Advances in Fractional Integral and Derivative Operators with Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 16534

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Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: mathematical analysis; applied mathematics; fractional calculus and its applications
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Special Issue Information

Dear Colleagues,

The usefulness of operational techniques, which are based upon a large variety of operators of fractional integrals and fractional derivatives such as those that are named after Riemann–Liouville, Weyl, Hadamard, Grunwald–Letnikov, Riesz, Erdélyi–Kober, Liouville–Caputo, and so on, in a number of diverse areas including mathematical, physical, chemical, engineering, and statistical sciences cannot be overemphasized. Many of these and other fractional-order operators provide interesting and potentially useful tools and techniques for solving ordinary and partial differential equations, as well as integral, differ-integral, and integro-differential equations, the fractional-calculus analogues and extensions of each of these equations, and various other problems involving higher transcendental functions including special functions of mathematical physics, applicable analysis and applied mathematics, and their univariate and multivariate extensions and generalizations.

In this Special Issue, we cordially invite and welcome review, expository, topical survey, and original research articles which aim at presenting the recent advances in the theory of fractional-order operators of integrals and derivatives and their multidisciplinary applications, especially in the fractional-order modelling and analysis of applied problems in diverse areas of the mathematical, physical, chemical, engineering, and statistical sciences.

Prof. Dr. Rekha Srivastava
Guest Editor

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Keywords

  • theoretical advances involving operators of fractional integrals and fractional derivatives
  • applications involving chaos and dynamical systems based upon fractional calculus
  • applications based upon fractional-order ordinary and partial differential equations
  • fractional-order differ-integral and integro-differential equations
  • fractional-order integrals and derivatives associated with higher transcendental functions including special functions of mathematical physics and applied mathematics
  • inequalities and identities involving integrals and derivatives of a fractional order

Published Papers (15 papers)

