Advanced Calculus in Problems with Symmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (28 February 2021) | Viewed by 22628

Special Issue Editors

Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals
1. Mathematics, Anand International College of Engineering, Jaipur, Rajasthan 303012, India
2. Nonlinear Dynamics Research Center (NDRC), Ajman University, Al Jerf 1, Ajman, United Arab Emirates
Interests: special functions; fractional calculus; integral transform; control theory
Special Issues, Collections and Topics in MDPI journals
Department of Mathematics, Poornima College of Engineering, ISI-6, RI- ICO Institutional Area, Sitapura, Jaipur, Rajasthan 302022, India
Interests: special functions and fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Computer vision (e.g., for the autonomous driving of cars), artificial intelligence, robotics, machine learning, and computational and mathematical modeling of biological, engineering, physical systems are becoming increasingly attractive and suitable methods for the solution of daily complex non-linear problems which, however, are showing the existence of some symmetries. These problems can be easily solved using methods of modern advanced calculus and original theories and algorithms that have recently been discovered. The most efficient computations which follow from the increasingly powerful hardware and software thus enable us to handle more and more complex problems. However, it has also been recognized that very often also in some problems there exists some kind of symmetry which reduces the complexity. Therefore in the analysis of radioactive decay in chemistry, in the prediction of birth and death rates, as well as in the study of gravity and planetary motion, fluid flow, ship design, geometric curves, and bridge engineering, we can combine advanced calculus with symmetry properties to easily handle and solve such problems. This demand requires that all application areas maintain great accuracy and increasing efficiency in all the systems by using advanced calculus.

This Special Issue aims to focus on symmetry in advanced calculus in the development, design, implementation, modeling, and validation of systems in all areas of science, computer science, engineering applications, finance, and natural science. Authors are encouraged to submit both original research articles and surveys. Research articles should address the originality, as well as practical aspects and implementation, of the work in the field, while surveys should provide an overview and up-to-date information. Topics of interest include:

  • Advanced calculus in mathematics;
  • Advanced methods, computations, and algorithms in artificial intelligence and applications;
  • Advanced methods, computations, and algorithms in robotics and machine learning;
  • Advanced calculus in mathematical modeling of physical, biological, engineering, and financial problems;
  • Advanced calculus in sequence and series in functional analysis;
  • Advanced calculus in generalized fractional calculus and applications;
  • Advanced calculus in the theory of special functions related to fractional (non-integer) order control systems and equations;
  • Advanced applications of calculus in mechanics;
  • Advanced applications of calculus in physics;
  • Advanced methods for the analysis of special functions arising in the fractional diffusion-wave equations;
  • Advanced operational methods in calculus;
  • Advanced applications of inequalities for classical and fractional differential equations.

Prof. Dr. Carlo Cattani
Prof. Dr. Praveen Agarwal
Prof. Dr. Shilpi Jain
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. Symmetry 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.

Published Papers (13 papers)

