Research

27 pages, 512 KiB

Open AccessArticle

Stability and Hopf Bifurcation Analysis for a Phage Therapy Model with and without Time Delay

by Ei Ei Kyaw, Hongchan Zheng and Jingjing Wang

Axioms 2023, 12(8), 772; https://doi.org/10.3390/axioms12080772 - 09 Aug 2023

Cited by 1 | Viewed by 752

This study proposes a mathematical model that accounts for the interaction of bacteria, phages, and the innate immune response with a discrete time delay. First, for the non-delayed model we determine the local and global stability of various equilibria and the existence of [...] Read more.

This study proposes a mathematical model that accounts for the interaction of bacteria, phages, and the innate immune response with a discrete time delay. First, for the non-delayed model we determine the local and global stability of various equilibria and the existence of Hopf bifurcation at the positive equilibrium. Second, for the delayed model we provide sufficient conditions for the local stability of the positive equilibrium by selecting the discrete time delay as a bifurcation parameter; Hopf bifurcation happens when the time delay crosses a critical threshold. Third, based on the normal form method and center manifold theory, we derive precise expressions for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to verify our theoretical analysis. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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18 pages, 349 KiB

Open AccessArticle

Some Hadamard-Type Integral Inequalities Involving Modified Harmonic Exponential Type Convexity

by Asif Ali Shaikh, Evren Hincal, Sotiris K. Ntouyas, Jessada Tariboon and Muhammad Tariq

Axioms 2023, 12(5), 454; https://doi.org/10.3390/axioms12050454 - 05 May 2023

Cited by 1 | Viewed by 868

Abstract

The term convexity and theory of inequalities is an enormous and intriguing domain of research in the realm of mathematical comprehension. Due to its applications in multiple areas of science, the theory of convexity and inequalities have recently attracted a lot of attention [...] Read more.

The term convexity and theory of inequalities is an enormous and intriguing domain of research in the realm of mathematical comprehension. Due to its applications in multiple areas of science, the theory of convexity and inequalities have recently attracted a lot of attention from historians and modern researchers. This article explores the concept of a new group of modified harmonic exponential s-convex functions. Some of its significant algebraic properties are elegantly elaborated to maintain the newly described idea. A new sort of Hermite–Hadamard-type integral inequality using this new concept of the function is investigated. In addition, several new estimates of Hermite–Hadamard inequality are presented to improve the study. These new results illustrate some generalizations of prior findings in the literature. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

13 pages, 306 KiB

Open AccessArticle

Gamma-Nabla Hardy–Hilbert-Type Inequalities on Time Scales

by Barakah Almarri and Ahmed A. El-Deeb

Axioms 2023, 12(5), 449; https://doi.org/10.3390/axioms12050449 - 01 May 2023

Cited by 2 | Viewed by 779

Abstract

We investigated several novel conformable fractional gamma-nabla dynamic Hardy–Hilbert inequalities on time scales in this study. Several continuous inequalities and their corresponding discrete analogues in the literature are combined and expanded by these inequalities. Hölder’s inequality on time scales and a few algebraic [...] Read more.

We investigated several novel conformable fractional gamma-nabla dynamic Hardy–Hilbert inequalities on time scales in this study. Several continuous inequalities and their corresponding discrete analogues in the literature are combined and expanded by these inequalities. Hölder’s inequality on time scales and a few algebraic inequalities are used to demonstrate our findings. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

11 pages, 369 KiB

Open AccessArticle

A Study of the Monotonic Properties of Solutions of Neutral Differential Equations and Their Applications

by Osama Moaaz and Abtehal E. Alhgilan

Axioms 2023, 12(4), 346; https://doi.org/10.3390/axioms12040346 - 31 Mar 2023

Cited by 2 | Viewed by 830

Abstract

In this paper, we aim to study the monotonic properties of the solutions of a class of neutral delay differential equations. The importance of this study lies in the fact that the monotonic properties largely control the study of the oscillation and asymptotic [...] Read more.

In this paper, we aim to study the monotonic properties of the solutions of a class of neutral delay differential equations. The importance of this study lies in the fact that the monotonic properties largely control the study of the oscillation and asymptotic behaviour of the solutions to delay differential equations. Then, by using the new properties, we create improved criteria for testing the oscillation of solutions to the studied equation. We also find new criteria that can be applied more than once. Moreover, we discuss the importance and novelty of the results through the application to a special case of the studied equation. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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24 pages, 3696 KiB

Open AccessArticle

Mathematical Models of the Processes of Operation, Renewal and Degradation of a Fleet of Complex Technical Systems with Metrological Support

by Rustam Khayrullin, Denis Ershov, Alexander Malahov and Tatyana Levina

Axioms 2023, 12(3), 300; https://doi.org/10.3390/axioms12030300 - 15 Mar 2023

Cited by 1 | Viewed by 1068

Abstract

(1) Background: The aim of the study is to develop a set of models for managing a fleet of complex technical systems with metrological support, allowing the simulation and management at all the stages of the life cycle of the complex technical systems, [...] Read more.

