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Fractional Calculus and the Applied Analysis
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Fractional Calculus and the Applied Analysis

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 28 August 2024 | Viewed by 5663

Special Issue Editor

Dr. Trushit Patel

E-Mail Website
Guest Editor
Department of General Studies and Science, University of the People, Pasadena, CA, USA
Interests: fractional differential equations; heat and mass transfer; fractional partial derivative equations; fractional physical equations

Special Issue Information

Dear Colleagues,

We are pleased to announce this Special Issue on “Fractional Calculus and the Applied Analysis (FCA)” in the specialized international journal of Axioms, which invites submissions on real-world applications of mathematical analysis, both at the level of its applications and the theoretical level. In essence, fractional calculus theory is a mathematical analysis tool applied to studying integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. That is why applying fractional calculus theory has become a focus of international academic research. 

The prominent members of the Editorial Board and the expertise of invited external reviewers ensure the high standards of its content. Since its inception, the journal has always aspired to be the most prestigious and suitable forum for publishing high-quality original results and surveys on special topics such as physics, biology, chemistry, heat transfer, fluid mechanics, signal processing, viscoelasticity, dynamical systems, or entropy theory, as well as for the exchange of ideas and discussion of open problems. 

Dr. Trushit Patel
Guest Editor

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. Axioms 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.

Keywords

  • fractional calculus
  • multivariable fractional calculus
  • fractional integral and derivatives
  • fractional ordinary and partial differential equations
  • problems of mathematical physics
  • control theory
  • fractional stochastic processes

Published Papers (6 papers)