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Research

13 pages, 282 KiB  
Article
Common Best Proximity Point Results for T-GKT Cyclic ϕ-Contraction Mappings in Partial Metric Spaces with Some Applications
Symmetry 2021, 13(6), 1098; https://doi.org/10.3390/sym13061098 - 21 Jun 2021
Cited by 1 | Viewed by 1411
Abstract
The aim of this paper is to derive some common best proximity point results in partial metric spaces defining a new class of symmetric mappings, which is a generalization of cyclic ϕ-contraction mappings. With the help of these symmetric mappings, the characterization [...] Read more.
The aim of this paper is to derive some common best proximity point results in partial metric spaces defining a new class of symmetric mappings, which is a generalization of cyclic ϕ-contraction mappings. With the help of these symmetric mappings, the characterization of completeness of metric spaces given by Cobzas (2016) is extended here for partial metric spaces. The existence of a solution to the Fredholm integral equation is also obtained here via a fixed-point formulation for such mappings. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
14 pages, 416 KiB  
Article
Dynamics of a COVID-19 Model with a Nonlinear Incidence Rate, Quarantine, Media Effects, and Number of Hospital Beds
Symmetry 2021, 13(6), 947; https://doi.org/10.3390/sym13060947 - 26 May 2021
Cited by 6 | Viewed by 1826
Abstract
In many countries the COVID-19 pandemic seems to witness second and third waves with dire consequences on human lives and economies. Given this situation the modeling of the transmission of the disease is still the subject of research with the ultimate goal of [...] Read more.
In many countries the COVID-19 pandemic seems to witness second and third waves with dire consequences on human lives and economies. Given this situation the modeling of the transmission of the disease is still the subject of research with the ultimate goal of understanding the dynamics of the disease and assessing the efficacy of different mitigation strategies undertaken by the affected countries. We propose a mathematical model for COVID-19 transmission. The model is structured upon five classes: an individual can be susceptible, exposed, infectious, quarantined or removed. The model is based on a nonlinear incidence rate, takes into account the influence of media on public behavior, and assumes the recovery rate to be dependent on the hospital-beds to population ratio. A detailed analysis of the proposed model is carried out, including the existence and uniqueness of solutions, stability analysis of the disease-free equilibrium (symmetry) and sensitivity analysis. We found that if the basic reproduction number is less than unity the system can exhibit Hopf and backward bifurcations for some range of parameters. Numerical simulations using parameter values fitted to Saudi Arabia are carried out to support the theoretical proofs and to analyze the effects of hospital-beds to population ratio, quarantine, and media effects on the predicted nonlinear behavior. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
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14 pages, 282 KiB  
Article
Relaxed Modulus-Based Matrix Splitting Methods for the Linear Complementarity Problem
Symmetry 2021, 13(3), 503; https://doi.org/10.3390/sym13030503 - 19 Mar 2021
Cited by 7 | Viewed by 1554
Abstract
In this paper, we obtain a new equivalent fixed-point form of the linear complementarity problem by introducing a relaxed matrix and establish a class of relaxed modulus-based matrix splitting iteration methods for solving the linear complementarity problem. Some sufficient conditions for guaranteeing the [...] Read more.
In this paper, we obtain a new equivalent fixed-point form of the linear complementarity problem by introducing a relaxed matrix and establish a class of relaxed modulus-based matrix splitting iteration methods for solving the linear complementarity problem. Some sufficient conditions for guaranteeing the convergence of relaxed modulus-based matrix splitting iteration methods are presented. Numerical examples are offered to show the efficacy of the proposed methods. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
17 pages, 313 KiB  
Article
Convergence Analysis of a Three-Step Iterative Algorithm for Generalized Set-Valued Mixed-Ordered Variational Inclusion Problem
Symmetry 2021, 13(3), 444; https://doi.org/10.3390/sym13030444 - 09 Mar 2021
Cited by 11 | Viewed by 1355
Abstract
This manuscript aims to study a generalized, set-valued, mixed-ordered, variational inclusion problem involving H(·,·)-compression XOR-αM-non-ordinary difference mapping and relaxed cocoercive mapping in real-ordered Hilbert spaces. The resolvent operator associated with [...] Read more.