(1) Background: The aim of the study is to develop a set of models for managing a fleet of complex technical systems with metrological support, allowing the simulation and management at all the stages of the life cycle of the complex technical systems, as well as to simulate the functioning of large fleets of complex technical systems, including up to several hundred thousand samples; (2) Methods: The authors use methods of mathematical modeling, methods of the theory of Markov and semi-Markov processes, methods of optimization, methods of reliability theory, and methods of probability theory and mathematical statistics; (3) Results: an interconnected set of mathematical models for managing a fleet of complex technical systems with metrological support was developed and the applied software was developed; (4) Conclusions: The set of models presented in the article allows for the adequate simulation of all the stages of the life cycle of large complex technical systems fleets, including up to several hundreds of thousands of samples, to optimize the functioning processes of a fleet of complex technical systems, to form strategies for fleet development, and to assess the risks associated with false and undetected failures, as well as the risks associated with the degradation of complex technical systems. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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15 pages, 308 KiB

Open AccessArticle

Further Closed Formulae of Exotic ₃F₂-Series

by Wenchang Chu

Axioms 2023, 12(3), 291; https://doi.org/10.3390/axioms12030291 - 10 Mar 2023

Viewed by 788

Abstract

By making use of the linearization method, we examine a class of nonterminating

_{3} F_{2}

-series with five free integer parameters that yields twenty summation formulae. Under the Kummer and Thomae transformations, six classes of exotic

_{3} F_{2}

-series are consequently [...] Read more.

By making use of the linearization method, we examine a class of nonterminating

_{3} F_{2}

-series with five free integer parameters that yields twenty summation formulae. Under the Kummer and Thomae transformations, six classes of exotic

_{3} F_{2}

-series are consequently evaluated in closed forms. There are overall 100 identities recorded in the present paper Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

10 pages, 288 KiB

Open AccessArticle

A Note on the Volume Conserving Solution to Simultaneous Aggregation and Collisional Breakage Equation

by Farel William Viret Kharchandy, Arijit Das, Vamsinadh Thota, Jitraj Saha and Mehakpreet Singh

Axioms 2023, 12(2), 181; https://doi.org/10.3390/axioms12020181 - 09 Feb 2023

Viewed by 1016

Abstract

A new population balance model is introduced, in which a pair of particles can coagulate into a larger one if their encounter is a completely inelastic collision; otherwise, one of them breaks into multiple fragments (two or more) due to the elastic collision. [...] Read more.

A new population balance model is introduced, in which a pair of particles can coagulate into a larger one if their encounter is a completely inelastic collision; otherwise, one of them breaks into multiple fragments (two or more) due to the elastic collision. Mathematically, coagulation and breakage models both manifest nonlinearity behavior. We prove the global existence and uniqueness of the solution to this model for the compactly supported kinetic kernels and an unbounded breakage distribution function. A further investigation dealt with the volume conservation property (necessary condition) of the solution. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

11 pages, 704 KiB

Open AccessArticle

Properties of Differential Subordination and Superordination for Multivalent Functions Associated with the Convolution Operators

by Luminiţa-Ioana Cotîrlă and Abdul Rahman S. Juma

Axioms 2023, 12(2), 169; https://doi.org/10.3390/axioms12020169 - 07 Feb 2023

Cited by 2 | Viewed by 951

Abstract

Using convolution (or Hadamard product), we define the El-Ashwah and Drbuk linear operator, which is a multivalent function in the unit disk

U = [w : |w| < 1 and w \in ₵]

, and satisfy its specific relationship to derive [...] Read more.

Using convolution (or Hadamard product), we define the El-Ashwah and Drbuk linear operator, which is a multivalent function in the unit disk

U = [w : |w| < 1 and w \in ₵]

, and satisfy its specific relationship to derive the subordination, superordination, and sandwich results for this operator by using properties of subordination and superordination concepts. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

27 pages, 4819 KiB

Open AccessArticle

Analysis of a Modified System of Infectious Disease in a Closed and Convex Subset of a Function Space with Numerical Study

by Tahira Sumbal Shaikh, Ali Akgül, Muhammad Aziz ur Rehman, Nauman Ahmed, Muhammad Sajid Iqbal, Naveed Shahid, Muhammad Rafiq and Manuel De la Sen

Axioms 2023, 12(1), 79; https://doi.org/10.3390/axioms12010079 - 12 Jan 2023

Cited by 1 | Viewed by 1249

Abstract

In this article, the transmission dynamical model of the deadly infectious disease named Ebola is investigated. This disease identified in the Democratic Republic of Congo (DRC) and Sudan (now South Sudan) and was identified in 1976. The novelty of the model under discussion [...] Read more.

