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: 31 October 2024 | Viewed by 5852

Special Issue Editors


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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
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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

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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 (6 papers)

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Research

17 pages, 1266 KiB  
Article
Synchronization of Fractional Delayed Memristive Neural Networks with Jump Mismatches via Event-Based Hybrid Impulsive Controller
by Huiyu Wang, Shutang Liu, Xiang Wu, Jie Sun and Wei Qiao
Fractal Fract. 2024, 8(5), 297; https://doi.org/10.3390/fractalfract8050297 - 18 May 2024
Viewed by 289
Abstract
This study investigates the asymptotic synchronization in fractional memristive neural networks of the Riemann–Liouville type, considering mixed time delays and jump mismatches. Addressing the challenges associated with discrepancies in the circuit switching speed and the accuracy of the memristor, this paper introduces an [...] Read more.
This study investigates the asymptotic synchronization in fractional memristive neural networks of the Riemann–Liouville type, considering mixed time delays and jump mismatches. Addressing the challenges associated with discrepancies in the circuit switching speed and the accuracy of the memristor, this paper introduces an enhanced model that effectively navigates these complexities. We propose two novel event-based hybrid impulsive controllers, each characterized by unique triggering conditions. Utilizing advanced techniques in inequality and hybrid impulsive control, we establish the conditions necessary for achieving synchronization through innovative Lyapunov functions. Importantly, the developed controllers are theoretically optimized to minimize control costs, an essential consideration for their practical deployment. Finally, the effectiveness of our proposed approach is demonstrated through two illustrative simulation examples. Full article
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24 pages, 656 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 530
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
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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 - 6 Mar 2024
Viewed by 881
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
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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 944
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
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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 1588
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
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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 - 1 Jul 2023
Cited by 7 | Viewed by 813
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
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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.

saleemabdullah@awkum.edu.pk; Title: Fractional Aggregation Operators ...

ghlmfarid@cuiatk.edu.pk

faical@ua.pt

tadesse.abdi@aau.edu.et

jzm22@mails.tsinghua.edu.cn

deepak.sarve@mathematics.mu.ac.in

maleraxa@gmail.com

zmalek@kau.edu.sa

 

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