This manuscript aims to study a generalized, set-valued, mixed-ordered, variational inclusion problem involving H(·,·)-compression XOR-αM-non-ordinary difference mapping and relaxed cocoercive mapping in real-ordered Hilbert spaces. The resolvent operator associated with H(·,·)-compression XOR-αM-non-ordinary difference mapping is defined, and some of its characteristics are discussed. We prove existence and uniqueness results for the considered generalized, set-valued, mixed-ordered, variational inclusion problem. Further, we put forward a three-step iterative algorithm using a ⊕ operator, and analyze the convergence of the suggested iterative algorithm under some mild assumptions. Finally, we reconfirm the existence and convergence results by an illustrative numerical example. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
23 pages, 382 KiB  
Article
Coefficient Bounds for Certain Classes of Analytic Functions Associated with Faber Polynomial
Symmetry 2021, 13(2), 302; https://doi.org/10.3390/sym13020302 - 10 Feb 2021
Cited by 12 | Viewed by 1892
Abstract
In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of nth(n3) Taylor–Maclaurin coefficients an is [...] Read more.
In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of nth(n3) Taylor–Maclaurin coefficients an is obtained. Furthermore, the bounds value of the first two coefficients of such functions is established. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
18 pages, 370 KiB  
Article
Synchronous Steady State Solutions of a Symmetric Mixed Cubic-Superlinear Schrödinger System
Symmetry 2021, 13(2), 190; https://doi.org/10.3390/sym13020190 - 26 Jan 2021
Cited by 1 | Viewed by 1520
Abstract
Systems of coupled nonlinear PDEs are applied in many fields as suitable models for many natural and physical phenomena. This makes them active and attractive subjects for both theoretical and numerical investigations. In the present paper, a symmetric nonlinear Schrödinger (NLS) system is [...] Read more.
Systems of coupled nonlinear PDEs are applied in many fields as suitable models for many natural and physical phenomena. This makes them active and attractive subjects for both theoretical and numerical investigations. In the present paper, a symmetric nonlinear Schrödinger (NLS) system is considered for the existence of the steady state solutions by applying a minimizing problem on some modified Nehari manifold. The nonlinear part is a mixture of cubic and superlinear nonlinearities and cubic correlations. Some numerical simulations are also illustrated graphically to confirm the theoretical results. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
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8 pages, 278 KiB  
Article
Fixed Point of Interpolative Rus–Reich–Ćirić Contraction Mapping on Rectangular Quasi-Partial b-Metric Space
Symmetry 2021, 13(1), 32; https://doi.org/10.3390/sym13010032 - 28 Dec 2020
Cited by 15 | Viewed by 1935
Abstract
The purpose of this study is to introduce a new type of extended metric space, i.e., the rectangular quasi-partial b-metric space, which means a relaxation of the symmetry requirement of metric spaces, by including a real number s in the definition of the [...] Read more.
The purpose of this study is to introduce a new type of extended metric space, i.e., the rectangular quasi-partial b-metric space, which means a relaxation of the symmetry requirement of metric spaces, by including a real number s in the definition of the rectangular metric space defined by Branciari. Here, we obtain a fixed point theorem for interpolative Rus–Reich–Ćirić contraction mappings in the realm of rectangular quasi-partial b-metric spaces. Furthermore, an example is also illustrated to present the applicability of our result. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
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16 pages, 271 KiB  
Article
Analysis of Optical Solitons for Nonlinear Schrödinger Equation with Detuning Term by Iterative Transform Method
Symmetry 2020, 12(11), 1850; https://doi.org/10.3390/sym12111850 - 10 Nov 2020
Cited by 57 | Viewed by 2056
Abstract
In this article, the iteration transform method is used to evaluate the solution of a fractional-order dark optical soliton, bright optical soliton, and periodic solution of the nonlinear Schrödinger equations. The Caputo operator describes the fractional-order derivatives. The solutions of some illustrative examples [...] Read more.
In this article, the iteration transform method is used to evaluate the solution of a fractional-order dark optical soliton, bright optical soliton, and periodic solution of the nonlinear Schrödinger equations. The Caputo operator describes the fractional-order derivatives. The solutions of some illustrative examples are presented to show the validity of the proposed technique without using any polynomials. The proposed method provides the series form solutions, which converge to the exact results. Using the present methodology, the solutions of fractional-order problems as well as integral-order problems are calculated. The present method has less computational costs and a higher rate of convergence. Therefore, the suggested algorithm is constructive to solve other fractional-order linear and nonlinear partial differential equations. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
20 pages, 320 KiB  
Article
Subordination and Superordination Properties for Certain Family of Analytic Functions Associated with Mittag–Leffler Function
