.3390/axioms12020202 - 14 Feb 2023

Viewed by 1406

Abstract

If T is a bounded linear operator on a Hilbert space

H

and V is a given linear isometry on a Hilbert space

K

, we present necessary and sufficient conditions on T in order to ensure the existence of a linear isometry [...] Read more.

If T is a bounded linear operator on a Hilbert space

H

and V is a given linear isometry on a Hilbert space

K

, we present necessary and sufficient conditions on T in order to ensure the existence of a linear isometry

π : H \to K

such that

π T = V^{*} π

(i.e.,

(π, V^{*})

extends T). We parametrize the set of all solutions

π

of this equation. We show, for example, that for a given unitary operator U on a Hilbert space

E

and for the multiplication operator by the independent variable

M_{z}

on the Hardy space

H_{D}^{2} (D)

, there exists an isometric operator

π : H \to E \oplus H_{D}^{2} (D)

such that

(π, {(U \oplus M_{z})}^{*})

extends T if and only if T is a contraction, the defect index

δ_{T} \leq dim D

and, for some

Y : A_{T} \to E

(Y, U^{*})

extends the isometric operator

A_{T}^{1 / 2} h \mapsto A_{T}^{1 / 2} T h

on the space

A_{T} = \bar{A_{T} H}

, where

A_{T}

is the asymptotic limit associated with T. We also prove that if T is isometric and V is unitary, there exists an isometric operator

π : H \to K

such that

(π, V)

extends T if and only if (a) the spectral measures of the unitary part of T (in its Wold decomposition) and the restriction of V to one of its reducing subspaces

K_{0}

possess identical multiplicity functions and (b)

dim (ker T^{*}) = dim (K_{1} ⊖ V K_{1})

for a certain subspace

K_{1}

K

that contains

K_{0}

and is invariant under V. The precise form of

π

, in each situation, and characterizations of the minimality conditions are also included. Several examples are given for illustrative purposes. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

16 pages, 516 KiB

Open AccessArticle

A Nonlinear System of Differential Equations in Supercritical Flow Spread Problem and Its Solution Technique

by Sergej Evtushenko

Axioms 2023, 12(1), 11; https://doi.org/10.3390/axioms12010011 - 22 Dec 2022

Cited by 5 | Viewed by 1534

Abstract

A nonlinear system of differential equations in the problem of free flowing of supercritical flow is considered and a method of its solution is proposed. The analytical method is based on the introduction of the velocity hodograph plane and the obtaining of analytical solutions for the system of partial differential equations. It is pointed out that apart from being purely analytical, the potential flow model has a great practical demand due to its use as a base for the further research of the flow resistance forces. The proposed model can be developed by taking into account flow resistance and gradient, the bottom of the diverting channel flow. The theoretical results are complemented by numerical experiments and compared with experimental data. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

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12 pages, 2985 KiB

Open AccessArticle

Mathematical Computations of Long-Term Settlement and Bearing Capacity of Soil Bases and Foundations near Vertical Excavation Pits

by Zaven G. Ter-Martirosyan, Armen Z. Ter-Martirosyan and Yulia V. Vanina

Axioms 2022, 11(12), 679; https://doi.org/10.3390/axioms11120679 - 29 Nov 2022

Cited by 5 | Viewed by 1829

Abstract

The present paper describes and provides an analytical solution for the problem of evaluating the settlement and load-bearing capacity of weighty soil layers of limited thickness resting upon incompressible soil bases and an excavation pit wall, upon exposure of the foundation to a distributed load in the vicinity of a wall. The authors developed a method for determining the stressed state component in the reduced engineering problem based on the Ribere–Faylon trigonometric series, accounting for the nonlinear deformation properties of soils. To determine the settlement over time of the foundation near the pit, we used the A.Z. Ter-Martirosyan’s model to describe shear deformations and the Kelvin–Voigt model to describe volume deformations, assuming that ε·_z(t) = ε·_v(t) + ε·_γ(t), according to the Hencky’s system of physical equations. The obtained solutions make it possible to assess the long-term deformation of soil bases and the long-term load-bearing capacity with respect to nonlinear rheological properties in a way that accurately corresponds to the actual performance of subsoils exposed to loading. The theoretical results were followed by numerical experiments to prove their validity. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

