Fractional Calculus: Theory and Applications

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 the study of 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 a myriad of applied sciences. Linear, nonlinear, and complex dynamical systems have attracted researchers from many areas of science and technology, involved in systems modelling and control, with applications to real-world problems. Despite the extraordinary advances in FC, addressing both systems’ modelling 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. The Special Issue 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. Alireza Alfi
Prof. Dr. Liping Chen 
Prof. Dr. Sergio Adriani David
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Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Axioms axioms	2.0	-	2012	21.8 Days	CHF 2400
Fractal and Fractional fractalfract	5.4	3.6	2017	18.9 Days	CHF 2700
Mathematical and Computational Applications mca	1.9	-	1996	22.5 Days	CHF 1400
Mathematics mathematics	2.4	3.5	2013	16.9 Days	CHF 2600
Symmetry symmetry	2.7	4.9	2009	16.2 Days	CHF 2400

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15 pages, 443 KiB  

Open AccessArticle

A Second-Order Accurate Numerical Approximation for a Two-Sided Space-Fractional Diffusion Equation

by Taohua Liu, Xiucao Yin, Yinghao Chen and Muzhou Hou

Mathematics 2023, 11(8), 1838; https://doi.org/10.3390/math11081838 - 12 Apr 2023

Viewed by 995

Abstract

In this paper, we investigate a practical numerical method for solving a one-dimensional two-sided space-fractional diffusion equation with variable coefficients in a finite domain, which is based on the classical Crank-Nicolson (CN) method combined with Richardson extrapolation. Second-order exact numerical estimates in time [...] Read more.

In this paper, we investigate a practical numerical method for solving a one-dimensional two-sided space-fractional diffusion equation with variable coefficients in a finite domain, which is based on the classical Crank-Nicolson (CN) method combined with Richardson extrapolation. Second-order exact numerical estimates in time and space are obtained. The unconditional stability and convergence of the method are tested. Two numerical examples are also presented and compared with the exact solution. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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9 pages, 280 KiB  

Open AccessArticle

Lyapunov Inequalities for Two Dimensional Fractional Boundary-Value Problems with Mixed Fractional Derivatives

by Tatiana Odzijewicz

Axioms 2023, 12(3), 301; https://doi.org/10.3390/axioms12030301 - 15 Mar 2023

Viewed by 991

Abstract

We consider two types of partial fractional differential equations in two dimensions with mixed fractional derivatives. Appropriate Lyapunov-type inequalities are proved, and applications to the certain eigenvalue problems are presented. Moreover, some connections with the fractional variational problems are highlighted. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

17 pages, 480 KiB  

Open AccessArticle

Refinable Trapezoidal Method on Riemann–Stieltjes Integral and Caputo Fractional Derivatives for Non-Smooth Functions

by Gopalakrishnan Karnan and Chien-Chang Yen

Fractal Fract. 2023, 7(3), 263; https://doi.org/10.3390/fractalfract7030263 - 15 Mar 2023

Viewed by 1383

Abstract

The Caputo fractional $\alpha$-derivative, $0<\alpha <1$, for non-smooth functions with $1+\alpha$ regularity is calculated by numerical computation. Let I be an interval and ${\mathcal{D}}_{\alpha }\left(I\right)$ be the set of all functions [...] Read more.

The Caputo fractional $\alpha$-derivative, $0<\alpha <1$, for non-smooth functions with $1+\alpha$ regularity is calculated by numerical computation. Let I be an interval and ${\mathcal{D}}_{\alpha }\left(I\right)$ be the set of all functions $f\left(x\right)$ which satisfy $f\left(x\right)=f\left(c\right)+f^{\prime }\left(c\right)(x-a)+g_c\left(x\right)(x-c){\left|(x-c)\right|}^{\alpha }$, where $x,c\in I$ and $g_c\left(x\right)$ is a continuous function for each c. We first extend the trapezoidal method on the set ${\mathcal{D}}_{\alpha }\left(I\right)$ and rewrite the integrand of the Caputo fractional integral as a product of two differentiable functions. In this approach, the non-smooth function and the singular kernel could have the same impact. The trapezoidal method using the Riemann–Stieltjes integral (TRSI) depends on the regularity of the two functions in the integrand. Numerical simulations demonstrated that the order of accuracy cannot be increased as the number of zones increases using the uniform discretization. However, for a fixed coarsest grid discretization, a refinable mesh approach was employed; the corresponding results show that the order of accuracy is $k\alpha$, where k is a refinable scale. Meanwhile, the application of the product of two differentiable functions can also be applied to some Riemann–Liouville fractional differential equations. Finally, the stable numerical scheme is shown. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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13 pages, 310 KiB  

