Advanced Computational Methods for Fractional Calculus

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

Deadline for manuscript submissions: closed (30 October 2022) | Viewed by 11065

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,

Fractional calculus governs many phenomena that occur in nature and plays an important role in the progress of engineering and technology. Fractional differential equations describe various phenomena such as fluid flow in a porous material, anomalous diffusion transport, signal processing, control theory of dynamical systems, viscoelasticity, etc.

In the last few years, various numerical and computational methods and simulations have been developed and especially designed to handle symmetrical fractal and fractional problems, where the nonlocal properties and recursive algorithms play a fundamental role. In particular, Artificial Neural Network (ANN) methods have also been established as an additional powerful technique to solve a variety of real-world issues described by non-integer dimensional order operators. Therefore, ANN-based approaches have been recently used by various researchers to solve different types of symmetrical fractional order differential equations, also under the name of physics informed neural network (PINN).

Fractional calculus has been credited as being the natural mathematical model for power–law relations. These relations are often observed as accurate descriptors for natural phenomena. This Special Issue seeks to explore potential advantages that might be gained in applying advanced numerical and computational methods, including ANN and PINN to fractional calculus and non-integer order problems.

Potential topics include, but are not limited to:

  • Non-Linear Dynamics;
  • Fractional Calculus;
  • Neural Network Design, PINN;
  • Real Life Applications;
  • Mathematical Modeling of Neural Networks;
  • VLSI implementation of Neural Networks;
  • 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;
  • Advanced control systems;
  • Fractional calculus and its applications;
  • Evolutionary computing;
  • Finance and economy dynamics;
  • Fractals and chaos;
  • Biological systems and bioinformatics;
  • Nonlinear waves and acoustics;
  • Image and signal processing;
  • Transportation systems;
  • Geosciences;
  • Astronomy and cosmology;
  • Nuclear physics.

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 (7 papers)