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9 pages, 266 KiB

Open AccessArticle

Nearly Sasakian Manifolds of Constant Type

by Aligadzhi Rustanov

Axioms 2022, 11(12), 673; https://doi.org/10.3390/axioms11120673 - 26 Nov 2022

Viewed by 1260

Abstract

The article deals with nearly Sasakian manifolds of a constant type. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the nearly Sasakian manifold is a nearly Kähler manifold. It is proved that the class of nearly Sasakian manifolds of the zero constant type coincides with the class of Sasakian manifolds. The concept of constancy of the type of an almost contact metric manifold is introduced through its Nijenhuis tensor, and the criterion of constancy of the type of an almost contact metric manifold is proved. The coincidence of both concepts of type constancy for the nearly Sasakian manifold is proved. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the almost contact metric manifold of the zero constant type is the Hermitian structure. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

8 pages, 277 KiB

Open AccessArticle

Analytic Approximate Solution in the Neighborhood of a Moving Singular Point of a Class of Nonlinear Equations

by Victor Orlov and Magomedyusuf Gasanov

Axioms 2022, 11(11), 637; https://doi.org/10.3390/axioms11110637 - 12 Nov 2022

Cited by 5 | Viewed by 1394

Abstract

The paper considers a class of nonlinear differential equations which are not solvable in quadratures in general case. The author’s technology for solving such equations contains six problems. In this article, the solution to one of these problems is given, a real area in which it is possible to calculate an analytical approximate solution in the case of an approximate value of a moving singular point is obtained. Obtained results are based on the application of elements of differential calculus in finding estimates for the approximate solution error. Theoretical provisions are confirmed by numerical calculations, which characterize their reliability. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

16 pages, 306 KiB

Open AccessArticle

On One Approximate Method of a Boundary Value Problem for a One-Dimensional Advection–Diffusion Equation

by Temirkhan Aleroev and Victor Orlov

Axioms 2022, 11(10), 541; https://doi.org/10.3390/axioms11100541 - 10 Oct 2022

Cited by 2 | Viewed by 1545

Abstract

This article discusses the author’s version of the technology for solving a one-dimensional boundary value problem for a one-dimensional advection–diffusion equation based on the method of separation of variables, as well as the theory of eigenvalues and eigenfunctions when constructing a solution to a differential equation. This problem is solved in two stages. Firstly, we illustrate the technology of separating variables for equations with fractional derivatives, and then apply the theory of eigenvalues and eigenfunctions to obtain an exact solution in the form of an infinite series. Since this series converges very quickly, it is natural to replace it with the sum of the first few terms. The approximate solution obtained in this way is quite suitable for numerical calculations in practice. The article provides a listing of the program for performing calculations, as well as the results of calculations themselves. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

12 pages, 2842 KiB

Open AccessArticle

Mathematical Analysis for the Evaluation of Settlement and Load-Bearing Capacity of a Soil Base Adjacent to an Excavation Pit

by Zaven G. Ter-Martirosyan, Armen Z. Ter-Martirosyan and Yulia V. Vanina

Axioms 2022, 11(8), 353; https://doi.org/10.3390/axioms11080353 - 22 Jul 2022

Cited by 3 | Viewed by 1941

Abstract

The present paper states and provides an analytical solution for the problem of evaluating the settlement and load-bearing capacity of weighty soil layers of limited thickness resting upon incompressible soil bases and an excavation pit wall upon exposure of the foundation to a distributed load in the vicinity of a wall. The authors develop a method for determining the stressed state component in the reduced engineering problem based on the Ribere–Faylon trigonometric series and for accounting for the nonlinear deformation properties of soils, building on the analytical dependencies of S.S. Grigoryan and S.P. Timoshenko. In order to determine the relationship between stress and strain, the Hencky’s physical equation systems were used. They factor in the impact of average stresses σ_m on the shear modulus of elasticity G (σ_m) and volumetric modulus of elasticity K (σ_m). The obtained solutions make it possible to assess the deformation of soil bases and the load-bearing capacity with respect to nonlinear properties in a way that accurately corresponds to the actual performance of subsoils exposed to loading. The theoretical results are followed by numerical experiments to prove their validity. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