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Research

17 pages, 5398 KiB  
Article
High-Frequency Fractional Predictions and Spatial Distribution of the Magnetic Loss in a Grain-Oriented Magnetic Steel Lamination
by Benjamin Ducharne, Hamed Hamzehbahmani, Yanhui Gao, Patrick Fagan and Gael Sebald
Fractal Fract. 2024, 8(3), 176; https://doi.org/10.3390/fractalfract8030176 - 19 Mar 2024
Viewed by 916
Abstract
Grain-oriented silicon steel (GO FeSi) laminations are vital components for efficient energy conversion in electromagnetic devices. While traditionally optimized for power frequencies of 50/60 Hz, the pursuit of higher frequency operation (f ≥ 200 Hz) promises enhanced power density. This paper introduces [...] Read more.
Grain-oriented silicon steel (GO FeSi) laminations are vital components for efficient energy conversion in electromagnetic devices. While traditionally optimized for power frequencies of 50/60 Hz, the pursuit of higher frequency operation (f ≥ 200 Hz) promises enhanced power density. This paper introduces a model for estimating GO FeSi laminations’ magnetic behavior under these elevated operational frequencies. The proposed model combines the Maxwell diffusion equation and a material law derived from a fractional differential equation, capturing the viscoelastic characteristics of the magnetization process. Remarkably, the model’s dynamical contribution, characterized by only two parameters, achieves a notable 4.8% Euclidean relative distance error across the frequency spectrum from 50 Hz to 1 kHz. The paper’s initial section offers an exhaustive description of the model, featuring comprehensive comparisons between simulated and measured data. Subsequently, a methodology is presented for the localized segregation of magnetic losses into three conventional categories: hysteresis, classical, and excess, delineated across various tested frequencies. Further leveraging the model’s predictive capabilities, the study extends to investigating the very high-frequency regime, elucidating the spatial distribution of loss contributions. The application of proportional–iterative learning control facilitates the model’s adaptation to standard characterization conditions, employing sinusoidal imposed flux density. The paper deliberates on the implications of GO FeSi behavior under extreme operational conditions, offering insights and reflections essential for understanding and optimizing magnetic core performance in high-frequency applications. Full article
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10 pages, 1773 KiB  
Article
The El Niño Southern Oscillation Recharge Oscillator with the Stochastic Forcing of Long-Term Memory
by Xiaofeng Li and Yaokun Li
Fractal Fract. 2024, 8(2), 121; https://doi.org/10.3390/fractalfract8020121 - 18 Feb 2024
Viewed by 906
Abstract
The influence of the fast-varying variables that have a long-term memory on the El Niño Southern Oscillation (ENSO) is investigated by adding a fractional Ornstein–Uhlenbeck (FOU) process stochastic noise on the simple recharge oscillator (RO) model. The FOU process noise converges to zero [...] Read more.
The influence of the fast-varying variables that have a long-term memory on the El Niño Southern Oscillation (ENSO) is investigated by adding a fractional Ornstein–Uhlenbeck (FOU) process stochastic noise on the simple recharge oscillator (RO) model. The FOU process noise converges to zero very slowly with a negative power law. The corresponding non-zero ensemble mean during the integration period can exert a pronounced influence on the ensemble-mean dynamics of the RO model. The state-dependent noise, also called the multiplicative noise, can present its influence by reducing the relaxation coefficient and by introducing periodic external forcing. The decreasing relaxation coefficient can enhance the oscillation amplitude and shorten the oscillation period. The forced frequency is close to the natural frequency. The two mechanisms together can further amplify the amplitude and shorten the period, compared with the state-independent noise or additive noise, which only exhibits its influence by introducing non-periodic external forcing. These two mechanisms explicitly elucidate the influence of the stochastic forcing on the ensemble-mean dynamics of the RO model. It provides comprehensive knowledge to better understand the interaction between the fast-varying stochastic forcing and the slow-varying deterministic system and deserves further investigation. Full article
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24 pages, 12369 KiB  
Article
Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration
by Rekha Srivastava, Asifa Tassaddiq and Ruhaila Md Kasmani
Fractal Fract. 2024, 8(2), 116; https://doi.org/10.3390/fractalfract8020116 - 15 Feb 2024
Cited by 1 | Viewed by 1047
Abstract
Fractals are a common characteristic of many artificial and natural networks having topological patterns of a self-similar nature. For example, the Mandelbrot set has been investigated and extended in several ways since it was first introduced, whereas some authors characterized it using various [...] Read more.
Fractals are a common characteristic of many artificial and natural networks having topological patterns of a self-similar nature. For example, the Mandelbrot set has been investigated and extended in several ways since it was first introduced, whereas some authors characterized it using various complex functions or polynomials, others generalized it using iterations from fixed-point theory. In this paper, we generate Mandelbrot sets using the hybrid Picard S-iterations. Therefore, an escape criterion involving complex functions is proved and used to provide numerical and graphical examples. We produce a wide range of intriguing fractal patterns with the suggested method, and we compare our findings with the classical S-iteration. It became evident that the newly proposed iteration method produces novel images that are more spontaneous and fascinating than those produced by the S-iteration. Therefore, the generated sets behave differently based on the parameters involved in different iteration schemes. Full article
