Fractional Calculus: Theory and Applications, 2nd Edition

Topic Information

Dear Colleagues,

The fractional calculus (FC) generalizes the operations of differentiation and integration to non-integer orders. FC emerged as an important tool for studying dynamical systems since fractional order operators are non-local and capture the history of dynamics. Moreover, FC and fractional processes have become one of the most useful approaches to deal with particular properties of (long) memory effects in various applied sciences. Linear, nonlinear, and complex dynamical systems have attracted researchers from many areas of science and technology involved in systems modeling and control, with applications to real-world problems. Despite the extraordinary advances in FC, addressing both systems’ modeling and control, new theoretical developments and applications are still needed in order to describe or control accurately many systems and signals characterized by chaos, bifurcations, criticality, symmetry, memory, scale invariance, fractality, fractionality, and other rich features.

This Topic focuses on original and new research results on fractional calculus in science and engineering. Manuscripts addressing novel theoretical issues, as well as those on more specific applications, are welcome. Topics of interest include (but are not limited to) fractional calculus theory, methods for fractional differential and integral equations, nonlinear dynamical systems, advanced control systems, fractals and chaos, complex dynamics, evolutionary computing, finance and economy dynamics, biological systems and bioinformatics, nonlinear waves and acoustics, image and signal processing, transportation systems, geosciences, astronomy and cosmology, nuclear physics, fractional modeling in econophysics, and fractional modeling for time series.

Dr. António Lopes
Prof. Dr. Liping Chen
Prof. Dr. Sergio Adriani David
Prof. Dr. Alireza Alfi
Topic Editors

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Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Axioms axioms	1.9	-	2012	22.8 Days	CHF 2400	Submit
Computation computation	1.9	3.5	2013	18.6 Days	CHF 1800	Submit
Fractal and Fractional fractalfract	3.6	4.6	2017	23.7 Days	CHF 2700	Submit
Mathematics mathematics	2.3	4.0	2013	18.3 Days	CHF 2600	Submit
Symmetry symmetry	2.2	5.4	2009	17.3 Days	CHF 2400	Submit

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All Axioms Computation Fractal Fract Mathematics Symmetry

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23 pages, 1276 KiB  

Open AccessArticle

Fractional and Higher Integer-Order Moments for Fractional Stochastic Differential Equations

by Arsalane Chouaib Guidoum, Fatimah A. Almulhim, Mohammed Bassoudi, Kamal Boukhetala and Mohammed B. Alamari

Symmetry 2025, 17(5), 665; https://doi.org/10.3390/sym17050665 (registering DOI) - 27 Apr 2025

Viewed by 61

Abstract

This study investigates the computation of fractional and higher integer-order moments for a stochastic process governed by a one-dimensional, non-homogeneous linear stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). Unlike conventional approaches relying on moment-generating functions or Fokker–Planck equations, which often [...] Read more.

This study investigates the computation of fractional and higher integer-order moments for a stochastic process governed by a one-dimensional, non-homogeneous linear stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). Unlike conventional approaches relying on moment-generating functions or Fokker–Planck equations, which often yield intractable expressions, we derive explicit closed-form formulas for these moments. Our methodology leverages the Wick–Itô calculus (fractional Itô formula) and the properties of Hermite polynomials to express moments efficiently. Additionally, we establish a recurrence relation for moment computation and propose an alternative approach based on generalized binomial expansions. To validate our findings, Monte Carlo simulations are performed, demonstrating a high degree of accuracy between theoretical and empirical results. The proposed framework provides novel insights into stochastic processes with long-memory properties, with potential applications in statistical inference, mathematical finance, and physical modeling of anomalous diffusion. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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16 pages, 1576 KiB  

Open AccessArticle

Comparison Principle Based Synchronization Analysis of Fractional-Order Chaotic Neural Networks with Multi-Order and Its Circuit Implementation

by Rongbo Zhang, Kun Qiu, Chuang Liu, Hongli Ma and Zhaobi Chu

Fractal Fract. 2025, 9(5), 273; https://doi.org/10.3390/fractalfract9050273 - 23 Apr 2025

Viewed by 87

Abstract

This article investigates non-fragile synchronization control and circuit implementation for incommensurate fractional-order (IFO) chaotic neural networks with parameter uncertainties. In this paper, we explore three aspects of the research challenges, i.e., theoretical limitations of uncertain IFO systems, the fragility of the synchronization controller, [...] Read more.

