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Fractal Fract
Special Issues
Numerical Methods and Simulations in Fractal and Fractional Problems
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Numerical Methods and Simulations in Fractal and Fractional Problems

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: closed (15 September 2021) | Viewed by 8096

Special Issue Editors

Prof. Dr. Carlo Cattani

E-Mail Website
Guest Editor
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
Prof. Dr. Mustafa Inc

E-Mail Website
Guest Editor
Department of Mathematics, Science Faculty, Firat University, Elazig 23119, Turkey
Interests: numerical and analytical solutions of fractional-stochastic differential equations; lie symmetry; conservation laws; optical soliton
Prof. Dr. Mehmet Ali Akinlar

E-Mail Website
Guest Editor
Engineering Sciences Department, Faculty of Engineering and Natural Sciences, Bandirma Onyedi Eylul University, 10200, Bandirma, Balikesir, Turkey
Interests: solutions; chaos analysis and optimal control of fractional-stochastic differential equations

Special Issue Information

Dear Colleagues,

The fast development of new technologies in almost all scientific fields and engineering applications has led to the investigation of challenging problems with high nonlinearities, singularities, parametric dependence, and randomness, so that analytical and rigorous solutions can not satisfy the increasing expectations. On the other hand, the many facilities offered by more and more sophisticated tools enable us to compute the numerical solution of complex problems with high accuracy and robustness. Special attention must be paid to those fractal-like problems or fractional problems where self-similarity, scale invariance, and fractional order of operators might add some extra difficulties to the numerical approximation methods.

The main aim of this Special Issue is to collect research papers to illustrate the numerical and computational methods in modeling, simulating, and implementing fractal and fractional problems.

Potential topics include but are not limited to:

  • Non-Linear dynamics, nonlinear evolution equation
  • Numerical method, computational method
  • Partial differential equation
  • Integral equation
  • Ordinary differential equation
  • Stochastic fractional partial differential equation models and applications
  • Numerical and computational methods in fractional differential equations
  • Quantitative theory of differential equations
  • Fractional calculus-based control systems
  • Complex dynamics—nonlinear dynamical systems
  • Fractional calculus and its applications
  • Finance and economy dynamics
  • Fractals and chaos
  • Biological systems and bioinformatics
  • Image and signal processing
  • Stability
  • High-order numerical differential formulas for the fractional derivatives
  • High-order numerical algorithms for fractional differential equations

Prof. Dr. Carlo Cattani
Prof. Dr. Mustafa Inc
Prof. Dr. Mehmet Ali Akinlar
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. Fractal and Fractional 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 2700 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 (4 papers)

