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Fractal Fract
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Application of Fractional Calculus as an Interdisciplinary Modeling Framework
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Application of Fractional Calculus as an Interdisciplinary Modeling Framework

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (20 March 2023) | Viewed by 17018

Special Issue Editors

Dr. Antonela Toma

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Guest Editor
Center for Research and Training in Innovative Techniques of Applied Mathematics in Engineering, Faculty of Applied Sciences, University Politehnica of Bucharest, 060042 Bucharest, Romania
Interests: applied mathematics; fractional calculus; distribution theory; partial differential equations
Special Issues, Collections and Topics in MDPI journals
Dr. Dorota Mozyrska

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Guest Editor
Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15-351 Bialystok, Poland
Interests: fractional-order systems; dynamical systems; numerical analysis; stability analysis; mathematical modeling
Special Issues, Collections and Topics in MDPI journals
Dr. Octavian Postavaru

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Guest Editor
Center for Research and Training in Innovative Techniques of Applied Mathematics in Engineering, University Politehnica of Bucharest, 060042 Bucharest, Romania
Interests: fractional-order partial differential equations; hybrid functions; block-pulse; non-orthogonal polynomials
Special Issues, Collections and Topics in MDPI journals
Dr. Mihai Rebenciuc

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Guest Editor
Center for Research and Training in Innovative Techniques of Applied Mathematics in Engineering, Faculty of Applied Sciences, University Politehnica of Bucharest, 060042 Bucharest, Romania
Interests: algebraic logic; relationship theory; rough sets; random variables; stochastic processes; game theory; differential equations; fractional calculus; grammatical evolution; set theory; mathematical analysis
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Dr. Simona Mihaela Bibic

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Guest Editor
Center for Research and Training in Innovative Techniques of Applied Mathematics in Engineering, Faculty of Applied Sciences, University Politehnica of Bucharest, 060042 Bucharest, Romania
Interests: applied mathematics; fractional calculus; wavelet analysis; operations research; graph theory; bio-inspired computing
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

From a mathematical fantasy to a complex and rigorous mathematical theory, the subject of fractional calculus has applications in diverse and widespread fields of engineering and science, having a rapid growth of its applications.

One of the greatest ways to make discoveries in math and science is finding answers to many new questions and interesting results. Even if fractional calculus has found an important place in science and engineering as a powerful tool for modeling complex phenomena with many excellent results, there are still some unresolved challenges.

The aim of this special issue is to bring together researchers of diverse fields of Physics, Medicine, Biology, Biosciences, Engineering, Robotics and Signal Processing, including Applied Mathematics and to create an international and interdisciplinary framework for sharing innovative research work related to fractional calculus.

This special issue will cover all theoretical and applied aspects of the fractional calculus and related approaches. Original research articles submissions dealing with topics mentioned bellow are encourage.

TOPIC

Fractional Differential Theory and Application

Fractional Differential Equation Numerical Solution and Application

Fractional Integral Theory and Application

Fractional Integral Equation Numerical Solution and Application

Local Fractional Calculus Theory and Application

Applications of fractional differentiation in signal analysis, chaos, bioengineering, economics, finance, fractal theory, optics, control systems, artificial intelligence, mathematical biology, nanotechnology and medicine, physics, mechanics, engineering, probability and statistics.

Dr. Antonela Toma
Prof. Dr. Dorota Mozyrska
Dr. Octavian Postavaru
Dr. Mihai Rebenciuc
Dr. Simona Mihaela Bibic
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.

Keywords

  • general fractional calculus
  • special functions
  • integral transforms
  • harmonic analysis
  • fractional variational calculus
  • ODEs, PDEs and integral equations and systems
  • wave equation
  • evolution equation
  • mathematical models of phenomena
  • fractional quantum fields
  • nonlinear control methods
  • fractional-order controllers
  • bio-medical applications
  • economic models with memory
  • numerical and approximation methods
  • computational procedures and algorithms

Published Papers (9 papers)