This article investigates non-fragile synchronization control and circuit implementation for incommensurate fractional-order (IFO) chaotic neural networks with parameter uncertainties. In this paper, we explore three aspects of the research challenges, i.e., theoretical limitations of uncertain IFO systems, the fragility of the synchronization controller, and the lack of circuit implementation. First, we establish an IFO chaotic neural network model incorporating parametric uncertainties, extending beyond conventional commensurate-order architectures. Second, a novel, non-fragile state-error feedback controller is designed. Through the formulation of FO Lyapunov functions and the application of inequality scaling techniques, sufficient conditions for asymptotic synchronization of master–slave systems are rigorously derived via the multi-order fractional comparison principle. Third, an analog circuit implementation scheme utilizing FO impedance units is developed to experimentally validate synchronization efficacy and accurately replicate the system’s dynamic behavior. Numerical simulations and circuit experiments substantiate the theoretical findings, demonstrating both robustness against parameter perturbations and the feasibility of circuit realization. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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33 pages, 641 KiB  

Open AccessArticle

Computational Representation of Fractional Inequalities Through 2D and 3D Graphs with Applications

by Muhammad Younis, Ahsan Mehmood, Muhammad Samraiz, Gauhar Rahman, Salma Haque, Ahmad Aloqaily and Nabil Mlaiki

Computation 2025, 13(2), 46; https://doi.org/10.3390/computation13020046 - 7 Feb 2025

Viewed by 534

Abstract

The aim of this research article is to use the extended fractional operators involving the multivariate Mittag–Leffler (M-M-L) function, we provide the generalization of the Hermite–Hadamard–Fejer (H-H-F) inequalities. We relate these inequalities to previously published disparities in the literature by making appropriate substitutions. [...] Read more.

The aim of this research article is to use the extended fractional operators involving the multivariate Mittag–Leffler (M-M-L) function, we provide the generalization of the Hermite–Hadamard–Fejer (H-H-F) inequalities. We relate these inequalities to previously published disparities in the literature by making appropriate substitutions. In the last section, we analyze several inequalities related to the H-H-F inequalities, focusing on generalized h-convexity associated with extended fractional operators involving the M-M-L function. To achieve this, we derive two identities for locally differentiable functions, which allows us to provide specific estimates for the differences between the left, middle, and right terms in the H-H-F inequalities. Also, we have constructed specific inequalities and visualized them through graphical representations to facilitate their applications in analysis. The research bridges theoretical advancements with practical applications, providing high-accuracy bounds for complex systems involving fractional calculus. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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13 pages, 491 KiB  

Open AccessArticle

Linearly Implicit Conservative Schemes for the Nonlocal Schrödinger Equation

by Yutong Zhang, Bin Li and Mingfa Fei

Mathematics 2024, 12(21), 3339; https://doi.org/10.3390/math12213339 - 24 Oct 2024

Viewed by 855

Abstract

This paper introduces two high-accuracy linearly implicit conservative schemes for solving the nonlocal Schrödinger equation, employing the extrapolation technique. These schemes are based on the generalized scalar auxiliary variable approach and the symplectic Runge–Kutta method. By integrating these advanced methods, the proposed schemes [...] Read more.

This paper introduces two high-accuracy linearly implicit conservative schemes for solving the nonlocal Schrödinger equation, employing the extrapolation technique. These schemes are based on the generalized scalar auxiliary variable approach and the symplectic Runge–Kutta method. By integrating these advanced methods, the proposed schemes aim to significantly enhance computational accuracy and efficiency, while maintaining the essential conservative properties necessary for accurate physical modeling. This offers a structured approach to handle auxiliary variables, ensuring stability and conservation, while the symplectic Runge–Kutta method provides a robust framework with high accuracy. Together, these techniques offer a powerful and reliable approach for researchers dealing with complex quantum mechanical systems described by the nonlocal Schrödinger equation, ensuring both accuracy and stability in their numerical simulations. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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11 pages, 242 KiB  

Open AccessArticle

A Note on a Min–Max Method for a Singular Kirchhoff Problem of Fractional Type

by Ramzi Alsaedi

Mathematics 2024, 12(20), 3269; https://doi.org/10.3390/math12203269 - 18 Oct 2024

Cited by 1 | Viewed by 669

Abstract

In the present work, we study a fractional elliptic Kirchhoff-type problem that has a singular term. More precisely, we start by proving some properties related to the energy functional associated with the studied problem. Then, we use the variational method combined with the [...] Read more.