In this article, the transmission dynamical model of the deadly infectious disease named Ebola is investigated. This disease identified in the Democratic Republic of Congo (DRC) and Sudan (now South Sudan) and was identified in 1976. The novelty of the model under discussion is the inclusion of advection and diffusion in each compartmental equation. The addition of these two terms makes the model more general. Similar to a simple population dynamic system, the prescribed model also has two equilibrium points and an important threshold, known as the basic reproductive number. The current work comprises the existence and uniqueness of the solution, the numerical analysis of the model, and finally, the graphical simulations. In the section on the existence and uniqueness of the solutions, the optimal existence is assessed in a closed and convex subset of function space. For the numerical study, a nonstandard finite difference (NSFD) scheme is adopted to approximate the solution of the continuous mathematical model. The main reason for the adoption of this technique is delineated in the form of the positivity of the state variables, which is necessary for any population model. The positivity of the applied scheme is verified by the concept of M-matrices. Since the numerical method gives a discrete system of difference equations corresponding to a continuous system, some other relevant properties are also needed to describe it. In this respect, the consistency and stability of the designed technique are corroborated by using Taylor’s series expansion and Von Neumann’s stability criteria, respectively. To authenticate the proposed NSFD method, two other illustrious techniques are applied for the sake of comparison. In the end, numerical simulations are also performed that show the efficiency of the prescribed technique, while the existing techniques fail to do so. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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12 pages, 1827 KiB

Open AccessArticle

A Variational Formulation for Fins with Nonzero Contact Thermal Resistance at the Base

by Rogério Martins Saldanha da Gama, Rogério Pazetto Saldanha da Gama, Vinicius Vendas Sarmento and Maria Laura Martins-Costa

Axioms 2023, 12(1), 54; https://doi.org/10.3390/axioms12010054 - 03 Jan 2023

Cited by 2 | Viewed by 873

Abstract

This paper considers the steady-state heat transfer process in a fin with a Robin boundary condition at the base (instead of the usual Dirichlet boundary condition at the base). Robin boundary condition models the effect of the thermal resistance between the base of [...] Read more.

This paper considers the steady-state heat transfer process in a fin with a Robin boundary condition at the base (instead of the usual Dirichlet boundary condition at the base). Robin boundary condition models the effect of the thermal resistance between the base of the fin and the surface on which the fin is placed. This work presents an equivalent minimum principle, represented by a convex and coercive functional, ensuring the solution’s existence and uniqueness. In order to illustrate the use of the proposed functional for reaching approximations, the heat-transfer process in a trapezoidal fin considering a piecewise linear approximation is simulated. The Appendix presents a case in which the exact solution in a closed form has been achieved. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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16 pages, 351 KiB

Open AccessArticle

On the Iterative Multivalued ⊥-Preserving Mappings and an Application to Fractional Differential Equation

by Muhammad Nazam, Sumit Chandok, Aftab Hussain and Hamed H. Al Sulmi

Axioms 2023, 12(1), 53; https://doi.org/10.3390/axioms12010053 - 03 Jan 2023

Cited by 1 | Viewed by 950

Abstract

In this paper, we introduce orthogonal multivalued contractions, which are based on the recently introduced notion of orthogonality in the metric spaces. We construct numerous fixed point theorems for these contractions. We show how these fixed point theorems aid in the generalization of [...] Read more.

In this paper, we introduce orthogonal multivalued contractions, which are based on the recently introduced notion of orthogonality in the metric spaces. We construct numerous fixed point theorems for these contractions. We show how these fixed point theorems aid in the generalization of a number of recently published findings. Additionally, we offer a theorem that establishes the existence of a fractional differential equation’s solution. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

11 pages, 289 KiB

Open AccessArticle

Hyers Stability in Generalized Intuitionistic P-Pseudo Fuzzy 2-Normed Spaces

by Ehsan Movahednia and Manuel De la Sen

Axioms 2023, 12(1), 28; https://doi.org/10.3390/axioms12010028 - 26 Dec 2022

Viewed by 1222

Abstract

In this article, we defined the generalized intuitionistic P-pseudo fuzzy 2-normed spaces and investigated the Hyers stability of m-mappings in this space. The m-mappings are interesting functional equations; these functional equations are additive for m = 1, quadratic for m = [...] Read more.

