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Symmetry
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Golden Mean Number System and Fractal Theory for Physics and...
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Golden Mean Number System and Fractal Theory for Physics and Materials Science

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

Deadline for manuscript submissions: 31 December 2024 | Viewed by 15684

Special Issue Editors

Prof. Dr. Ji-Huan He

E-Mail Website
Guest Editor
National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou, China
Interests: fractal calculus; fractional calculus; nonlinear science; nanotechnology; biomechanics
Special Issues, Collections and Topics in MDPI journals
Dr. Muhammad Nadeem

E-Mail Website
Guest Editor
Faculty of Science, Yibin University, Yibin 644000, China
Interests: approximate and numerical solution of PDE's; fractal and fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The golden mean (0.618) is nature’s greatest secret (Scott Olsen, 2006) and can be found everywhere in a variety of contexts with vastly differing scales—from a small virus to our vast universe. The whole universe moves symmetrically and harmoniously, a small virus has a fractal symmetry, and all physical laws might be related to the golden mean and, thus, interest in the symmetry and harmonism of the golden mean has skyrocketed in physics and materials science.

This Special Issue focuses on the golden mean number system and fractal theory for physics and materials science, and submissions on the following specific topics are welcome:

1) Golden mean in nature materials;

2) Golden mean in physical phenomena;

3) Golden mean in music and clothing design;

4) Golden mean in mechanical and architectural designs;

5) Fractal patterns in natural phenomena and chaotic structures;

6) Golden mean and fractal patterns in image identification;

7) Golden mean and fractal models for metamaterials and other artificial materials;

8) Physical laws in fractal space or fractal spacetime;

9) Dark energy and dark matter in a fractal universe;

10) Continued fraction of the golden mean and its applications;

11) Fractal–fractional differential models for fractal phenomena.

Prof. Dr. Ji-Huan He
Dr. Muhammad Nadeem
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.

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Published Papers (8 papers)