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7 pages, 242 KiB

Open AccessArticle

On a Class of Positive Definite Operators and Their Application in Fractional Calculus

by Temirkhan Aleroev

Axioms 2022, 11(6), 272; https://doi.org/10.3390/axioms11060272 - 5 Jun 2022

Cited by 2 | Viewed by 2069

Abstract

This paper is devoted to the spectral analysis of one class of integral operators, associated with the boundary value problems for differential equations of fractional order. Approximation matrices are also investigated. In particular, the positive definiteness of the studied operators is shown, which [...] Read more.

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

16 pages, 1784 KiB

Open AccessArticle

Solvability of Conformable Type Frictionless Contact Problem via Hemivariational Inequalities

by Jianwei Hao, Jinrong Wang and Jiangfeng Han

Axioms 2022, 11(6), 271; https://doi.org/10.3390/axioms11060271 - 5 Jun 2022

Viewed by 1883

Abstract

In this paper, we study a class of conformable frictionless contact problems with the surface traction driven by the conformable impulsive differential equation. The existence of a mild solution for conformable impulsive hemivariational inequality is obtained by the Rothe method, subjectivity of multivalued pseudomonotone operators and the property of the conformable derivative. Notice that we imply some new fractional viscoelastic constitutive laws. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

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8 pages, 241 KiB

Open AccessArticle

Exact Criteria for the Existence of a Moving Singular Point in a Complex Domain for a Nonlinear Differential Third-Degree Equation with a Polynomial Seventh-Degree Right-Hand Side

by Victor Orlov and Magomedyusuf Gasanov

Axioms 2022, 11(5), 222; https://doi.org/10.3390/axioms11050222 - 10 May 2022

Cited by 2 | Viewed by 1774

Abstract

Earlier, the authors formulated and proved interval and point criteria for the existence of moving singular points of a third-degree nonlinear differential equation with a polynomial seventh-degree right-hand side for a real domain. For the complex domain, these criteria are associated with specificity of transition to phase spaces. Necessary as well as necessary and sufficient conditions for the existence of moving singular points are obtained. The results of these studies are the basis of the algorithm for obtaining moving singular points of nonlinear differential equations in the complex domain. The results of this paper allow us to expand the classes of the studied nonlinear differential equations with moving peculiarities. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

10 pages, 628 KiB

Open AccessArticle

The Boundary Value Problem with Stationary Inhomogeneities for a Hyperbolic-Type Equation with a Fractional Derivative

by Ludmila Vladimirovna Kirianova

Axioms 2022, 11(5), 207; https://doi.org/10.3390/axioms11050207 - 28 Apr 2022

Cited by 1 | Viewed by 1750

Abstract

The paper presents an analytical solution of a partial differential equation of hyperbolic-type, containing both second-order partial derivatives and fractional derivatives of order below the second. Examples of applying the solution of a boundary value problem with stationary inhomogeneities for a hyperbolic-type equation [...] Read more.

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

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6 pages, 245 KiB

Open AccessArticle

Existence and Uniqueness Theorem for a Solution to a Class of a Third-Order Nonlinear Differential Equation in the Domain of Analyticity

by Victor Orlov and Magomedyusuf Gasanov

Axioms 2022, 11(5), 203; https://doi.org/10.3390/axioms11050203 - 25 Apr 2022

Cited by 3 | Viewed by 2177

Abstract

The paper considers the specifics of nonlinear differential equations that have applications in different areas. Earlier, the authors proved the existence and uniqueness theorem for a solution to a class of non-linear differential equations in a neighborhood of a moving singular point. In this paper, we consider the first problem of studying a third-order nonlinear differential equation in the domain of analyticity. An analytical approximate solution is built, taking into account the solution search area. A priori estimates of the analytical approximate solution are obtained, and the technology of their optimization using a posteriori ones is illustrated. The result of a numerical experiment is presented. The presented results allow to expand the class of nonlinear differential equations for describing various phenomena and processes. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