In the present work, we study a fractional elliptic Kirchhoff-type problem that has a singular term. More precisely, we start by proving some properties related to the energy functional associated with the studied problem. Then, we use the variational method combined with the min–max method to prove that the energy functional reaches its global minimum. Finally, since the energy functional has a singularity, we use the implicit function theorem to show that the point where the minimum is reached is a weak solution for the main problem. To illustrate our main result, we give an example at the end of this paper. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

15 pages, 355 KiB  

Open AccessArticle

Multiple Solutions for a Critical Steklov Kirchhoff Equation

by Maryam Ahmad Alyami and Abdeljabbar Ghanmi

Fractal Fract. 2024, 8(10), 598; https://doi.org/10.3390/fractalfract8100598 - 11 Oct 2024

Viewed by 782

Abstract

In the present work, we study some existing results related to a new class of Steklov $p\left(x\right)$-Kirchhoff problems with critical exponents. More precisely, we propose and prove some properties of the associated energy functional. In the first existence result, [...] Read more.

In the present work, we study some existing results related to a new class of Steklov $p\left(x\right)$-Kirchhoff problems with critical exponents. More precisely, we propose and prove some properties of the associated energy functional. In the first existence result, we use the mountain pass theorem to prove that the energy functional admits a critical point, which is a weak solution for such a problem. In the second main result, we use a symmetric version of the mountain pass theorem to prove that the investigated problem has an infinite number of solutions. Finally, in the third existence result, we use a critical point theorem proposed by Kajikiya to prove the existence of a sequence of solutions that tend to zero. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

11 pages, 312 KiB  

Open AccessArticle

Existence of Solutions for a Coupled Hadamard Fractional System of Integral Equations in Local Generalized Morrey Spaces

by Asra Hadadfard, Mohammad Bagher Ghaemi and António M. Lopes

Axioms 2024, 13(10), 688; https://doi.org/10.3390/axioms13100688 - 3 Oct 2024

Viewed by 956

Abstract

This paper introduces a new measure of non-compactness within a bounded domain of ${\mathbb{R}}^{\mathbb{N}}$ in the generalized Morrey space. This measure is used to establish the existence of solutions for a coupled Hadamard fractional system of integral equations in generalized Morrey spaces. [...] Read more.

This paper introduces a new measure of non-compactness within a bounded domain of ${\mathbb{R}}^{\mathbb{N}}$ in the generalized Morrey space. This measure is used to establish the existence of solutions for a coupled Hadamard fractional system of integral equations in generalized Morrey spaces. To illustrate the application of the main result, an example is presented. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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13 pages, 273 KiB  

Open AccessFeature PaperEditor’s ChoiceArticle

Some Fractional Integral and Derivative Formulas Revisited

by Juan Luis González-Santander and Francesco Mainardi

Mathematics 2024, 12(17), 2786; https://doi.org/10.3390/math12172786 - 9 Sep 2024

Cited by 2 | Viewed by 1016

Abstract

In the most common literature about fractional calculus, we find that ${}_{a}D_t^{\alpha }f\left(t\right)={}_{a}I_t^{-\alpha }f\left(t\right)$ is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the [...] Read more.