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Research

12 pages, 864 KiB  
Article
Numerical Analysis of Time-Fractional Porous Media and Heat Transfer Equations Using a Semi-Analytical Approach
by Muhammad Nadeem, Asad Islam, Shazia Karim, Sorin Mureşan and Loredana Florentina Iambor
Symmetry 2023, 15(7), 1374; https://doi.org/10.3390/sym15071374 - 6 Jul 2023
Cited by 6 | Viewed by 1213
Abstract
In nature, symmetry is all around us. The symmetry framework represents integer partial differential equations and their fractional order in the sense of Caputo derivatives. This article suggests a semi-analytical approach based on Aboodh transform (AT) and the homotopy perturbation scheme [...] Read more.
In nature, symmetry is all around us. The symmetry framework represents integer partial differential equations and their fractional order in the sense of Caputo derivatives. This article suggests a semi-analytical approach based on Aboodh transform (AT) and the homotopy perturbation scheme (HPS) for achieving the approximate solution of time-fractional porous media and heat transfer equations. The AT converts the fractional problems into simple ones and obtains the recurrence relation without any discretization or assumption. This nonlinear recurrence relation can be decomposed via the use of the HPS to obtain the iterations in terms of series solutions. The initial conditions play an important role in determining the successive iterations and yields towards the exact solution. We provide some numerical applications to analyze the accuracy of this proposed scheme and show that the performance of our scheme has strong agreement with the exact results. Full article
(This article belongs to the Special Issue Golden Mean Number System and Fractal Theory for Physics and Materials Science)
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27 pages, 1063 KiB  
Article
A Soliton Solution for the Kadomtsev–Petviashvili Model Using Two Novel Schemes
by Asghar Ali, Sara Javed, Muhammad Nadeem, Loredana Florentina Iambor and Sorin Mureşan
Symmetry 2023, 15(7), 1364; https://doi.org/10.3390/sym15071364 - 4 Jul 2023
Cited by 7 | Viewed by 1339
Abstract
Symmetries are crucial to the investigation of nonlinear physical processes, particularly the evaluation of a differential problem in the real world. This study focuses on the investigation of the Kadomtsev–Petviashvili (KP) model within a (3+1)-dimensional domain, governing the behavior of wave [...] Read more.
Symmetries are crucial to the investigation of nonlinear physical processes, particularly the evaluation of a differential problem in the real world. This study focuses on the investigation of the Kadomtsev–Petviashvili (KP) model within a (3+1)-dimensional domain, governing the behavior of wave propagation in a medium characterized by both nonlinearity and dispersion. The inquiry employs two distinct analytical techniques to derive multiple soliton solutions and multiple solitary wave solutions. These methods include the modified Sardar sub-equation technique and the Darboux transformation (DT). The modified Sardar sub-equation technique is used to obtain multiple soliton solutions, while the DT is introduced to develop two bright and two dark soliton solutions. These solutions are presented alongside their corresponding constraint conditions and illustrated through 3-D, 2-D, and contour plots to physically portray the derived solutions. The results demonstrate that the employed analytical techniques are useful and have not yet been explored in the context of the analyzed models. The proposed methodologies are valuable and can be applied to additional nonlinear evolutionary models employed to describe nonlinear physical models within the domain of nonlinear science. Full article
(This article belongs to the Special Issue Golden Mean Number System and Fractal Theory for Physics and Materials Science)
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16 pages, 416 KiB  
Article
Analysis of the Fractional Differential Equations Using Two Different Methods
by Mohammad Partohaghighi, Ali Akgül, Esra Karatas Akgül, Nourhane Attia, Manuel De la Sen and Mustafa Bayram
Symmetry 2023, 15(1), 65; https://doi.org/10.3390/sym15010065 - 26 Dec 2022
Cited by 18 | Viewed by 2635
Abstract
Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, [...] Read more.
Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, a combination of the group preserving scheme and fictitious time integration method (FTIM) is considered to solve the problem. Firstly, we applied the FTIM role, and then the GPS came to integrate the obtained new system using initial conditions. Figure and tables containing the solutions are provided. The tabulated numerical simulations are compared with the reproducing kernel Hilbert space method (RKHSM) as well as the exact solution. The methodology of RKHSM mainly relies on the right choice of the reproducing kernel functions. The results confirm that the FTIM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed methods. Full article
(This article belongs to the Special Issue Golden Mean Number System and Fractal Theory for Physics and Materials Science)
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25 pages, 964 KiB  
Article
Crossover Dynamics of Rotavirus Disease under Fractional Piecewise Derivative with Vaccination Effects: Simulations with Real Data from Thailand, West Africa, and the US
by Surapol Naowarat, Shabir Ahmad, Sayed Saifullah, Manuel De la Sen and Ali Akgül
Symmetry 2022, 14(12), 2641; https://doi.org/10.3390/sym14122641 - 14 Dec 2022
Cited by 6 | Viewed by 1563
Abstract
Many diseases are caused by viruses of different symmetrical shapes. Rotavirus particles are approximately 75 nm in diameter. They have icosahedral symmetry and particles that possess two concentric protein shells, or capsids. In this research, using a piecewise derivative framework with singular and [...] Read more.
Many diseases are caused by viruses of different symmetrical shapes. Rotavirus particles are approximately 75 nm in diameter. They have icosahedral symmetry and particles that possess two concentric protein shells, or capsids. In this research, using a piecewise derivative framework with singular and non-singular kernels, we investigate the evolution of rotavirus with regard to the effect of vaccination. For the considered model, the existence of a solution of the piecewise rotavirus model is investigated via fixed-point results. The Adam–Bashforth numerical method along with the Newton polynomial is implemented to deduce the numerical solution of the considered model. Various versions of the stability of the solution of the piecewise rotavirus model are presented using the Ulam–Hyres concept and nonlinear analysis. We use MATLAB to perform the numerical simulation for a few fractional orders to study the crossover dynamics and evolution and effect of vaccination on rotavirus disease. To check the validity of the proposed approach, we compared our simulated results with real data from various countries. Full article
(This article belongs to the Special Issue Golden Mean Number System and Fractal Theory for Physics and Materials Science)
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15 pages, 315 KiB  
Article
Controllability of Impulsive Neutral Fractional Stochastic Systems
by Qura Tul Ain, Muhammad Nadeem, Ali Akgül and Manuel De la Sen
Symmetry 2022, 14(12), 2612; https://doi.org/10.3390/sym14122612 - 9 Dec 2022
Cited by 6 | Viewed by 1375
Abstract