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Research

15 pages, 1190 KiB  
Article
Fractional Order Sequential Minimal Optimization Classification Method
by Chunna Zhao, Licai Dai and Yaqun Huang
Fractal Fract. 2023, 7(8), 637; https://doi.org/10.3390/fractalfract7080637 - 21 Aug 2023
Cited by 2 | Viewed by 1494
Abstract
Sequential minimal optimization (SMO) method is an algorithm for solving optimization problems arising from the training process of support vector machines (SVM). The SMO algorithm is mainly used to solve the optimization problem of the objective function of SVM, and it can have [...] Read more.
Sequential minimal optimization (SMO) method is an algorithm for solving optimization problems arising from the training process of support vector machines (SVM). The SMO algorithm is mainly used to solve the optimization problem of the objective function of SVM, and it can have high accuracy. However, its optimization accuracy can be improved. Fractional order calculus is an extension of integer order calculus, which can more accurately describe the actual system and get more accurate results. In this paper, the fractional order sequential minimal optimization (FOSMO) method is proposed based on the SMO method and fractional order calculus for classification. Firstly, an objective function is expressed by a fractional order function using the FOSMO method. The representation and meaning of fractional order terms in the objective function are studied. Then the fractional derivative of Lagrange multipliers is obtained according to fractional order calculus. Lastly, the objective function is optimized based on fractional order Lagrange multipliers, and then some experiments are carried out on the linear and nonlinear classification cases. Some experiments are carried out on two-classification and multi-classification situations, and experimental results show that the FOSMO method can obtain better accuracy than the normal SMO method. Full article
(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework)
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23 pages, 847 KiB  
Article
Solving and Numerical Simulations of Fractional-Order Governing Equation for Micro-Beams
by Aimin Yang, Qunwei Zhang, Jingguo Qu, Yuhuan Cui and Yiming Chen
Fractal Fract. 2023, 7(2), 204; https://doi.org/10.3390/fractalfract7020204 - 18 Feb 2023
Cited by 4 | Viewed by 1443
Abstract
This paper applies a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. A nonlinear-fractional order model for viscoelastic micro-beams is constructed. Before solving the governing equations, the operator matrices of shifted Chebyshev polynomials [...] Read more.
This paper applies a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. A nonlinear-fractional order model for viscoelastic micro-beams is constructed. Before solving the governing equations, the operator matrices of shifted Chebyshev polynomials are derived first. Shifted Chebyshev polynomials are used to approximate the deflection function, and the nonlinear fractional order governing equation is expressed in the form of operator matrices. Next, the collocation method is used to discretize the equations into the form of algebraic equations for solution. It simplifies the calculation. MATLAB software was used to program this algorithm to simulate the numerical solution of the deflection. The effectiveness and accuracy of the algorithm are verified by the numerical example. Finally, numerical simulations are carried out on the viscoelastic micro-beams. It is found that the viscous damping coefficient is inversely proportional to the deflection, and the length scale parameter of the micro-beam is also inversely proportional to the deflection. In addition, the stress and strain of micro-beam, the change of deflection under different simple harmonic loads, and potential energy of micro-beam are discussed. The results of the study fully demonstrated that the shifted Chebyshev polynomial algorithm is effective for the numerical simulations of viscoelastic micro-beams. Full article
(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework)
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12 pages, 336 KiB  
Article
Inflation and Fractional Quantum Cosmology
by Seyed Meraj Mousavi Rasouli, Emanuel W. de Oliveira Costa, Paulo Moniz and Shahram Jalalzadeh
Fractal Fract. 2022, 6(11), 655; https://doi.org/10.3390/fractalfract6110655 - 5 Nov 2022
Cited by 14 | Viewed by 1246
Abstract
The Wheeler–DeWitt equation for a flat and compact Friedmann–Lemaître–Robertson–Walker cosmology at the pre-inflation epoch is studied in the contexts of the standard and fractional quantum cosmology. Working within the semiclassical regime and applying the Wentzel-Kramers-Brillouin (WKB) approximation, we show that some fascinating consequences [...] Read more.
The Wheeler–DeWitt equation for a flat and compact Friedmann–Lemaître–Robertson–Walker cosmology at the pre-inflation epoch is studied in the contexts of the standard and fractional quantum cosmology. Working within the semiclassical regime and applying the Wentzel-Kramers-Brillouin (WKB) approximation, we show that some fascinating consequences are obtained for our simple fractional scenario that are completely different from their corresponding standard counterparts: (i) The conventional de Sitter behavior of the inflationary universe for constant potential is replaced by a power-law inflation. (ii) The non-locality of the Riesz’s fractional derivative produces a power-law inflation that depends on the fractal dimension of the compact spatial section of space-time, independent of the energy scale of the inflaton. Full article