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14 pages, 303 KiB  
Article
(ω,c)-Periodic Solution to Semilinear Integro-Differential Equations with Hadamard Derivatives
by Ahmad Al-Omari, Hanan Al-Saadi and Fawaz Alharbi
Fractal Fract. 2024, 8(2), 86; https://doi.org/10.3390/fractalfract8020086 - 28 Jan 2024
Viewed by 975
Abstract
This study aims to prove the existence and uniqueness of the (ω,c)-periodic solution as a specific solution to Hadamard impulsive boundary value integro-differential equations with fixed lower limits. The results are proven using the Banach contraction, Schaefer’s fixed [...] Read more.
This study aims to prove the existence and uniqueness of the (ω,c)-periodic solution as a specific solution to Hadamard impulsive boundary value integro-differential equations with fixed lower limits. The results are proven using the Banach contraction, Schaefer’s fixed point theorem, and the Arzelà–Ascoli theorem. Furthermore, we establish the necessary conditions for a set of solutions to the explored boundary values with impulsive fractional differentials. Finally, we present two examples as applications for our results. Full article
22 pages, 360 KiB  
Article
Boundary Value Problem for a Coupled System of Nonlinear Fractional q-Difference Equations with Caputo Fractional Derivatives
by Saleh S. Redhwan, Maoan Han, Mohammed A. Almalahi, Mona Alsulami and Maryam Ahmed Alyami
Fractal Fract. 2024, 8(1), 73; https://doi.org/10.3390/fractalfract8010073 - 22 Jan 2024
Cited by 2 | Viewed by 1219
Abstract
This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic [...] Read more.
This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic friction. To investigate this system, we introduce an operator that exhibits fixed points corresponding to the solutions of the problem, effectively transforming the system into an equivalent fixed-point problem. We established the necessary conditions for the existence and uniqueness of solutions using the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle, respectively. Stability results in the Ulam sense for the coupled system are also discussed, along with a sensitivity analysis of the range parameters. To support the validity of their findings, we provide illustrative examples. Overall, the paper offers a thorough examination and analysis of the considered coupled system, making important contributions to the understanding of multi-atomic systems and their mathematical modeling. Full article
20 pages, 382 KiB  
Article
New Applications of Fractional q-Calculus Operator for a New Subclass of q-Starlike Functions Related with the Cardioid Domain
by Mohammad Faisal Khan and Mohammed AbaOud
Fractal Fract. 2024, 8(1), 71; https://doi.org/10.3390/fractalfract8010071 - 22 Jan 2024
Cited by 1 | Viewed by 1106
Abstract
Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical [...] Read more.
Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical biology, plasma physics, electromagnetic theory, and many more. This article explores some mathematical applications of the fractional q-differential and integral operator in the field of geometric function theory. By using the linear multiplier fractional q-differintegral operator Dq,λmρ,σ and subordination, we define and develop a collection of q-starlike functions that are linked to the cardioid domain. This study also investigates sharp inequality problems like initial coefficient bounds, the Fekete–Szego problems, and the coefficient inequalities for a new class of q-starlike functions in the open unit disc U. Furthermore, we analyze novel findings with respect to the inverse function (μ1) within the class of q-starlike functions in U. The findings in this paper are easy to understand and show a connection between present and past studies. Full article
28 pages, 412 KiB  
Article
Positive Solutions for a System of Fractional q-Difference Equations with Multi-Point Boundary Conditions
by Rodica Luca
Fractal Fract. 2024, 8(1), 70; https://doi.org/10.3390/fractalfract8010070 - 21 Jan 2024
Cited by 1 | Viewed by 906
Abstract
We explore the existence, uniqueness, and multiplicity of positive solutions to a system of fractional q-difference equations that include fractional q-integrals. This investigation is carried out under coupled multi-point boundary conditions featuring q-derivatives and fractional q-derivatives of various orders. [...] Read more.
We explore the existence, uniqueness, and multiplicity of positive solutions to a system of fractional q-difference equations that include fractional q-integrals. This investigation is carried out under coupled multi-point boundary conditions featuring q-derivatives and fractional q-derivatives of various orders. The proofs of our principal findings employ a range of fixed-point theorems, including the Guo–Krasnosel’skii fixed-point theorem, the Leggett–Williams fixed-point theorem, the Schauder fixed-point theorem, and the Banach contraction mapping principle. Full article
13 pages, 7880 KiB  
Article
Mandelbrot Set as a Particular Julia Set of Fractional Order, Equipotential Lines and External Rays of Mandelbrot and Julia Sets of Fractional Order
by Marius-F. Danca
Fractal Fract. 2024, 8(1), 69; https://doi.org/10.3390/fractalfract8010069 - 19 Jan 2024