Open AccessArticle

Instantaneous and Non-Instantaneous Impulsive Boundary Value Problem Involving the Generalized ψ-Caputo Fractional Derivative

by Dongping Li, Yankai Li, Fangqi Chen and Xiaozhou Feng

Fractal Fract. 2023, 7(3), 206; https://doi.org/10.3390/fractalfract7030206 - 21 Feb 2023

Cited by 5 | Viewed by 965

Abstract

This paper studies a new class of instantaneous and non-instantaneous impulsive boundary value problem involving the generalized $\psi$-Caputo fractional derivative with a weight. Depending on critical point theorems and some properties of $\psi$-Caputo-type fractional integration and differentiation, the variational construction and [...] Read more.

This paper studies a new class of instantaneous and non-instantaneous impulsive boundary value problem involving the generalized $\psi$-Caputo fractional derivative with a weight. Depending on critical point theorems and some properties of $\psi$-Caputo-type fractional integration and differentiation, the variational construction and multiplicity result of solutions are established. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

33 pages, 6636 KiB  

Open AccessArticle

Bifurcations and the Exact Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation with Parabolic Law Nonlinearity

by Wenjing Zhu, Zijie Ling, Yonghui Xia and Min Gao

Fractal Fract. 2023, 7(2), 201; https://doi.org/10.3390/fractalfract7020201 - 18 Feb 2023

Cited by 5 | Viewed by 1161

Abstract

This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, [...] Read more.

This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, which is different from the previous works, we obtain the phase portraits of the the corresponding traveling wave systems. In addition, we derive the exact parametric representations of solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, peakon solutions, periodic peakon solutions and compacton solutions under different parameter conditions. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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26 pages, 5707 KiB  

Open AccessArticle

Analytical Investigation of the Heat Transfer Effects of Non-Newtonian Hybrid Nanofluid in MHD Flow Past an Upright Plate Using the Caputo Fractional Order Derivative

by N. M. Lisha and A. G. Vijayakumar

Symmetry 2023, 15(2), 399; https://doi.org/10.3390/sym15020399 - 02 Feb 2023

Cited by 6 | Viewed by 1511

Abstract

The objective of this paper is to examine the augmentation of the heat transfer rate utilizing graphene (Gr) and multi-walled carbon nanotubes (MWCNTs) as nanoparticles, and water as a host fluid in magnetohydrodynamics (MHD) flow through an upright plate using Caputo fractional derivatives [...] Read more.

The objective of this paper is to examine the augmentation of the heat transfer rate utilizing graphene (Gr) and multi-walled carbon nanotubes (MWCNTs) as nanoparticles, and water as a host fluid in magnetohydrodynamics (MHD) flow through an upright plate using Caputo fractional derivatives with a Brinkman model on the convective Casson hybrid nanofluid flow. The performance of hybrid nanofluids is examined with various shapes of nanoparticles. The Caputo fractional derivative is utilized to describe the governing fractional partial differential equations with initial and boundary conditions on the flow model. Exact solutions are obtained for flow transport, temperature distribution besides that heat transfer rate and friction drag in terms of Mittag-Leffler function by using Fourier sine and Laplace techniques as hybrid methods. Further, we provided the limiting case solutions for classic partial differential equations on obtained governing fluid flow models. The influence of various physical parameters with different fractional orders are investigated on hybrid nanofluid’s fractional momentum and energy by plotting velocity and energy curves. Few of the findings suggest that fractional parameters have significant effect on flow parameters and that blade-shaped nanoparticles have a high heat transfer rate. The graphical results reveal that the Grashof number shows a symmetry effect in the case of cooling and heating the plate. Furthermore, the performance of hybrid nanofluid is considerably more effective with the Caputo-fractional derivatives rather than in the classic derivative approach. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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15 pages, 3864 KiB  