9 pages, 263 KiB

Open AccessArticle

Application of Fractional Grey Forecasting Model in Economic Growth of the Group of Seven

by Yumei Liao, Xu Wang and Jinrong Wang

Axioms 2022, 11(4), 155; https://doi.org/10.3390/axioms11040155 - 28 Mar 2022

Cited by 1 | Viewed by 2488

Abstract

This paper uses the idea of fractional order accumulation instead of the form of grey index, and applies the fractional order accumulation prediction model to the economic growth prediction of the member states of the Group of Seven from 1973 to 2016. By [...] Read more.

R^{2}

, MAD and BIC, it is found that the prediction performance of fractional order cumulative grey prediction model (GM(

α

,1)) is significantly improved in the medium and long term compared with the traditional grey prediction model (GM(1,1)). Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

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11 pages, 288 KiB

Open AccessArticle

Nearly Cosymplectic Manifolds of Constant Type

by Aligadzhi Rustanov

Axioms 2022, 11(4), 152; https://doi.org/10.3390/axioms11040152 - 25 Mar 2022

Cited by 3 | Viewed by 2227

Abstract

Fundamental identities characterizing a nearly cosymplectic structure and analytical expressions for the first and second structural tensors are obtained in this paper. An identity that is satisfied by the first structural tensor of a nearly cosymplectic structure is proved as well. A contact analog of nearly cosymplectic manifolds’ constancy of type is introduced in this paper. Pointwise constancy conditions of the type of nearly cosymplectic manifolds are obtained. It is proved that for nearly cosymplectic manifolds of dimension greater than three, pointwise constancy of type is equivalent to global constancy of type. A complete classification of nearly cosymplectic manifolds of constant type is obtained. It is also proved that a nearly cosymplectic manifold of dimension less than seven is a proper nearly cosymplectic manifold. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

10 pages, 280 KiB

Open AccessArticle

(ω,c)-Periodic Solutions to Fractional Differential Equations with Impulses

by Lulu Ren and JinRong Wang

Axioms 2022, 11(3), 83; https://doi.org/10.3390/axioms11030083 - 22 Feb 2022

Cited by 4 | Viewed by 1653

Abstract

This paper deals with the

(ω, c)

-periodic solutions to impulsive fractional differential equations with Caputo fractional derivative with a fixed lower limit. Firstly, a necessary and sufficient condition of the existence of

(ω, c)

-periodic solutions [...] Read more.

This paper deals with the

(ω, c)

-periodic solutions to impulsive fractional differential equations with Caputo fractional derivative with a fixed lower limit. Firstly, a necessary and sufficient condition of the existence of

(ω, c)

-periodic solutions to linear problem is given. Secondly, the existence and uniqueness of

(ω, c)

-periodic solutions to semilinear problem are proven. Lastly, two examples are given to demonstrate our results. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

15 pages, 307 KiB

Open AccessArticle

g-Expectation for Conformable Backward Stochastic Differential Equations

by Mei Luo, Michal Fečkan, Jin-Rong Wang and Donal O’Regan

Axioms 2022, 11(2), 75; https://doi.org/10.3390/axioms11020075 - 14 Feb 2022

Cited by 2 | Viewed by 2397

Abstract

In this paper, we study the applications of conformable backward stochastic differential equations driven by Brownian motion and compensated random measure in nonlinear expectation. From the comparison theorem, we introduce the concept of g-expectation and give related properties of g-expectation. In addition, we find that the properties of conformable backward stochastic differential equations can be deduced from the properties of the generator g. Finally, we extend the nonlinear Doob–Meyer decomposition theorem to more general cases. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

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