(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework)
46 pages, 15290 KiB  
Article
Proposal of a General Identification Method for Fractional-Order Processes Based on the Process Reaction Curve
by Juan J. Gude and Pablo García Bringas
Fractal Fract. 2022, 6(9), 526; https://doi.org/10.3390/fractalfract6090526 - 17 Sep 2022
Cited by 11 | Viewed by 1846
Abstract
This paper aims to present a general identification procedure for fractional first-order plus dead-time (FFOPDT) models. This identification method is general for processes having S-shaped step responses, where process information is collected from an open-loop step-test experiment, and has been conducted by fitting [...] Read more.
This paper aims to present a general identification procedure for fractional first-order plus dead-time (FFOPDT) models. This identification method is general for processes having S-shaped step responses, where process information is collected from an open-loop step-test experiment, and has been conducted by fitting three arbitrary points on the process reaction curve. In order to validate this procedure and check its effectiveness for the identification of fractional-order models from the process reaction curve, analytical expressions of the FFOPDT model parameters have been obtained for both situations: as a function of any three points and three points symmetrically located on the reaction curve, respectively. Some numerical examples are provided to show the simplicity and effectiveness of the proposed procedure. Good results have been obtained in comparison with other well-recognized identification methods, especially when simplicity is emphasized. This identification procedure has also been applied to a thermal-based experimental setup in order to test its applicability and to obtain insight into the practical issues related to its implementation in a microprocessor-based control hardware. Finally, some comments and reflections about practical issues relating to industrial practice are offered in this context. Full article
(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework)
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15 pages, 3882 KiB  
Article
Adaptive Intelligent High-Order Sliding Mode Fractional Order Control for Harmonic Suppression
by Yunmei Fang, Siyang Li and Juntao Fei
Fractal Fract. 2022, 6(9), 482; https://doi.org/10.3390/fractalfract6090482 - 30 Aug 2022
Cited by 3 | Viewed by 1362
Abstract
A second-order sliding mode control (SOSMC) with a fractional module using adaptive fuzzy controller is developed for an active power filter (APF). A second-order sliding surface using a fractional module which can decrease the discontinuities and chattering is designed to make the system [...] Read more.
A second-order sliding mode control (SOSMC) with a fractional module using adaptive fuzzy controller is developed for an active power filter (APF). A second-order sliding surface using a fractional module which can decrease the discontinuities and chattering is designed to make the system work stably and simplify the design process. In addition, a fuzzy logic control is utilized to estimate the parameter uncertainties. Simulation and experimental discussion illustrated that the designed fractional SOSMC with adaptive fuzzy controller is valid in satisfactorily eliminating harmonic, showing good robustness and stability compared with an integer order one. Full article
(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework)
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11 pages, 2901 KiB  
Article
Cluster Oscillation of a Fractional-Order Duffing System with Slow Variable Parameter Excitation
by Xianghong Li, Yanli Wang and Yongjun Shen
Fractal Fract. 2022, 6(6), 295; https://doi.org/10.3390/fractalfract6060295 - 28 May 2022
Cited by 1 | Viewed by 1512
Abstract
The complicated dynamic behavior of a fractional-order Duffing system with slow variable parameter excitation is investigated. The stability and bifurcation behavior of the fast subsystem are analyzed by using the dynamic theory of fractional-order systems. The pitchfork bifurcation, Hopf bifurcation and limit cycle [...] Read more.
The complicated dynamic behavior of a fractional-order Duffing system with slow variable parameter excitation is investigated. The stability and bifurcation behavior of the fast subsystem are analyzed by using the dynamic theory of fractional-order systems. The pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation are discussed in detail, and it was found that Hopf bifurcation only happens while the fractional order is bigger than 1. On the other hand, the influence of the amplitude of parametric excitation on cluster oscillation models is discussed. The results show that amplitude regulates cluster oscillation models with different bifurcation types. The point–point cluster oscillation only relates to pitchfork bifurcation. The point–cycle cluster oscillation includes pitchfork bifurcation and Hopf bifurcation. The point–cycle–cycle cluster oscillation involves three kinds of bifurcation, i.e., the pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation. The larger the amplitude, the more bifurcation types are involved. The research results of cluster oscillation and its generation mechanism will provide valuable theoretical basis for mechanical manufacturing and engineering practice. Full article