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Research

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22 pages, 416 KiB  
Article
Stability and Existence of Solutions for a Tripled Problem of Fractional Hybrid Delay Differential Equations
Symmetry 2022, 14(12), 2579; https://doi.org/10.3390/sym14122579 - 06 Dec 2022
Cited by 5 | Viewed by 1102
Abstract
The purpose of this paper is to determine the existence of tripled fixed point results for the tripled symmetry system of fractional hybrid delay differential equations. We obtain results which support the existence of at least one solution to our system by applying [...] Read more.
The purpose of this paper is to determine the existence of tripled fixed point results for the tripled symmetry system of fractional hybrid delay differential equations. We obtain results which support the existence of at least one solution to our system by applying hybrid fixed point theory. Similar types of stability analysis are presented, including Ulam–Hyers, generalized Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias. The necessary stipulations for obtaining the solution to our proposed problem are established. Finally, we provide a non-trivial illustrative example to support and enhance our analysis. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fractional Calculus)
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20 pages, 463 KiB  
Article
Ulam Stability of Fractional Hybrid Sequential Integro-Differential Equations with Existence and Uniqueness Theory
Symmetry 2022, 14(11), 2438; https://doi.org/10.3390/sym14112438 - 17 Nov 2022
Viewed by 859
Abstract
In this paper, a variety of boundary value problems (BVPs) known as hybrid fractional sequential integro-differential equations (HFSIDs) with two point orders (p,q) are investigated. The uniqueness and existence of the solution are discussed via Banach fixed-point theorems. Certain [...] Read more.
In this paper, a variety of boundary value problems (BVPs) known as hybrid fractional sequential integro-differential equations (HFSIDs) with two point orders (p,q) are investigated. The uniqueness and existence of the solution are discussed via Banach fixed-point theorems. Certain particular theorems associated with Hyers–Ulam and Hyers–Ulam–Rassias stability to the solution, as well as the uniqueness and existence of the solution of the BVPs are studied. The results are illustrated with some particular examples, and the numerical data are analyzed for confirmation of the results. The results obtained in this work are simple and can easily be applicable to physical systems. Furthermore, symmetry analysis of fractional differential equations and HFSIDs are also presented. This is due to the fact that the aforementioned analysis plays a significant role in both the optimization and qualitative theory of fractional differential equations. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fractional Calculus)
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25 pages, 397 KiB  
Article
Fractional Jensen-Mercer Type Inequalities Involving Generalized Raina’s Function and Applications
Symmetry 2022, 14(10), 2204; https://doi.org/10.3390/sym14102204 - 19 Oct 2022
Viewed by 1044
Abstract
The aim of this paper is to derive some new generalized fractional analogues of Mercer type inequalities, essentially using the convexity property of the functions and Raina’s function. We also discuss several new special cases which show that our results are, to an [...] Read more.
The aim of this paper is to derive some new generalized fractional analogues of Mercer type inequalities, essentially using the convexity property of the functions and Raina’s function. We also discuss several new special cases which show that our results are, to an extent, unifying. In order to illustrate the significance of our results, we offer some interesting applications of our results to special means, error bounds, and q-digamma functions. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fractional Calculus)
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28 pages, 5717 KiB  
Article
The Dynamics of a Fractional-Order Mathematical Model of Cancer Tumor Disease
Symmetry 2022, 14(8), 1694; https://doi.org/10.3390/sym14081694 - 15 Aug 2022
Cited by 15 | Viewed by 2047
Abstract
This article explores the application of the reduced differential transform method (RDTM) for the computational solutions of two fractional-order cancer tumor models in the Caputo sense: the model based on cancer chemotherapeutic effects which explain the relation between chemotherapeutic drugs, tumor cells, normal [...] Read more.
This article explores the application of the reduced differential transform method (RDTM) for the computational solutions of two fractional-order cancer tumor models in the Caputo sense: the model based on cancer chemotherapeutic effects which explain the relation between chemotherapeutic drugs, tumor cells, normal cells, and immune cells using a fractional partial differential equations, and the model that describes the different cases of killing rate K of cancer cells (the killing percentage of cancer cells K (I) is dependent on the number of cells, (II) is a function of time only, and (III) is a function of space only). The solutions are presented using Mathematica software as a convergent power series with elegantly computed terms using the suggested technique. The proposed method gives new series form results for various values of gamma. To clarify the complexity of the models, we plot the two- and three-dimensional and contour graphics of the obtained solutions at varied values of fractional-order gamma and the selected system parameters. The solutions are analyzed with fractional and reduced differential transform methods to obtain an idea of invariance regarding the computed solution of the designed mathematical model. The obtained results demonstrate the efficiency and preciseness of the proposed method to achieve a better understanding of chemotherapy effects. It is observed that chemotherapy drugs boost immunity against the specific cancer by decreasing the number of tumor cells, and the killing rate K of cancerous cells depend on the cells concentration. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fractional Calculus)
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14 pages, 323 KiB  
Article
Some Characteristics of Matrix Operators on Generalized Fibonacci Weighted Difference Sequence Space
Symmetry 2022, 14(7), 1283; https://doi.org/10.3390/sym14071283 - 21 Jun 2022
Cited by 6 | Viewed by 1148
Abstract
The forthcoming property of this manuscript is its calculating of the goal of norms and lower bounds of matrix operators taken from the weighted sequence space p(w) onto a novel one defined in the present article as the generalized [...] Read more.
The forthcoming property of this manuscript is its calculating of the goal of norms and lower bounds of matrix operators taken from the weighted sequence space p(w) onto a novel one defined in the present article as the generalized Fibonacci weighted difference sequence space. In this process, first of all the Fibonacci difference matrix F˜(r,s) and the space composed of sequences of which F˜(r,s)-transforms lie in p(w˜), where r,sR are defined. Additionaly, since the seminormed space p(w˜,F˜(r,s)) has the absolute homogeneous property, the topological characteristics on it are distributed symmetrically everywhere in the space. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fractional Calculus)
14 pages, 307 KiB  
Article
Generalization of Some Fractional Integral Operator Inequalities for Convex Functions via Unified Mittag–Leffler Function
Symmetry 2022, 14(5), 922; https://doi.org/10.3390/sym14050922 - 30 Apr 2022
Cited by 3 | Viewed by 1163
Abstract
This paper aims to obtain the bounds of a class of integral operators containing Mittag–Leffler functions in their kernels. A recently defined unified Mittag–Leffler function plays a vital role in connecting the results of this paper with the well-known bounds of fractional integral [...] Read more.
This paper aims to obtain the bounds of a class of integral operators containing Mittag–Leffler functions in their kernels. A recently defined unified Mittag–Leffler function plays a vital role in connecting the results of this paper with the well-known bounds of fractional integral operators published in the recent past. The symmetry of a function about a line is a fascinating property that plays an important role in mathematical inequalities. A variant of the Hermite–Hadamard inequality is established using the closely symmetric property for (α,m)-convex functions. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fractional Calculus)

Review

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15 pages, 294 KiB  
Review
Fractional Hypergeometric Functions
Symmetry 2022, 14(4), 714; https://doi.org/10.3390/sym14040714 - 01 Apr 2022
Cited by 4 | Viewed by 1627
Abstract
The fractional calculus of special functions has significant importance and applications in various fields of science and engineering. Here, we aim to find the fractional integral and differential formulas of the extended hypergeometric-type functions by using the Marichev–Saigo–Maeda operators. All the outcomes presented [...] Read more.
The fractional calculus of special functions has significant importance and applications in various fields of science and engineering. Here, we aim to find the fractional integral and differential formulas of the extended hypergeometric-type functions by using the Marichev–Saigo–Maeda operators. All the outcomes presented here are of general attractiveness and can yield a number of previous works as special cases due to the high degree of symmetry of the involved functions. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fractional Calculus)
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