The study of dynamic systems appears in various aspects of dynamical structures such as decomposition, decoupling, observability, and controllability. In the present research, we study the controllability of fractional stochastic systems (FSF) and examine the Poisson jumps in finite dimensional space where the [...] Read more.
The study of dynamic systems appears in various aspects of dynamical structures such as decomposition, decoupling, observability, and controllability. In the present research, we study the controllability of fractional stochastic systems (FSF) and examine the Poisson jumps in finite dimensional space where the fractional impulsive neutral stochastic system is controllable. Sufficient conditions are demonstrated with the aid of fixed point theory. The Mittag-Leffler (ML) matrix function defines the controllability of the Grammian matrix (GM). The relation to symmetry is clear since the controllability Grammian is a hermitian matrix (since the integrand in its definition is hermitian) and this is the complex version of a symmetric matrix. In fact, such a Grammian becomes a symmetric matrix in the specific scenario where the controllability Grammian is a real matrix. Some examples are provided to demonstrate the feasibility of the present theory. Full article
(This article belongs to the Special Issue Golden Mean Number System and Fractal Theory for Physics and Materials Science)
21 pages, 1990 KiB  
Article
Numerical Inverse Laplace Transform Methods for Advection-Diffusion Problems
by Kamran, Farman Ali Shah, Wael Hosny Fouad Aly, Hasan Aksoy, Fahad M. Alotaibi and Ibrahim Mahariq
Symmetry 2022, 14(12), 2544; https://doi.org/10.3390/sym14122544 - 1 Dec 2022
Cited by 7 | Viewed by 2463
Abstract
Partial differential equations arising in engineering and other sciences describe nature adequately in terms of symmetry properties. This article develops a numerical method based on the Laplace transform and the numerical inverse Laplace transform for numerical modeling of diffusion problems. This method transforms [...] Read more.
Partial differential equations arising in engineering and other sciences describe nature adequately in terms of symmetry properties. This article develops a numerical method based on the Laplace transform and the numerical inverse Laplace transform for numerical modeling of diffusion problems. This method transforms the time-dependent problem to a corresponding time-independent inhomogeneous problem by employing the Laplace transform. Then a local radial basis functions method is employed to solve the transformed problem in the Laplace domain. The main feature of the local radial basis functions method is the collocation on overlapping sub-domains of influence instead of on the whole domain, which reduces the size of the collocation matrix; hence, the problem of ill-conditioning in global radial basis functions is resolved. The Laplace transform is used in comparison with a finite difference technique to deal with the time derivative and avoid the effect of the time step on numerical stability and accuracy. However, using the Laplace transform sometimes leads to a solution in the Laplace domain that cannot be converted back into the real domain by analytic methods. Therefore, in such a case, the Laplace transform is inverted numerically. In this investigation, two inversion techniques are utilized; (i) the contour integration method, and (ii) the Stehfest method. Three test problems are used to evaluate the proposed numerical method. The numerical results demonstrate that the proposed method is computationally efficient and highly accurate. Full article
(This article belongs to the Special Issue Golden Mean Number System and Fractal Theory for Physics and Materials Science)
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8 pages, 456 KiB  
Article
Extraction of Exact Solutions of Higher Order Sasa-Satsuma Equation in the Sense of Beta Derivative
by Emad Fadhal, Arzu Akbulut, Melike Kaplan, Muath Awadalla and Kinda Abuasbeh
Symmetry 2022, 14(11), 2390; https://doi.org/10.3390/sym14112390 - 11 Nov 2022
Cited by 18 | Viewed by 1572
Abstract
Nearly every area of mathematics, natural, social, and engineering now includes research into finding exact answers to nonlinear fractional differential equations (NFDES). In order to discover the exact solutions to the higher order Sasa-Satsuma equation in the sense of the beta derivative, the [...] Read more.
Nearly every area of mathematics, natural, social, and engineering now includes research into finding exact answers to nonlinear fractional differential equations (NFDES). In order to discover the exact solutions to the higher order Sasa-Satsuma equation in the sense of the beta derivative, the paper will discuss the modified simple equation (MSE) and exponential rational function (ERF) approaches. In general, symmetry and travelling wave solutions of the Sasa-Satsuma equation have a common correlation with each other, thus we reduce equations from wave transformations to ordinary differential equations with the help of Lie symmetries. Actually, we can say that wave moves are symmetrical. The considered procedures are effective, accurate, simple, and straightforward to compute. In order to highlight the physical characteristics of the solutions, we also provide 2D and 3D plots of the results. Full article
(This article belongs to the Special Issue Golden Mean Number System and Fractal Theory for Physics and Materials Science)
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14 pages, 4280 KiB  
Article
Do Artificial Neural Networks Always Provide High Prediction Performance? An Experimental Study on the Insufficiency of Artificial Neural Networks in Capacitance Prediction of the 6H-SiC/MEH-PPV/Al Diode
by Andaç Batur Çolak, Tamer Güzel, Anum Shafiq and Kamsing Nonlaopon
Symmetry 2022, 14(8), 1511; https://doi.org/10.3390/sym14081511 - 23 Jul 2022
Cited by 4 | Viewed by 1873
Abstract
In this paper, we study a new model that represents the symmetric connection between capacitance–voltage and Schottky diode. This model has a symmetrical shape towards the horizontal direction. In recent times, works conducted on artificial neural network structure, which is one of the [...] Read more.
In this paper, we study a new model that represents the symmetric connection between capacitance–voltage and Schottky diode. This model has a symmetrical shape towards the horizontal direction. In recent times, works conducted on artificial neural network structure, which is one of the greatest actual artificial intelligence apparatuses used in various fields, stated that artificial neural networks are apparatuses that proposal very high forecast performance by the side of conventional structures. In the current investigation, an artificial neural network structure has been generated to guess the capacitance voltage productions of the Schottky diode with organic polymer edge, contingent on the frequency with a symmetrical shape. Of the dataset, 130 were grouped for training, 28 for validation, and 28 for testing. In order to evaluate the effect of the number of neurons on the prediction accuracy, three different models with different neuron numbers have been developed. This study, in which an artificial neural network model, although well-trained, could not predict the output values correctly, is a first in the literature. With this aspect, the study can be considered as a pioneering study that brings a novelty to the literature. Full article
(This article belongs to the Special Issue Golden Mean Number System and Fractal Theory for Physics and Materials Science)
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