Viewed by 1544
Abstract
This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the [...] Read more.
This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the differences between the fractional-order Mandelbrot set and Julia sets from their integer-order variants are revealed. Equipotential lines and external rays of a Mandelbrot set and Julia sets of fractional order are determined. Full article
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25 pages, 4345 KiB  
Article
A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications
by Seyeon Lee, Hyunju Kim and Bongsoo Jang
Fractal Fract. 2024, 8(1), 65; https://doi.org/10.3390/fractalfract8010065 - 17 Jan 2024
Viewed by 1399
Abstract
In this article, a considerably efficient predictor-corrector method (PCM) for solving Atangana–Baleanu Caputo (ABC) fractional differential equations (FDEs) is introduced. First, we propose a conventional PCM whose computational speed scales with quadratic time complexity O(N2) as the number of [...] Read more.
In this article, a considerably efficient predictor-corrector method (PCM) for solving Atangana–Baleanu Caputo (ABC) fractional differential equations (FDEs) is introduced. First, we propose a conventional PCM whose computational speed scales with quadratic time complexity O(N2) as the number of time steps N grows. A fast algorithm to reduce the computational complexity of the memory term is investigated utilizing a sum-of-exponentials (SOEs) approximation. The conventional PCM is equipped with a fast algorithm, and it only requires linear time complexity O(N). Truncation and global error analyses are provided, achieving a uniform accuracy order O(h2) regardless of the fractional order for both the conventional and fast PCMs. We demonstrate numerical examples for nonlinear initial value problems and linear and nonlinear reaction-diffusion fractional-order partial differential equations (FPDEs) to numerically verify the efficiency and error estimates. Finally, the fast PCM is applied to the fractional-order Rössler dynamical system, and the numerical results prove that the computational cost consumed to obtain the bifurcation diagram is significantly reduced using the proposed fast algorithm. Full article
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15 pages, 320 KiB  
Article
Solutions for Hilfer-Type Linear Fractional Integro-Differential Equations with a Variable Coefficient
by Sigang Zhu, Huiwen Wang and Fang Li
Fractal Fract. 2024, 8(1), 63; https://doi.org/10.3390/fractalfract8010063 - 17 Jan 2024
Viewed by 943
Abstract
In this paper, we derive an explicit formula of solutions to Hilfer linear fractional integro-differential equations with a variable coefficient in a weighted space, and obtain the existence and uniqueness of solutions for fractional kinetic equations and fractional integro-differential equations with a generalized [...] Read more.
In this paper, we derive an explicit formula of solutions to Hilfer linear fractional integro-differential equations with a variable coefficient in a weighted space, and obtain the existence and uniqueness of solutions for fractional kinetic equations and fractional integro-differential equations with a generalized Mittag–Leffler function. An example is given to illustrate the result obtained. Full article
17 pages, 3347 KiB  
Article
Fractional Diffusion Equation under Singular and Non-Singular Kernel and Its Stability
by Enrique C. Gabrick, Paulo R. Protachevicz, Ervin K. Lenzi, Elaheh Sayari, José Trobia, Marcelo K. Lenzi, Fernando S. Borges, Iberê L. Caldas and Antonio M. Batista
Fractal Fract. 2023, 7(11), 792; https://doi.org/10.3390/fractalfract7110792 - 30 Oct 2023
Cited by 2 | Viewed by 1047
Abstract
The fractional reaction–diffusion equation has been used in many real-world applications in fields such as physics, biology, and chemistry. Motivated by the huge application of fractional reaction–diffusion, we propose a numerical scheme to solve the fractional reaction–diffusion equation under different kernels. Our method [...] Read more.
The fractional reaction–diffusion equation has been used in many real-world applications in fields such as physics, biology, and chemistry. Motivated by the huge application of fractional reaction–diffusion, we propose a numerical scheme to solve the fractional reaction–diffusion equation under different kernels. Our method can be particularly employed for singular and non-singular kernels, such as the Riemann–Liouville, Caputo, Fabrizio–Caputo, and Atangana–Baleanu operators. Moreover, we obtained general inequalities that guarantee that the stability condition depends explicitly on the kernel. As an implementation of the method, we numerically solved the diffusion equation under the power-law and exponential kernels. For the power-law kernel, we solved by considering fractional time, space, and both operators. In another example, we considered the exponential kernel acting on the time derivative and compared the numerical results with the analytical ones. Our results showed that the numerical procedure developed in this work can be employed to solve fractional differential equations considering different kernels. Full article
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11 pages, 302 KiB  
Article
Positive Solutions for Some Semipositone Fractional Boundary Value Problems on the Half-Line
by Imed Bachar
Fractal Fract. 2023, 7(11), 774; https://doi.org/10.3390/fractalfract7110774 - 24 Oct 2023
Viewed by 711
Abstract
Our goal is to address the question of existence and uniqueness of a positive continuous solution to some semipositone fractional boundary value problems on the half-line. Global estimates on this solution are given. This kind of problems, where the nonlinearity is allowed to [...] Read more.