In this article, we defined the generalized intuitionistic P-pseudo fuzzy 2-normed spaces and investigated the Hyers stability of m-mappings in this space. The m-mappings are interesting functional equations; these functional equations are additive for m = 1, quadratic for m = 2, cubic for m = 3, and quartic for m = 4. We have investigated the stability of four types of functional equations in generalized intuitionistic P-pseudo fuzzy 2-normed spaces by the fixed point method. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

12 pages, 298 KiB

Open AccessFeature PaperArticle

Nonhomogeneous Dirichlet Problems with Unbounded Coefficient in the Principal Part

by Dumitru Motreanu

Axioms 2022, 11(12), 739; https://doi.org/10.3390/axioms11120739 - 17 Dec 2022

Cited by 2 | Viewed by 1194

Abstract

The main result of the paper establishes the existence of a bounded weak solution for a nonlinear Dirichlet problem exhibiting full dependence on the solution u and its gradient

\nabla u

in the reaction term, which is driven by a p-Laplacian-type operator [...] Read more.

The main result of the paper establishes the existence of a bounded weak solution for a nonlinear Dirichlet problem exhibiting full dependence on the solution u and its gradient

\nabla u

in the reaction term, which is driven by a p-Laplacian-type operator with a coefficient

G (u)

that can be unbounded. Through a special Moser iteration procedure, it is shown that the solution set is uniformly bounded. A truncated problem is formulated that drops that

G (u)

be unbounded. The existence of a bounded weak solution to the truncated problem is proven via the theory of pseudomonotone operators. It is noted that the bound of the solution for the truncated problem coincides with the uniform bound of the original problem. This estimate allows us to deduce that for an appropriate choice of truncation, one actually resolves the original problem. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

21 pages, 360 KiB

Open AccessArticle

Amended Criteria for Testing the Asymptotic and Oscillatory Behavior of Solutions of Higher-Order Functional Differential Equations

by Barakah Almarri, Fahd Masood, Osama Moaaz and Ali Muhib

Axioms 2022, 11(12), 718; https://doi.org/10.3390/axioms11120718 - 12 Dec 2022

Cited by 4 | Viewed by 811

Abstract

Our interest in this article is to develop oscillation conditions for solutions of higher order differential equations and to extend recent results in the literature to differential equations of several delays. We obtain new asymptotic properties of a class from the positive solutions [...] Read more.

Our interest in this article is to develop oscillation conditions for solutions of higher order differential equations and to extend recent results in the literature to differential equations of several delays. We obtain new asymptotic properties of a class from the positive solutions of an even higher order neutral delay differential equation. Then we use these properties to create more effective criteria for studying oscillation. Finally, we present some special cases of the studied equation and apply the new results to them. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

17 pages, 316 KiB

Open AccessArticle

Inexact Restoration Methods for Semivectorial Bilevel Programming Problem on Riemannian Manifolds

by Jiagen Liao and Zhongping Wan

Axioms 2022, 11(12), 696; https://doi.org/10.3390/axioms11120696 - 05 Dec 2022

Viewed by 921

Abstract

For a better understanding of the bilevel programming on Riemannian manifolds, a semivectorial bilevel programming scheme is proposed in this paper. The semivectorial bilevel programming is firstly transformed into a single-level programming problem by using the Karush–Kuhn–Tucker (KKT) conditions of the lower-level problem, [...] Read more.

For a better understanding of the bilevel programming on Riemannian manifolds, a semivectorial bilevel programming scheme is proposed in this paper. The semivectorial bilevel programming is firstly transformed into a single-level programming problem by using the Karush–Kuhn–Tucker (KKT) conditions of the lower-level problem, which is convex and satisfies the Slater constraint qualification. Then, the single-level programming is divided into two stages: restoration and minimization, based on which an Inexact Restoration algorithm is developed. Under certain conditions, the stability and convergence of the algorithm are analyzed. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

13 pages, 850 KiB

Open AccessArticle

A Novel Approach for the Approximate Solution of Wave Problems in Multi-Dimensional Orders with Computational Applications

by Muhammad Nadeem, Ali Akgül, Liliana Guran and Monica-Felicia Bota

Axioms 2022, 11(12), 665; https://doi.org/10.3390/axioms11120665 - 24 Nov 2022

Cited by 2 | Viewed by 1028

Abstract

The main goal of this paper is to introduce a new scheme, known as the Aboodh homotopy integral transform method (

A

HITM), for the approximate solution of wave problems in multi-dimensional orders. The Aboodh integral transform (

A

IT) removes the restriction [...] Read more.

