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Fractal Fract
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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 2025 | Viewed by 12326

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

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Published Papers (11 papers)

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Research

17 pages, 340 KiB  
Article
Some Results of R-Matrix Functions and Their Fractional Calculus
by Mohra Zayed and Ahmed Bakhet
Fractal Fract. 2025, 9(2), 82; https://doi.org/10.3390/fractalfract9020082 - 25 Jan 2025
Viewed by 284
Abstract
In this study, we explore various fractional integral properties of R-matrix functions using the Hilfer fractional derivative operator within the framework of fractional calculus. We introduce the θ integral operator and extend its definition to include the R matrix functions. The composition [...] Read more.
In this study, we explore various fractional integral properties of R-matrix functions using the Hilfer fractional derivative operator within the framework of fractional calculus. We introduce the θ integral operator and extend its definition to include the R matrix functions. The composition of Riemann–Liouville fractional integral and differential operators is determined using the θ-integral operator. Additionally, we investigate the compositional properties of θ-integral operators, and we establish their inversion, offering new insights into their structural and functional characteristics. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
25 pages, 477 KiB  
Article
Topology of Locally and Non-Locally Generalized Derivatives
by Dimiter Prodanov
Fractal Fract. 2025, 9(1), 53; https://doi.org/10.3390/fractalfract9010053 - 20 Jan 2025
Viewed by 460
Abstract
This article investigates the continuity of derivatives of real-valued functions from a topological perspective. This is achieved by the characterization of their sets of discontinuity. The same principle is applied to Gateaux derivatives and gradients in Euclidean spaces. This article also introduces a [...] Read more.
This article investigates the continuity of derivatives of real-valued functions from a topological perspective. This is achieved by the characterization of their sets of discontinuity. The same principle is applied to Gateaux derivatives and gradients in Euclidean spaces. This article also introduces a generalization of the derivatives from the perspective of the modulus of continuity and characterizes their sets of discontinuities. There is a need for such generalizations when dealing with physical phenomena, such as fractures, shock waves, turbulence, Brownian motion, etc. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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21 pages, 1282 KiB  
Article
Computational Study of a Fractional-Order HIV Epidemic Model with Latent Phase and Treatment
by Sana Abdulkream Alharbi and Nada A. Almuallem
Fractal Fract. 2025, 9(1), 28; https://doi.org/10.3390/fractalfract9010028 - 7 Jan 2025
Viewed by 490
Abstract
In this work, we propose and investigate a model of the dynamical behavior of HIV/AIDS transmission by considering a new compartment of the population with HIV: the latent asymptomatic class. The infection reproduction number that stabilizes the global dynamics of the model is [...] Read more.
In this work, we propose and investigate a model of the dynamical behavior of HIV/AIDS transmission by considering a new compartment of the population with HIV: the latent asymptomatic class. The infection reproduction number that stabilizes the global dynamics of the model is evaluated. We analyze the model’s global asymptotic stability using the Lyapunov function and LaSalle’s invariance principle. To identify the primary factors affecting the dynamics of HIV/AIDS, a sensitivity analysis of the model parameters is conducted. We also examine a fractional-order HIV model using the Caputo fractional differential operator. Through qualitative analysis and applications, we determine the existence and uniqueness of the model’s solutions. We derive some results from the fixed-point theorem and Ulam–Hyers stability. Ultimately, the obtained numerical simulation results are in agreement with the analytical outcomes obtained from the model analysis. Our findings illustrate the efficacy of the fractional model in depicting the dynamics of the HIV/AIDS epidemic and offering critical insights for the formulation of effective control strategies. The results show that early intervention and treatment in the latent phase of infection can decrease the spread of the disease and its progression to AIDS, as well as increase the success of treatment strategies. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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21 pages, 751 KiB  
Article
Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations
by Asifa Tassaddiq, Carlo Cattani, Rabab Alharbi, Ruhaila Md Kasmani and Sania Qureshi
Fractal Fract. 2024, 8(12), 749; https://doi.org/10.3390/fractalfract8120749 - 19 Dec 2024
Viewed by 620
Abstract
The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel [...] Read more.
The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel fractional kinetic equation containing Fermi–Dirac (FD) and Bose–Einstein (BE) functions. Several distributional properties of these functions and their proposed new generalizations are investigated in this article. In fact, it is proved that these functions belong to distribution space D′ while their Fourier transforms belong to Z′. Fourier and Laplace transforms of these functions are computed by using their distributional representation. Thanks to them, we can compute various new fractional calculus formulae and a new relation involving the Fox–Wright function. Some fractional kinetic equations containing the FD and BE functions are also formulated and solved. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
17 pages, 407 KiB  
Article
On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel
by Zelin Liu, Xiaobin Yu and Yajun Yin
Fractal Fract. 2024, 8(11), 653; https://doi.org/10.3390/fractalfract8110653 - 11 Nov 2024
Viewed by 901
Abstract
Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional [...] Read more.
Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the α-th power of a differential operator corresponds precisely to the α-th fractional derivative, ensuring both accuracy and consistency. The relationship between the fractional power operators and fractional calculus is not arbitrary, it must be determined by the specific operator form and the initial conditions. Furthermore, we provide operator representations of commonly used fractional derivatives and illustrate their applications with examples of fractional power operators in physical fractal spaces. A superposition principle is also introduced to simplify fractional differential equations with non-integer exponents by transforming them into zero-initial-condition problems. This framework offers new insights into the commutative properties of fractional calculus operators and their relevance in the study of fractal structures. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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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
Cited by 1 | Viewed by 834
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
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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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 998
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)
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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
Cited by 2 | Viewed by 1367
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 1450
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 8 | Viewed by 2234
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 - 1 Jul 2023
Cited by 16 | Viewed by 1175
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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