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Research

17 pages, 1952 KiB  
Article
Swarm Intelligence Procedures Using Meyer Wavelets as a Neural Network for the Novel Fractional Order Pantograph Singular System
by Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Juan L. G. Guirao and Tareq Saeed
Fractal Fract. 2021, 5(4), 277; https://doi.org/10.3390/fractalfract5040277 - 17 Dec 2021
Cited by 7 | Viewed by 1816
Abstract
The purpose of the current investigation is to find the numerical solutions of the novel fractional order pantograph singular system (FOPSS) using the applications of Meyer wavelets as a neural network. The FOPSS is presented using the standard form of the Lane–Emden equation [...] Read more.
The purpose of the current investigation is to find the numerical solutions of the novel fractional order pantograph singular system (FOPSS) using the applications of Meyer wavelets as a neural network. The FOPSS is presented using the standard form of the Lane–Emden equation and the detailed discussions of the singularity, shape factor terms along with the fractional order forms. The numerical discussions of the FOPSS are described based on the fractional Meyer wavelets (FMWs) as a neural network (NN) with the optimization procedures of global/local search procedures of particle swarm optimization (PSO) and interior-point algorithm (IPA), i.e., FMWs-NN-PSOIPA. The FMWs-NN strength is pragmatic and forms a merit function based on the differential system and the initial conditions of the FOPSS. The merit function is optimized, using the integrated capability of PSOIPA. The perfection, verification and substantiation of the FOPSS using the FMWs is pragmatic for three cases through relative investigations from the true results in terms of stability and convergence. Additionally, the statics’ descriptions further authorize the presentation of the FMWs-NN-PSOIPA in terms of reliability and accuracy. Full article
(This article belongs to the Special Issue Numerical Methods and Simulations in Fractal and Fractional Problems)
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13 pages, 2516 KiB  
Article
Numerical Solutions of a Heat Transfer for Fractional Maxwell Fluid Flow with Water Based Clay Nanoparticles; A Finite Difference Approach
by Arfan Ali, Muhammad Imran Asjad, Muhammad Usman and Mustafa Inc
Fractal Fract. 2021, 5(4), 242; https://doi.org/10.3390/fractalfract5040242 - 25 Nov 2021
Cited by 5 | Viewed by 1334
Abstract
Fractional-order mathematical modelling of physical phenomena is a hot topic among various researchers due to its many advantages over positive integer mathematical modelling. In this context, the appropriate solutions of such fractional-order physical modelling become a challenging task among scientists. This paper presents [...] Read more.
Fractional-order mathematical modelling of physical phenomena is a hot topic among various researchers due to its many advantages over positive integer mathematical modelling. In this context, the appropriate solutions of such fractional-order physical modelling become a challenging task among scientists. This paper presents a study of unsteady free convection fluid flow and heat transfer of Maxwell fluids with the presence of Clay nanoparticle modelling using fractional calculus. The obtained model was transformed into a set of linear nondimensional, partial differential equations (PDEs). The finite difference scheme is proposed to discretize the obtained set of nondimensional PDEs. The Maple code was developed and executed against the physical parameters and fractional-order parameter to explain the behavior of the velocity and temperature profiles. Some limiting solutions were obtained and compared with the latest existing ones in literature. The comparative study witnesses that the proposed scheme is a very efficient tool to handle such a physical model and can be extended to other diversified problems of a complex nature. Full article
(This article belongs to the Special Issue Numerical Methods and Simulations in Fractal and Fractional Problems)
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17 pages, 8022 KiB  
Article
A Spatially Distributed Network for Tracking the Pulsation Signal of Flow Field Based on CFD Simulation: Method and a Case Study
by Faye Jin, Ran Tao, Zhaoheng Lu and Ruofu Xiao
Fractal Fract. 2021, 5(4), 181; https://doi.org/10.3390/fractalfract5040181 - 23 Oct 2021
Cited by 7 | Viewed by 1415
Abstract
The pulsating characteristics in turbulent flow are very important physical quantities. There are many studies focused on the temporal characteristics of pulsation. However, the spatial distribution of temporal states with pulsations rarely receives attention. Therefore, the pulsation tracking network (PTN) method is proposed [...] Read more.
The pulsating characteristics in turbulent flow are very important physical quantities. There are many studies focused on the temporal characteristics of pulsation. However, the spatial distribution of temporal states with pulsations rarely receives attention. Therefore, the pulsation tracking network (PTN) method is proposed to track the pulsating characteristics of turbulence. Based on the computational fluid dynamics (CFD) simulation result, the PTN is arranged in a specific region of the flow domain. The fast Fourier Transform (FFT) method is used for time-frequency conversion. As shown in the example of trailing-edge vortex-shedding flow over NACA0009 hydrofoil, important pulsation quantities, including the total pulsation intensity, dominant frequencies, amplitude of frequencies, and the phase and phase difference, can be obtained with a high spatial resolution. The source, reason and attenuation of the vortex-shedding frequency fvs and the 2 fvs frequency caused by vortex-interaction are well indicated. The dominant regions of fvs and 2 fvs are shown and analysed. The propagation and attenuation of vortex-shedding induced pulsation are understood in detail. Based on the comparison against traditional analysis, PTN is found to function as a good supplement for the CFD post-processing by tracking unknown temporal and spatial characteristics. These findings represent a potential breakthrough in terms of solving actual pulsation-excited flow problems. Full article
(This article belongs to the Special Issue Numerical Methods and Simulations in Fractal and Fractional Problems)
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11 pages, 471 KiB  
Article
Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative
by Youssri Hassan Youssri
Fractal Fract. 2021, 5(3), 100; https://doi.org/10.3390/fractalfract5030100 - 17 Aug 2021
Cited by 33 | Viewed by 2085
Abstract
Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral [...] Read more.
Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature. Full article
(This article belongs to the Special Issue Numerical Methods and Simulations in Fractal and Fractional Problems)
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