The main goal of this paper is to introduce a new scheme, known as the Aboodh homotopy integral transform method (

A

HITM), for the approximate solution of wave problems in multi-dimensional orders. The Aboodh integral transform (

A

IT) removes the restriction of variables in the recurrence relation, whereas the homotopy perturbation method (HPM) derives the successive iterations using the initial conditions. The convergence analysis is provided to study a wave equation with multiple dimensions. Some computational applications are considered to show the efficiency of this scheme. Graphical representation between the approximate and the exact solution predicts the high rate of convergence of this approach. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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13 pages, 284 KiB

Open AccessArticle

Some New Generalized Inequalities of Hardy Type Involving Several Functions on Time Scale Nabla Calculus

by A. I. Saied, Ghada ALNemer, Mohammed Zakarya, Clemente Cesarano and Haytham M. Rezk

Axioms 2022, 11(12), 662; https://doi.org/10.3390/axioms11120662 - 22 Nov 2022

Cited by 6 | Viewed by 978

Abstract

In this article, we establish several new generalized Hardy-type inequalities involving several functions on time-scale nabla calculus. Furthermore, we derive some new multidimensional Hardy-type inequalities on time scales nabla calculus. The main results are proved by applying Minkowski’s inequality, Jensen’s inequality and Arithmetic [...] Read more.

In this article, we establish several new generalized Hardy-type inequalities involving several functions on time-scale nabla calculus. Furthermore, we derive some new multidimensional Hardy-type inequalities on time scales nabla calculus. The main results are proved by applying Minkowski’s inequality, Jensen’s inequality and Arithmetic Mean–Geometric Mean inequality. As a special case of our results, when

T = R,

we obtain refinements of some well-known continuous inequalities and when

T = N

, the results which are essentially new. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

13 pages, 297 KiB

Open AccessArticle

On Some New Dynamic Inequalities Involving C-Monotonic Functions on Time Scales

by Ghada AlNemer, A. I. Saied, A. M. Hassan, Clemente Cesarano, Haytham M. Rezk and Mohammed Zakarya

Axioms 2022, 11(11), 644; https://doi.org/10.3390/axioms11110644 - 15 Nov 2022

Cited by 3 | Viewed by 1081

Abstract

In this paper, we establish some new dynamic inequalities involving C-monotonic functions with

C \geq 1,

on time scales. As a special case of our results when

C = 1,

we obtain the inequalities involving increasing or decreasing functions (where [...] Read more.

In this paper, we establish some new dynamic inequalities involving C-monotonic functions with

C \geq 1,

on time scales. As a special case of our results when

C = 1,

we obtain the inequalities involving increasing or decreasing functions (where for

C = 1,

the 1-decreasing function is decreasing and the 1-increasing function is increasing). The main results are proved by applying the properties of C-monotonic functions and the chain rule formula on time scales. As a special case of our results, when

T = R,

we obtain refinements of some well-known continuous inequalities and when

T = N

, to the best of the authors’ knowledge, the results are essentially new. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

15 pages, 323 KiB

Open AccessArticle

New Fractional Integral Inequalities Pertaining to Caputo–Fabrizio and Generalized Riemann–Liouville Fractional Integral Operators

by Muhammad Tariq, Omar Mutab Alsalami, Asif Ali Shaikh, Kamsing Nonlaopon and Sotiris K. Ntouyas

Axioms 2022, 11(11), 618; https://doi.org/10.3390/axioms11110618 - 07 Nov 2022

Cited by 4 | Viewed by 1260

Abstract

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies of fractional calculus have become the subject of intensive research in historical and contemporary generations because of their applications in various branches of science. In [...] Read more.

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies of fractional calculus have become the subject of intensive research in historical and contemporary generations because of their applications in various branches of science. In this paper, we concentrate on establishing Hermite–Hadamard and Pachpatte-type integral inequalities with the aid of two different fractional operators. In particular, we acknowledge the critical Hermite–Hadamard and related inequalities for n-polynomial s-type convex functions and n-polynomial s-type harmonically convex functions. We practice these inequalities to consider the Caputo–Fabrizio and the k-Riemann–Liouville fractional integrals. Several special cases of our main results are also presented in the form of corollaries and remarks. Our study offers a better perception of integral inequalities involving fractional operators. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

11 pages, 1164 KiB

Open AccessArticle

Variational Iteration Method for Solving Fractional Integro-Differential Equations with Conformable Differointegration

by Mondher Damak and Zaid Amer Mohammed

Axioms 2022, 11(11), 586; https://doi.org/10.3390/axioms11110586 - 24 Oct 2022

Viewed by 1221

Abstract

Multidimensional integro-differential equations are obtained when the unknown function of several independent variable and/or its derivatives appear under an integral sign. When the differentiation or integration operators or both are of fractional order, the integral equation in this case is called a multidimensional [...] Read more.