Open AccessArticle

A Fractional-Order Improved Quantum Logistic Map: Chaos, 0-1 Testing, Complexity, and Control

by Birong Xu, Ximei Ye, Guangyi Wang, Zhongxian Huang and Changwu Zhang

Axioms 2023, 12(1), 94; https://doi.org/10.3390/axioms12010094 - 16 Jan 2023

Cited by 2 | Viewed by 1483

Abstract

Based on a quantum logistic map and a Caputo-like delta difference operator, a fractional-order improved quantum logistic map, which has hidden attractors, was constructed. Its dynamical behaviors are investigated by employing phase portraits, bifurcation diagrams, Lyapunov spectra, dynamical mapping, and 0-1 testing. It [...] Read more.

Based on a quantum logistic map and a Caputo-like delta difference operator, a fractional-order improved quantum logistic map, which has hidden attractors, was constructed. Its dynamical behaviors are investigated by employing phase portraits, bifurcation diagrams, Lyapunov spectra, dynamical mapping, and 0-1 testing. It is shown that the proposed fractional-order map is influenced by both the parameters and the fractional order. Then, the complexity of the map is explored through spectral entropy and approximate entropy. The results show that the fractional-order improved quantum logistic map has stronger robustness within chaos and higher complexity, so it is more suitable for engineering applications. In addition, the fractional-order chaotic map can be controlled for different periodic orbits by the improved nonlinear mapping on the wavelet function. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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24 pages, 4668 KiB  

Open AccessArticle

Thermoelastic Analysis of Functionally Graded Nanobeams via Fractional Heat Transfer Model with Nonlocal Kernels

by Doaa Atta, Ahmed E. Abouelregal and Fahad Alsharari

Mathematics 2022, 10(24), 4718; https://doi.org/10.3390/math10244718 - 12 Dec 2022

Cited by 5 | Viewed by 1374

Abstract

The small size and clever design of nanoparticles can result in large surface areas. This gives nanoparticles enhanced properties such as greater sensitivity, strength, surface area, responsiveness, and stability. This research delves into the phenomenon of a nanobeam vibrating under the influence of [...] Read more.

The small size and clever design of nanoparticles can result in large surface areas. This gives nanoparticles enhanced properties such as greater sensitivity, strength, surface area, responsiveness, and stability. This research delves into the phenomenon of a nanobeam vibrating under the influence of a time-varying heat flow. The nanobeam is hypothesized to have material properties that vary throughout its thickness according to a unique exponential distribution law based on the volume fractions of metal and ceramic components. The top of the FG nanobeam is made entirely of ceramic, while the bottom is made of metal. To address this issue, we employ a nonlocal modified thermoelasticity theory based on a Moore–Gibson–Thompson (MGT) thermoelastic framework. By combining the Euler–Bernoulli beam idea with nonlocal Eringen’s theory, the fundamental equations that govern the proposed model have been constructed based on the extended variation principle. The fractional integral form, utilizing Atangana–Baleanu fractional operators, is also used to formulate the heat transfer equation in the suggested model. The strength of a thermoelastic nanobeam is improved by performing detailed parametric studies to determine the effect of many physical factors, such as the fractional order, the small-scale parameter, the volume fraction indicator, and the periodic frequency of the heat flow. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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10 pages, 283 KiB  

Open AccessArticle

The Power Fractional Calculus: First Definitions and Properties with Applications to Power Fractional Differential Equations

by El Mehdi Lotfi, Houssine Zine, Delfim F. M. Torres and Noura Yousfi

Mathematics 2022, 10(19), 3594; https://doi.org/10.3390/math10193594 - 01 Oct 2022

Cited by 5 | Viewed by 1741

Abstract

Using the Laplace transform method and the convolution theorem, we introduce new and more general definitions for fractional operators with non-singular kernels, extending well-known concepts existing in the literature. The new operators are based on a generalization of the Mittag–Leffler function, characterized by [...] Read more.