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Research

21 pages, 748 KiB  
Article
Fractional Order Mathematical Modelling of HFMD Transmission via Caputo Derivative
by Aakash Mohandoss, Gunasundari Chandrasekar, Mutum Zico Meetei and Ahmed H. Msmali
Axioms 2024, 13(4), 213; https://doi.org/10.3390/axioms13040213 - 25 Mar 2024
Viewed by 523
Abstract
This paper studies a nonlinear fractional mathematical model for hand, foot, and mouth Disease (HFMD), incorporating a vaccinated compartment. Our initial focus involves establishing the non-negativity and boundedness of the fractional order dynamical model. The existence and uniqueness of the system are discussed [...] Read more.
This paper studies a nonlinear fractional mathematical model for hand, foot, and mouth Disease (HFMD), incorporating a vaccinated compartment. Our initial focus involves establishing the non-negativity and boundedness of the fractional order dynamical model. The existence and uniqueness of the system are discussed using the Caputo derivative operator formulation. Applying a fixed-point approach, we obtain results that confirm the presence of at least one solution. We analyze the stability behavior at the two equilibrium points (disease-free and endemic states) of the model and derive the basic reproduction number. Numerical simulations are conducted using the fractional Euler approach, and the simulation results confirm our analytical conclusions. This comprehensive approach enhances the understanding of HFMD dynamics and facilitates the policy making of health care centers to control the further spread of this disease. Full article
(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)
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16 pages, 4276 KiB  
Article
Time-Domain Fractional Behaviour Modelling with Rational Non-Singular Kernels
by Jocelyn Sabatier and Christophe Farges
Axioms 2024, 13(2), 99; https://doi.org/10.3390/axioms13020099 - 31 Jan 2024
Viewed by 773
Abstract
This paper proposes a solution to model fractional behaviours with a convolution model involving non-singular kernels and without using fractional calculus. The non-singular kernels considered are rational functions of time. The interest of this class of kernel is demonstrated with a pure power [...] Read more.
This paper proposes a solution to model fractional behaviours with a convolution model involving non-singular kernels and without using fractional calculus. The non-singular kernels considered are rational functions of time. The interest of this class of kernel is demonstrated with a pure power law function that can be approximated in the time domain by a rational function whose pole and zeros are interlaced and linked by geometric laws. The Laplace transform and frequency response of this class of kernel is given and compared with an approximation found in the literature. The comparison reveals less phase oscillation with the solution proposed by the authors. A parameter estimation method is finally proposed to obtain the rational kernel model for general fractional behaviour. An application performed with this estimation method demonstrates the interest in non-singular rational kernels to model fractional behaviours. Another interest is the physical interpretation fractional behaviours that can be implemented with delay distributions. Full article
(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)
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14 pages, 300 KiB  
Article
General Fractional Calculus Operators of Distributed Order
by Mohammed Al-Refai and Yuri Luchko
Axioms 2023, 12(12), 1075; https://doi.org/10.3390/axioms12121075 - 24 Nov 2023
Cited by 1 | Viewed by 745
Abstract
In this paper, two types of general fractional derivatives of distributed order and a corresponding fractional integral of distributed type are defined, and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of one-parametric [...] Read more.
In this paper, two types of general fractional derivatives of distributed order and a corresponding fractional integral of distributed type are defined, and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of one-parametric Sonin kernels with power law singularities at the origin. The conventional fractional derivatives of distributed order based on the Riemann–Liouville and Caputo fractional derivatives are particular cases of the general fractional derivatives of distributed order introduced in this paper. Full article
(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)
14 pages, 1046 KiB  
Article
Solution of Two-Dimensional Solute Transport Model for Heterogeneous Porous Medium Using Fractional Reduced Differential Transform Method
by Manan A. Maisuria, Priti V. Tandel and Trushitkumar Patel
Axioms 2023, 12(11), 1039; https://doi.org/10.3390/axioms12111039 - 08 Nov 2023
Viewed by 738
Abstract
This study contains a two-dimensional mathematical model of solute transport in a river with temporally and spatially dependent flow, explicitly focusing on pulse-type input point sources with a fractional approach. This model is analyzed by assuming an initial concentration function as a declining [...] Read more.
This study contains a two-dimensional mathematical model of solute transport in a river with temporally and spatially dependent flow, explicitly focusing on pulse-type input point sources with a fractional approach. This model is analyzed by assuming an initial concentration function as a declining exponential function in both the longitudinal and transverse directions. The governing equation is a time-fractional two-dimensional advection–dispersion equation with a variable form of dispersion coefficients, velocities, decay constant of the first order, production rate coefficient for the solute at the zero-order level, and retardation factor. The solution of the present problem is obtained by the fractional reduced differential transform method (FRDTM). The analysis of the initial retardation factor has been carried out via plots. Also, the influence of initial longitudinal and transverse dispersion coefficients and velocities has been examined by graphical analysis. The impact of fractional parameters on pollution levels is also analyzed numerically and graphically. The study of convergence for the FRDTM technique has been conducted to assess its efficacy and accuracy. Full article
(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)
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15 pages, 692 KiB  
Article
New Results Achieved for Fractional Differential Equations with Riemann–Liouville Derivatives of Nonlinear Variable Order
by Hallouz Abdelhamid, Gani Stamov, Mohammed Said Souid and Ivanka Stamova
Axioms 2023, 12(9), 895; https://doi.org/10.3390/axioms12090895 - 20 Sep 2023
Cited by 2 | Viewed by 771
Abstract
This paper proposes new existence and uniqueness results for an initial value problem (IVP) of fractional differential equations of nonlinear variable order. Riemann–Liouville-type fractional derivatives are considered in the problem. The new fundamental results achieved in this work are obtained by using the [...] Read more.
This paper proposes new existence and uniqueness results for an initial value problem (IVP) of fractional differential equations of nonlinear variable order. Riemann–Liouville-type fractional derivatives are considered in the problem. The new fundamental results achieved in this work are obtained by using the inequalities technique and the fixed point theory. In addition, uniform stability criteria for the solutions are derived. The accomplished results are new and complement the scientific research in the field. A numerical example is composed to show the efficacy and potency of the proposed criteria. Full article
(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)
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13 pages, 1294 KiB  
Article
Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation
by Nidhish Kumar Mishra, Mashael M. AlBaidani, Adnan Khan and Abdul Hamid Ganie
Axioms 2023, 12(4), 400; https://doi.org/10.3390/axioms12040400 - 20 Apr 2023
Cited by 9 | Viewed by 961
Abstract
This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV [...] Read more.
This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV equation was constructed to represent a wide range of physical processes involving the evolution and interaction of nonlinear waves. In the Caputo sense, the fractional derivative is considered. We employed the Yang transform, the Adomian decomposition method and the homotopy perturbation method to obtain the solution to the time-fractional Lax’s Korteweg–de Vries problem. We examined and compared a particular example with the actual result to verify the approaches. By utilizing these methods, we can construct recurrence relations that represent the solution to the problem that is being proposed, and we are then able to present graphical representations that enable us to visually examine all of the results in the proposed case for different fractional order values. Furthermore, the results of the current approach exhibit a good correlation with the precise solution to the problem being studied. Furthermore, the present study offers an example of error analysis. The numerical outcomes obtained by applying the provided approaches demonstrate that the techniques are easy to use and have superior computational performance. Full article
(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)
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