Multidimensional integro-differential equations are obtained when the unknown function of several independent variable and/or its derivatives appear under an integral sign. When the differentiation or integration operators or both are of fractional order, the integral equation in this case is called a multidimensional fractional integro-differential equation. Such equations are difficult to solve analytically; therefore, as the main objective of this paper, an approximate method—which is the variational iteration method—will be used to solve this type of equation with conformable fractional-order derivatives and integrals. First, we drive the iterative sequence of approximate solutions using the proposed method, and then, under certain conditions over the kernel of the integro-differential equation, prove its convergence to the exact solution. Two illustrative examples, linear and nonlinear, are given, and their approximated solutions are simulated using computer programs in order to verify from the reliability and applicability of the proposed method. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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14 pages, 333 KiB

Open AccessArticle

Some New Refinements of Trapezium-Type Integral Inequalities in Connection with Generalized Fractional Integrals

by Muhammad Tariq, Soubhagya Kumar Sahoo, Sotiris K. Ntouyas, Omar Mutab Alsalami, Asif Ali Shaikh and Kamsing Nonlaopon

Axioms 2022, 11(10), 508; https://doi.org/10.3390/axioms11100508 - 27 Sep 2022

Cited by 1 | Viewed by 1086

Abstract

The main objective of this article is to introduce a new notion of convexity, i.e., modified exponential type convex function, and establish related fractional inequalities. To strengthen the argument of the paper, we introduce two new lemmas as auxiliary results and discuss some [...] Read more.

The main objective of this article is to introduce a new notion of convexity, i.e., modified exponential type convex function, and establish related fractional inequalities. To strengthen the argument of the paper, we introduce two new lemmas as auxiliary results and discuss some algebraic properties of the proposed notion. Considering a generalized fractional integral operator and differentiable mappings, whose initial absolute derivative at a given power is a modified exponential type convex, various improvements of the Hermite–Hadamard inequality are presented. Thanks to the main results, some generalizations about the earlier findings in the literature are recovered. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

10 pages, 277 KiB

Open AccessArticle

Analytical Solutions of Temperature Distribution in a Rectangular Parallelepiped

by Dinesh Kumar, Frédéric Yves Ayant and Clemente Cesarano

Axioms 2022, 11(9), 488; https://doi.org/10.3390/axioms11090488 - 19 Sep 2022

Cited by 1 | Viewed by 1495

Abstract

In the present article, we give analytical solutions for temperature distribution in a rectangular parallelepiped with the help of a multivariable I-function. The results established in this paper are of a general character from which several known and new results can be [...] Read more.

In the present article, we give analytical solutions for temperature distribution in a rectangular parallelepiped with the help of a multivariable I-function. The results established in this paper are of a general character from which several known and new results can be deduced. We also give the special and particular cases of our main findings. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

11 pages, 262 KiB

Open AccessArticle

Existence and Uniqueness of Solution to a Terminal Value Problem of First-Order Differential Equation

by Yuqiang Feng, Qian Pan and Jun Jiang

Axioms 2022, 11(9), 435; https://doi.org/10.3390/axioms11090435 - 26 Aug 2022

Viewed by 1052

Abstract

The terminal value problem of differential equations has an important application background. In this paper, we are concerned with the terminal value problem of a first-order differential equation. Some sufficient conditions are given to obtain the existence and uniqueness results of solutions to [...] Read more.

The terminal value problem of differential equations has an important application background. In this paper, we are concerned with the terminal value problem of a first-order differential equation. Some sufficient conditions are given to obtain the existence and uniqueness results of solutions to the problem. Firstly, some comparison lemmas are established; secondly, an iterative technique and fixed point method are used to set up the main results; Finally, an example is provided to illustrate the application of the main results. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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6 pages, 242 KiB

Open AccessArticle

Dirichlet Problem with L¹(S) Boundary Values

by Alexander G. Ramm

Axioms 2022, 11(8), 371; https://doi.org/10.3390/axioms11080371 - 28 Jul 2022

Viewed by 971

Abstract

Let D be a connected bounded domain in

R^{2}

, S be its boundary, which is closed and

C^{2}

-smooth. Consider the Dirichlet problem

{Δ u = 0 in D, u |}_{S} = h

, where [...] Read more.