(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework)
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14 pages, 810 KiB  
Article
Enhancing the Accuracy of Solving Riccati Fractional Differential Equations
by Antonela Toma, Flavius Dragoi and Octavian Postavaru
Fractal Fract. 2022, 6(5), 275; https://doi.org/10.3390/fractalfract6050275 - 20 May 2022
Cited by 6 | Viewed by 1929
Abstract
In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing x with xα, with positive α. Fractional derivatives are in the Caputo sense. With the help of [...] Read more.
In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing x with xα, with positive α. Fractional derivatives are in the Caputo sense. With the help of incomplete beta functions, we are able to build exactly the Riemann–Liouville fractional integral operator associated with FOHBPB. This operator, together with the Newton–Cotes collocation method, allows the reduction of fractional differential equations to a system of algebraic equations, which can be solved by Newton’s iterative method. The simplicity of the method recommends it for applications in engineering and nature. The accuracy of this method is illustrated by five examples, and there are situations in which we obtain accuracy eleven orders of magnitude higher than if α=1. Full article
(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework)
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18 pages, 7555 KiB  
Article
An Investigation of Fractional One-Dimensional Groundwater Recharge by Spreading Using an Efficient Analytical Technique
by Rekha Javare Gowda, Sandeep Singh, Suma Seethakal Padmarajaiah, Umair Khan, Aurang Zaib and Wajaree Weera
Fractal Fract. 2022, 6(5), 249; https://doi.org/10.3390/fractalfract6050249 - 30 Apr 2022
Cited by 1 | Viewed by 1556
Abstract
In the present work, the q-homotopy analysis transform method (q-HATM) was used to generate an analytical solution for the moisture content distribution in a one-dimensional vertical groundwater recharge problem. Three scenarios for the Brooks–Corey model are studied based on linear [...] Read more.
In the present work, the q-homotopy analysis transform method (q-HATM) was used to generate an analytical solution for the moisture content distribution in a one-dimensional vertical groundwater recharge problem. Three scenarios for the Brooks–Corey model are studied based on linear and nonlinear diffusivity and conductivity functions. The governing nonlinear fractional partial differential equations are solved effectively by the combination of a hybrid analytical technique, which is the combination of the q-homotopy analysis method and the Laplace transform method. Figures and tables are used to discuss the outcomes for fractional values of the time derivative. Mathematica software is used to plot the figures. The examples used in this paper demonstrate the accuracy and competence of the considered algorithm. The acquired results demonstrate the efficiency and reliability of the projected scheme and are also suitable to carry out the highly nonlinear complex problems in a real-world scenario. Full article
(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework)
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23 pages, 6571 KiB  
Article
Front Propagation of Exponentially Truncated Fractional-Order Epidemics
by Afshin Farhadi and Emmanuel Hanert
Fractal Fract. 2022, 6(2), 53; https://doi.org/10.3390/fractalfract6020053 - 21 Jan 2022
Cited by 3 | Viewed by 2238
Abstract
The existence of landscape constraints in the home range of living organisms that adopt Lévy-flight movement patterns, prevents them from making arbitrarily large displacements. Their random movements indeed occur in a finite space with an upper bound. In order to make realistic models, [...] Read more.
The existence of landscape constraints in the home range of living organisms that adopt Lévy-flight movement patterns, prevents them from making arbitrarily large displacements. Their random movements indeed occur in a finite space with an upper bound. In order to make realistic models, by introducing exponentially truncated Lévy flights, such an upper bound can thus be taken into account in the reaction-diffusion models. In this work, we have investigated the influence of the λ-truncated fractional-order diffusion operator on the spatial propagation of the epidemics caused by infectious diseases, where λ is the truncation parameter. Analytical and numerical simulations show that depending on the value of λ, different asymptotic behaviours of the travelling-wave solutions can be identified. For small values of λ (λ≳0), the tails of the infective waves can decay algebraically leading to an exponential growth of the epidemic speed. In that case, the truncation has no impact on the superdiffusive epidemics. By increasing the value of λ, the algebraic decaying tails can be tamed leading to either an upper bound on the epidemic speed representing the maximum speed value or the generation of the infective waves of a constant shape propagating at a minimum constant speed as observed in the classical models (second-order diffusion epidemic models). Our findings suggest that the truncated fractional-order diffusion equations have the potential to model the epidemics of animals performing Lévy flights, as the animal diseases can spread more smoothly than the exponential acceleration of the human disease epidemics. Full article
(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework)
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