Our goal is to address the question of existence and uniqueness of a positive continuous solution to some semipositone fractional boundary value problems on the half-line. Global estimates on this solution are given. This kind of problems, where the nonlinearity is allowed to be sign-changing, are often difficult to solve analytically and becomes more challenging specially when we are looking for positive solutions. The main result is obtained by means of the properties of the Green function and fixed point theorem. Full article
18 pages, 484 KiB  
Article
Operator Kernel Functions in Operational Calculus and Applications in Fractals with Fractional Operators
by Xiaobin Yu and Yajun Yin
Fractal Fract. 2023, 7(10), 755; https://doi.org/10.3390/fractalfract7100755 - 13 Oct 2023
Cited by 1 | Viewed by 940
Abstract
In this study, we delve into the general theory of operator kernel functions (OKFs) in operational calculus (OC). We established the rigorous mapping relation between the kernel function and the corresponding operator through the primary translation operator ept, which [...] Read more.
In this study, we delve into the general theory of operator kernel functions (OKFs) in operational calculus (OC). We established the rigorous mapping relation between the kernel function and the corresponding operator through the primary translation operator ept, which bears a striking resemblance to the Laplace transform. Our research demonstrates the uniqueness of the kernel function, determined by the rules of operational calculus and its integral representation. This discovery provides a novel perspective on how the operational calculus can be understood and applied, particularly through convolution with kernel functions. We substantiate the accuracy of the proposed method by demonstrating the consistency between the operator solution and the classical solution for the heat conduction problem. Subsequently, on the fractal tree, fractal loop, and fractal ladder structures, we illustrate the application of operational calculus in viscoelastic constitutive and hemodynamics confirming that the method proposed unifies the OKFs in the existing OC theory and can be extended to the operator field. These results underscore the practical significance of our results and open up new possibilities for future research. Full article
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15 pages, 512 KiB  
Article
Numerical Solutions of the Multi-Space Fractional-Order Coupled Korteweg–De Vries Equation with Several Different Kernels
by Khaled Mohammed Saad and Hari Mohan Srivastava
Fractal Fract. 2023, 7(10), 716; https://doi.org/10.3390/fractalfract7100716 - 29 Sep 2023
Cited by 10 | Viewed by 837
Abstract
In this article, the authors propose to investigate the numerical solutions of several fractional-order models of the multi-space coupled Korteweg–De Vries equation involving many different kernels. In order to transform these models into a set or system of differential equations, various properties of [...] Read more.
In this article, the authors propose to investigate the numerical solutions of several fractional-order models of the multi-space coupled Korteweg–De Vries equation involving many different kernels. In order to transform these models into a set or system of differential equations, various properties of the first-kind Chebyshev polynomial are used in this study. The main objective of the present study is to apply the spectral collocation approach for the multi-space fractional-order coupled Korteweg–De Vries equation with different kernels. We use finite differences to numerically solve these differential equations by reducing them to algebraic equations. The Newton (or, more precisely, the Newton–Raphson) method is then used to solve these resulting algebraic equations. By calculating the error involved in our approach, the precision of the numerical solution is verified. The use of spectral methods, which provide excellent accuracy and exponential convergence for issues with smooth solutions, is shown to be a benefit of the current study. Full article
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14 pages, 967 KiB  
Article
A Fractional (q,q) Non-Extensive Information Dimension for Complex Networks
by Aldo Ramirez-Arellano, Jazmin-Susana De-la-Cruz-Garcia and Juan Bory-Reyes
Fractal Fract. 2023, 7(10), 702; https://doi.org/10.3390/fractalfract7100702 - 24 Sep 2023
Viewed by 783
Abstract
This article introduces a new fractional approach to the concept of information dimensions in complex networks based on the (q,q)-entropy proposed in the literature. The q parameter measures how far the number of sub-systems (for a given size [...] Read more.
This article introduces a new fractional approach to the concept of information dimensions in complex networks based on the (q,q)-entropy proposed in the literature. The q parameter measures how far the number of sub-systems (for a given size ε) is from the mean number of overall sizes, whereas q (the interaction index) measures when the interactions between sub-systems are greater (q>1), lesser (q<1), or equal to the interactions into these sub-systems. Computation of the proposed information dimension is carried out on several real-world and synthetic complex networks. The results for the proposed information dimension are compared with those from the classic information dimension based on Shannon entropy. The obtained results support the conjecture that the fractional (q,q)-information dimension captures the complexity of the topology of the network better than the information dimension. Full article
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