Let D be a connected bounded domain in

R^{2}

, S be its boundary, which is closed and

C^{2}

-smooth. Consider the Dirichlet problem

{Δ u = 0 in D, u |}_{S} = h

, where

h \in L^{1} (S)

. The aim of this paper is to prove that the above problem has a solution for an arbitrary

h \in L^{1} (S)

, and this solution is unique. The result is new. The method of its proof is new. The definition of the

L^{1} (S)

-boundary value of a harmonic in the D function is given. No embedding theorems are used. The history of the Dirichlet problem goes back to 1828. The result in this paper is, to the author’s knowledge, the first result in the 194 years of research (since 1828) that yields the existence and uniqueness of the solution to the Dirichlet problem with the boundary values in

L^{1} (S)

. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

10 pages, 2060 KiB

Open AccessArticle

The Influence of Multiplicative Noise and Fractional Derivative on the Solutions of the Stochastic Fractional Hirota–Maccari System

by Farah M. Al-Askar, Wael W. Mohammed, Clemente Cesarano and M. El-Morshedy

Axioms 2022, 11(8), 357; https://doi.org/10.3390/axioms11080357 - 23 Jul 2022

Cited by 7 | Viewed by 1230

Abstract

We address here the space-fractional stochastic Hirota–Maccari system (SFSHMs) derived by the multiplicative Brownian motion in the Stratonovich sense. To acquire innovative elliptic, trigonometric and rational stochastic fractional solutions, we employ the Jacobi elliptic functions method. The attained solutions are useful in describing [...] Read more.

We address here the space-fractional stochastic Hirota–Maccari system (SFSHMs) derived by the multiplicative Brownian motion in the Stratonovich sense. To acquire innovative elliptic, trigonometric and rational stochastic fractional solutions, we employ the Jacobi elliptic functions method. The attained solutions are useful in describing certain fascinating physical phenomena due to the significance of the Hirota–Maccari system in optical fibers. We use MATLAB programm to draw our figures and exhibit several 3D graphs in order to demonstrate how the multiplicative Brownian motion and fractional derivative affect the exact solutions of the SFSHMs. We prove that the solutions of SFSHMs are stabilized by the multiplicative Brownian motion around zero. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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11 pages, 254 KiB

Open AccessArticle

New Type of Degenerate Changhee–Genocchi Polynomials

by Maryam Salem Alatawi and Waseem Ahmad Khan

Axioms 2022, 11(8), 355; https://doi.org/10.3390/axioms11080355 - 23 Jul 2022

Cited by 8 | Viewed by 1058

Abstract

A remarkably large number of polynomials and their extensions have been presented and studied. In this paper, we consider a new type of degenerate Changhee–Genocchi numbers and polynomials which are different from those previously introduced by Kim. We investigate some properties of these [...] Read more.

A remarkably large number of polynomials and their extensions have been presented and studied. In this paper, we consider a new type of degenerate Changhee–Genocchi numbers and polynomials which are different from those previously introduced by Kim. We investigate some properties of these numbers and polynomials. We also introduce a higher-order new type of degenerate Changhee–Genocchi numbers and polynomials which can be represented in terms of the degenerate logarithm function. Finally, we derive their summation formulae. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

14 pages, 319 KiB

Open AccessArticle

Iterative Approximate Solutions for Variational Problems in Hadamard Manifold

by Mohammad Dilshad, Doaa Filali, Sumit Chandok and Mohammad Akram

Axioms 2022, 11(7), 352; https://doi.org/10.3390/axioms11070352 - 21 Jul 2022

Viewed by 1385

Abstract

The goal of this paper is to propose and investigate new iterative methods for examining an approximate solution of a fixed-point problem, an equilibrium problem, and a finite collection of variational inclusions in the Hadamard manifold’s structure. Operating under some assumptions, we extend [...] Read more.

The goal of this paper is to propose and investigate new iterative methods for examining an approximate solution of a fixed-point problem, an equilibrium problem, and a finite collection of variational inclusions in the Hadamard manifold’s structure. Operating under some assumptions, we extend the proximal point algorithm to estimate the common solution of stated problems and obtain a strong convergence theorem for the common solution. We also present several consequences of the proposed iterative methods and their convergence results. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

18 pages, 320 KiB

Open AccessArticle

Numerical Processes for Approximating Solutions of Nonlinear Equations

by Samundra Regmi, Ioannis K. Argyros, Santhosh George and Christopher I. Argyros

Axioms 2022, 11(7), 307; https://doi.org/10.3390/axioms11070307 - 24 Jun 2022

Cited by 3 | Viewed by 1391

Abstract

In this article, we present generalized conditions of three-step iterative schemes for solving nonlinear equations. The convergence order is shown using Taylor series, but the existence of high-order derivatives is assumed. However, only the first derivative appears on these schemes. Therefore, the hypotheses [...] Read more.