In the most common literature about fractional calculus, we find that ${}_{a}D_t^{\alpha }f\left(t\right)={}_{a}I_t^{-\alpha }f\left(t\right)$ is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of ${}_{a}I_t^{\alpha }f\left(t\right)$ and ${}_{a}D_t^{\alpha }f\left(t\right)$. In this sense, we prove that ${}_{0}D_{t}f\left(t\right)={}_{0}I_t^{-\alpha }f\left(t\right)$ is true for $f\left(t\right)=t^{\nu -1}\log t$, and $f\left(t\right)=e^{\lambda t}$, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for ${}_{-\infty }D_t^{\alpha }{\left|t\right|}^{-\delta }$ and ${}_{-\infty }I_t^{\alpha }{\left|t\right|}^{-\delta }$ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that ${}_{-\infty }D_t^{\alpha }{\left|t\right|}^{-\delta }={}_{-\infty }I_t^{-\alpha }{\left|t\right|}^{-\delta }$ holds true. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

22 pages, 344 KiB  

Open AccessArticle

Study of a Coupled Ψ–Liouville–Riemann Fractional Differential System Characterized by Mixed Boundary Conditions

by Brahim Tellab, Abdelkader Amara, Mohammed El-Hadi Mezabia, Khaled Zennir and Loay Alkhalifa

Fractal Fract. 2024, 8(9), 510; https://doi.org/10.3390/fractalfract8090510 - 29 Aug 2024

Viewed by 1159

Abstract

This research is concerned with the existence and uniqueness of solutions for a coupled system of $\Psi$–Riemann–Liouville fractional differential equations. To achieve this objective, we establish a set of necessary conditions by formulating the problem as an integral equation and utilizing well-known [...] Read more.

This research is concerned with the existence and uniqueness of solutions for a coupled system of $\Psi$–Riemann–Liouville fractional differential equations. To achieve this objective, we establish a set of necessary conditions by formulating the problem as an integral equation and utilizing well-known fixed-point theorems. By employing these mathematical tools, we demonstrate the existence and uniqueness of solutions for the proposed system. Additionally, to illustrate the practical implications of our findings, we provide several examples that showcase the main results obtained in this study. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

19 pages, 348 KiB  

Open AccessArticle

Polynomial Decay of the Energy of Solutions of the Timoshenko System with Two Boundary Fractional Dissipations

by Suleman Alfalqi, Hamid Khiar, Ahmed Bchatnia and Abderrahmane Beniani

Fractal Fract. 2024, 8(9), 507; https://doi.org/10.3390/fractalfract8090507 - 28 Aug 2024

Viewed by 718

Abstract

In this study, we examine Timoshenko systems with boundary conditions featuring two types of fractional dissipations. By applying semigroup theory, we demonstrate the existence and uniqueness of solutions. Our analysis shows that while the system exhibits strong stability, it does not achieve uniform [...] Read more.

In this study, we examine Timoshenko systems with boundary conditions featuring two types of fractional dissipations. By applying semigroup theory, we demonstrate the existence and uniqueness of solutions. Our analysis shows that while the system exhibits strong stability, it does not achieve uniform stability. Consequently, we derive a polynomial decay rate for the system. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

28 pages, 1196 KiB  

Open AccessArticle

Advanced Observation-Based Bipartite Containment Control of Fractional-Order Multi-Agent Systems Considering Hostile Environments, Nonlinear Delayed Dynamics, and Disturbance Compensation

by Asad Khan, Muhammad Awais Javeed, Saadia Rehman, Azmat Ullah Khan Niazi and Yubin Zhong

Fractal Fract. 2024, 8(8), 473; https://doi.org/10.3390/fractalfract8080473 - 13 Aug 2024

Cited by 1 | Viewed by 1004

Abstract

This paper introduces an advanced observer-based control strategy designed for fractional multi-agent systems operating in hostile environments. We take into account the dynamic nature of the agents with nonlinear delayed dynamics and consider external disturbances affecting the system. The manuscript presents an improved [...] Read more.

This paper introduces an advanced observer-based control strategy designed for fractional multi-agent systems operating in hostile environments. We take into account the dynamic nature of the agents with nonlinear delayed dynamics and consider external disturbances affecting the system. The manuscript presents an improved observation-based control approach tailored for fractional-order multi-agent systems functioning in challenging conditions. We also establish various applicable conditions governing the creation of observers and disturbance compensation controllers using the fractional Razmikhin technique, signed graph theory, and matrix transformation. Furthermore, our investigation includes observation-based control on switching networks by employing a typical Lyapunov function approach. Finally, the effectiveness of the proposed strategy is demonstrated through the analysis of two simulation examples. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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