Symmetry 2020, 12(10), 1724; https://doi.org/10.3390/sym12101724 - 19 Oct 2020
Cited by 22 | Viewed by 1852
Abstract
We obtain new outcomes of analytic functions linked with operator Hα,βη,k(f) defined by Mittag–Leffler function. Moreover, new theorems of differential sandwich-type are obtained. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
19 pages, 2093 KiB  
Article
A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel
Symmetry 2020, 12(10), 1653; https://doi.org/10.3390/sym12101653 - 09 Oct 2020
Cited by 29 | Viewed by 2162
Abstract
The time–fractional reaction–diffusion (TFRD) model has broad physical perspectives and theoretical interpretation, and its numerical techniques are of significant conceptual and applied importance. A numerical technique is constructed for the solution of the TFRD model with the non-singular kernel. The Caputo–Fabrizio operator is [...] Read more.
The time–fractional reaction–diffusion (TFRD) model has broad physical perspectives and theoretical interpretation, and its numerical techniques are of significant conceptual and applied importance. A numerical technique is constructed for the solution of the TFRD model with the non-singular kernel. The Caputo–Fabrizio operator is applied for the discretization of time levels while the extended cubic B-spline (ECBS) function is applied for the space direction. The ECBS function preserves geometrical invariability, convex hull and symmetry property. Unconditional stability and convergence analysis are also proved. The projected numerical method is tested on two numerical examples. The theoretical and numerical results demonstrate that the order of convergence of 2 in time and space directions. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
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23 pages, 312 KiB  
Article
Some Dynamic Hilbert-Type Inequalities on Time Scales
Symmetry 2020, 12(9), 1410; https://doi.org/10.3390/sym12091410 - 25 Aug 2020
Cited by 19 | Viewed by 1739
Abstract
Throughout this article, we will demonstrate some new generalizations of dynamic Hilbert type inequalities, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. From these inequalities, as particular cases, we formulate [...] Read more.
Throughout this article, we will demonstrate some new generalizations of dynamic Hilbert type inequalities, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. From these inequalities, as particular cases, we formulate some integral and discrete inequalities that have been demonstrated in the literature and also extend some of the dynamic inequalities that have been achieved in time scales. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
15 pages, 264 KiB  
Article
Limit Cycles of a Class of Polynomial Differential Systems Bifurcating from the Periodic Orbits of a Linear Center
Symmetry 2020, 12(8), 1346; https://doi.org/10.3390/sym12081346 - 12 Aug 2020
Cited by 8 | Viewed by 1364
Abstract
In this paper, we study the number of limit cycles of a new class of polynomial differential systems, which is an extended work of two families of differential systems in systems considered earlier. We obtain the maximum number of limit cycles that bifurcate [...] Read more.
In this paper, we study the number of limit cycles of a new class of polynomial differential systems, which is an extended work of two families of differential systems in systems considered earlier. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a center using the averaging theory of first and second order. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
19 pages, 1629 KiB  
Article
Novel Numerical Approach Based on Modified Extended Cubic B-Spline Functions for Solving Non-Linear Time-Fractional Telegraph Equation
Symmetry 2020, 12(7), 1154; https://doi.org/10.3390/sym12071154 - 10 Jul 2020
Cited by 32 | Viewed by 2908
Abstract
The telegraph model describes that the current and voltage waves can be reflected on a wire, that symmetrical wave patterns can form along a line. A numerical study of these voltage and current waves on a transferral line has been proposed via a [...] Read more.
The telegraph model describes that the current and voltage waves can be reflected on a wire, that symmetrical wave patterns can form along a line. A numerical study of these voltage and current waves on a transferral line has been proposed via a modified extended cubic B-spline (MECBS) method. The B-spline functions have the flexibility and high order accuracy to approximate the solutions. These functions also preserve the symmetrical property. The MECBS and Crank Nicolson technique are employed to find out the solution of the non-linear time fractional telegraph equation. The time direction is discretized in the Caputo sense while the space dimension is discretized by the modified extended cubic B-spline. The non-linearity in the equation is linearized by Taylor’s series. The proposed algorithm is unconditionally stable and convergent. The numerical examples are displayed to verify the authenticity and implementation of the method. Full article
(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)
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