In this article, we present generalized conditions of three-step iterative schemes for solving nonlinear equations. The convergence order is shown using Taylor series, but the existence of high-order derivatives is assumed. However, only the first derivative appears on these schemes. Therefore, the hypotheses limit the utilization of the schemes to operators that are at least nine times differentiable, although the schemes may converge. To the best of our knowledge, no semi-local convergence has been given in the setting of a Banach space. Our goal is to extend the applicability of these schemes in both the local and semi-local convergence cases. Moreover, we use our idea of recurrent functions and conditions only on the derivative or divided differences of order one that appear in these schemes. This idea can be applied to extend other high convergence multipoint and multistep schemes. Numerical applications where the convergence criteria are tested complement this article. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

26 pages, 2137 KiB

Open AccessArticle

Development of Patent Technology Prediction Model Based on Machine Learning

by Chih-Wei Lee, Feng Tao, Yu-Yu Ma and Hung-Lung Lin

Axioms 2022, 11(6), 253; https://doi.org/10.3390/axioms11060253 - 26 May 2022

Cited by 3 | Viewed by 2766

Abstract

Intellectual property rights have a great impact on the development of the automobile industry. Issues related to the timeliness of patent applications often arise, such as the inability of firms to predict new technologies and patents developed by peers. To find the proper [...] Read more.

Intellectual property rights have a great impact on the development of the automobile industry. Issues related to the timeliness of patent applications often arise, such as the inability of firms to predict new technologies and patents developed by peers. To find the proper direction of product development, the R&D departments of enterprises need to accurately predict the technology trends. Machine learning adopts calculation through a large amount of data through mathematical models and methods and finds the best solution at the fastest speed through repeated simulation and experiments, to provide decision makers with a reference basis. Therefore, this paper provides accurate forecasts through established models. In terms of the significance of management, the planning of future enterprise strategy can be divided into three stages as a short-term plan of 1–3 years, a medium-term plan of 3–5 years, and a long-term plan of 5–10 years. This study will give appropriate suggestions for the development of automobile industry technology. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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31 pages, 3047 KiB

Open AccessArticle

Modeling Preferences through Personality and Satisfaction to Guide the Decision Making of a Virtual Agent

by Jorge Castro-Rivera, María Lucila Morales-Rodríguez, Nelson Rangel-Valdez, Claudia Gómez-Santillán and Luciano Aguilera-Vázquez

Axioms 2022, 11(5), 232; https://doi.org/10.3390/axioms11050232 - 17 May 2022

Cited by 1 | Viewed by 2248

Abstract

Satisfaction is relevant for decision makers (DM, Decision Makers). Satisfaction is the feeling produced in individuals by executing actions to satisfy their needs, for example, the payment of debts, jobs, or academic achievements, and the acquisition of goods or services. In the satisfaction [...] Read more.

Satisfaction is relevant for decision makers (DM, Decision Makers). Satisfaction is the feeling produced in individuals by executing actions to satisfy their needs, for example, the payment of debts, jobs, or academic achievements, and the acquisition of goods or services. In the satisfaction literature, some theories model the satisfaction of individuals from job and customer approaches. However, considering personality elements to influence satisfaction and define preferences in strategies that optimize decision making provides the unique characteristics of a DM. These characteristics favor the scope of solutions closer to the satisfaction expectation. Satisfaction theories do not include specific elements of personality and preferences, so integrating these elements will offer more efficient decisions in computable models. In this work, a model of satisfaction with personality characteristics that influence the preferences of a DM is proposed. The proposed model is integrated into a preference-based optimizer that improves the decision-making process of a Virtual Decision Maker (VDM) in an optimization context. The optimization context addressed in this work is the product selection process within a food product shopping problem. An experimental design is proposed that compares two configurations that represent the cognitive part of an agent’s decision process to validate the operation of the proposed model in the context of optimization: (1) satisfaction, personality, and preferences, and (2) personality and preferences. The results show that considering satisfaction and personality in combination with preferences provides solutions closer to the interests of an individual, reflecting a more realistic behavior. Furthermore, this work demonstrates that it is possible to create a configurable model that allows adapting to different aptitudes and reflecting them in a computable model. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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14 pages, 311 KiB

Open AccessFeature PaperArticle

The Classification of All Singular Nonsymmetric Macdonald Polynomials

by Charles F. Dunkl

Axioms 2022, 11(5), 208; https://doi.org/10.3390/axioms11050208 - 29 Apr 2022

Viewed by 1271

Abstract

The affine Hecke algebra of type A has two parameters

(q, t)

and acts on polynomials in N variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements whose simultaneous eigenfunctions are [...] Read more.

The affine Hecke algebra of type A has two parameters

(q, t)

and acts on polynomials in N variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values, it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form

q^{m} = t^{- n}

with

2 \leq n \leq N .

For a fixed parameter, the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of N) of the Hecke algebra module is

(d n - 1, n - 1, \dots, n - 1, r)

for some

d \geq 1

. In the present paper, it is shown that there are no other singular polynomials. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

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