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Fractal Fract
Special Issues
Fractional Differential Operators with Classical and New Memory Kernels
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Fractional Differential Operators with Classical and New Memory Kernels

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

Deadline for manuscript submissions: 30 April 2024 | Viewed by 4871

Special Issue Editors

Dr. Dimiter Prodanov

E-Mail Website
Guest Editor
1. EHS and NERF, Interuniversity Microelectronics Center (Imec), 3001 Leuven, Belgium
2. IICT, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Interests: fractional calculus; local fractional calculus; computer algebra tools; numerical techniques; special functions; modeling of biophysical phenomena; image processing
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Jordan Hristov

E-Mail Website1 Website2
Guest Editor
Department of Chemical Engineering, University of Chemical Technology and Metallurgy, 1756 Sofia, Bulgaria
Interests: mathematical modeling; fractional calculus; non-linear diffusion; viscoelasticity
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus has a rich history in the modelling of nonlinear problems in physics and engineering. Formally, the apparatus of fractional calculus includes a variety of fractional-order differintegral operators, such as the ones named after Riemann, Liouville, Weyl, Caputo, Riesz, Erdelyi, Kober, etc., which give rise to a variety of special functions. Beyond this, some new trends in modelling involve integral operators with nonsingular kernels, as well as operators defined on fractal sets. These were proposed to model dissipative phenomena that cannot be adequately modelled by classical operators. This Special Issue addresses contemporary modeling problems in science and engineering involving fractional differential operators with classical and new memory kernels. This is a call to authors involved in modeling with new and classical fractional differential operators to share their results in fractional modelling theory and applications. We will cover a broad range of applied topics and multidisciplinary applications of fractional-order differential operators with classical and new kernels in science and engineering.

Dr. Dimiter Prodanov
Prof. Dr. Jordan Hristov
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

  • fractional operators
  • memory kernels
  • biomechanical and medical models
  • analysis, special functions and kernels
  • numerical and computational methods
  • analytical solution methods: exact and approximate
  • modeling approaches with nonlocal (fractional) operators
  • probability and statistics based on non-local approaches
  • mathematical physics: heat, mass and momentum transfer
  • engineering applications and image processing
  • life science, biophysics and complexity

Published Papers (5 papers)

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Research

24 pages, 652 KiB  
Article
On Theoretical and Numerical Results of Serum Hepatitis Disease Using Piecewise Fractal–Fractional Perspectives
by Zareen A. Khan, Arshad Ali, Ateeq Ur Rehman Irshad, Burhanettin Ozdemir and Hussam Alrabaiah
Fractal Fract. 2024, 8(5), 260; https://doi.org/10.3390/fractalfract8050260 - 26 Apr 2024
Viewed by 168
Abstract
In the present research, we consider a biological model of serum hepatitis disease. We carry out a detailed analysis of the mentioned model along with a class with asymptomatic carriers to explore its theoretical and numerical aspects. We initiate the study by using [...] Read more.
In the present research, we consider a biological model of serum hepatitis disease. We carry out a detailed analysis of the mentioned model along with a class with asymptomatic carriers to explore its theoretical and numerical aspects. We initiate the study by using the piecewise fractal–fractional derivative (FFD) by which the crossover effects within the model are examined. We split the time interval into subintervals. In one subinterval, FFD with a power law kernel is taken, while in the second one, FFD with an exponential decay kernel of the proposed model is considered. This model is then studied for its disease-free equilibrium point, existence, and Hyers–Ulam (H-U) stability results. For numerical results of the model and a visual presentation, we apply the Lagrange interpolation method and an extended Adams–Bashforth–Moulton (ABM) method, respectively. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
14 pages, 1510 KiB  
Article
Convolution Kernel Function and Its Invariance Properties of Bone Fractal Operators
by Zhimo Jian, Gang Peng, Chaoqian Luo, Tianyi Zhou and Yajun Yin
Fractal Fract. 2024, 8(3), 151; https://doi.org/10.3390/fractalfract8030151 - 06 Mar 2024
Viewed by 817
Abstract
This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing [...] Read more.
This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: p+α2-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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21 pages, 662 KiB  
Article
Finite Representations of the Wright Function
by Dimiter Prodanov
Fractal Fract. 2024, 8(2), 88; https://doi.org/10.3390/fractalfract8020088 - 29 Jan 2024
Viewed by 881
Abstract
The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function [...] Read more.
The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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20 pages, 954 KiB  
Article
Numerical Solution of Advection–Diffusion Equation of Fractional Order Using Chebyshev Collocation Method
by Farman Ali Shah, Kamran, Wadii Boulila, Anis Koubaa and Nabil Mlaiki
Fractal Fract. 2023, 7(10), 762; https://doi.org/10.3390/fractalfract7100762 - 17 Oct 2023
Cited by 1 | Viewed by 1503
Abstract
This work presents a highly accurate method for the numerical solution of the advection–diffusion equation of fractional order. In our proposed method, we apply the Laplace transform to handle the time-fractional derivative and utilize the Chebyshev spectral collocation method for spatial discretization. The [...] Read more.
This work presents a highly accurate method for the numerical solution of the advection–diffusion equation of fractional order. In our proposed method, we apply the Laplace transform to handle the time-fractional derivative and utilize the Chebyshev spectral collocation method for spatial discretization. The primary motivation for using the Laplace transform is its ability to avoid the classical time-stepping scheme and overcome the adverse effects of time steps on numerical accuracy and stability. Our method comprises three primary steps: (i) reducing the time-dependent equation to a time-independent equation via the Laplace transform, (ii) employing the Chebyshev spectral collocation method to approximate the solution of the transformed equation, and (iii) numerically inverting the Laplace transform. We discuss the convergence and stability of the method and assess its accuracy and efficiency by solving various problems in two dimensions. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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12 pages, 718 KiB  
Article
The Analytical Solutions to the Fractional Kraenkel–Manna–Merle System in Ferromagnetic Materials
by Mohammad Alshammari, Amjad E. Hamza, Clemente Cesarano, Elkhateeb S. Aly and Wael W. Mohammed
Fractal Fract. 2023, 7(7), 523; https://doi.org/10.3390/fractalfract7070523 - 01 Jul 2023
Cited by 5 | Viewed by 752
Abstract
In this article, we examine the Kraenkel–Manna–Merle system (KMMS) with an M-truncated derivative (MTD). Our goal is to obtain rational, hyperbolic, and trigonometric solutions by using the F-expansion technique with the Riccati equation. To our knowledge, no one has studied the exact [...] Read more.
In this article, we examine the Kraenkel–Manna–Merle system (KMMS) with an M-truncated derivative (MTD). Our goal is to obtain rational, hyperbolic, and trigonometric solutions by using the F-expansion technique with the Riccati equation. To our knowledge, no one has studied the exact solutions to the KMMS in the presence/absence of a damping effect with an M-truncated derivative, using the F-expansion technique. The magnetic field propagation in a zero-conductivity ferromagnet is described by the KMMS; hence, solutions to this equation may provide light on several fascinating scientific phenomena. We use MATLAB to display figures in a variety of 3D and 2D formats to demonstrate the influence of the M-truncated derivative on the exact solutions to the KMMS. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

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