Using the Laplace transform method and the convolution theorem, we introduce new and more general definitions for fractional operators with non-singular kernels, extending well-known concepts existing in the literature. The new operators are based on a generalization of the Mittag–Leffler function, characterized by the presence of a key parameter p. This power parameter p is important to enable researchers to choose an adequate notion of the derivative that properly represents the reality under study, to provide good mathematical models, and to predict future dynamic behaviors. The fundamental properties of the new operators are investigated and rigorously proved. As an application, we solve a Caputo and a Riemann–Liouville fractional differential equation. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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9 pages, 269 KiB  

Open AccessArticle

Some Results on a New Refinable Class Suitable for Fractional Differential Problems

by Laura Pezza and Luca Tallini

Fractal Fract. 2022, 6(9), 521; https://doi.org/10.3390/fractalfract6090521 - 15 Sep 2022

Cited by 1 | Viewed by 1054

Abstract

In recent years, we found that some multiscale methods applied to fractional differential problems, are easy and efficient to implement, when we use some fractional refinable functions introduced in the literature. In fact, these functions not only generate a multiresolution on $\mathbb{R}$, [...] Read more.

In recent years, we found that some multiscale methods applied to fractional differential problems, are easy and efficient to implement, when we use some fractional refinable functions introduced in the literature. In fact, these functions not only generate a multiresolution on $\mathbb{R}$, but also have fractional (non-integer) derivative satisfying a very convenient recursive relation. For this reason, in this paper, we describe this class of refinable functions and focus our attention on their approximating properties. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

13 pages, 780 KiB  

Open AccessEditor’s ChoiceArticle

Analytical Investigations into Anomalous Diffusion Driven by Stress Redistribution Events: Consequences of Lévy Flights

by Josiah D. Cleland and Martin A. K. Williams

Mathematics 2022, 10(18), 3235; https://doi.org/10.3390/math10183235 - 06 Sep 2022

Cited by 2 | Viewed by 1250

Abstract

This research is concerned with developing a generalised diffusion equation capable of describing diffusion processes driven by underlying stress-redistributing type events. The work utilises the development of an appropriate continuous time random walk framework as a foundation to consider a new generalised diffusion [...] Read more.

This research is concerned with developing a generalised diffusion equation capable of describing diffusion processes driven by underlying stress-redistributing type events. The work utilises the development of an appropriate continuous time random walk framework as a foundation to consider a new generalised diffusion equation. While previous work has explored the resulting generalised diffusion equation for jump-timings motivated by stick-slip physics, here non-Gaussian probability distributions of the jump displacements are also considered, specifically Lévy flights. This work illuminates several features of the analytic solution to such a generalised diffusion equation using several known properties of the Fox H function. Specifically demonstrated are the temporal behaviour of the resulting position probability density function, and its normalisation. The reduction of the proposed form to expected known solutions upon the insertion of simplifying parameter values, as well as a demonstration of asymptotic behaviours, is undertaken to add confidence to the validity of this equation. This work describes the analytical solution of such a generalised diffusion equation for the first time, and additionally demonstrates the capacity of the Fox H function and its properties in solving and studying generalised Fokker–Planck equations. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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29 pages, 752 KiB  

Open AccessArticle

Hermite–Hadamard-Type Inequalities Involving Harmonically Convex Function via the Atangana–Baleanu Fractional Integral Operator

by Muhammad Amer Latif, Humaira Kalsoom and Muhammad Zainul Abidin

Symmetry 2022, 14(9), 1774; https://doi.org/10.3390/sym14091774 - 25 Aug 2022

Cited by 3 | Viewed by 1176

Abstract

Fractional integrals and inequalities have recently become quite popular and have been the prime consideration for many studies. The results of many different types of inequalities have been studied by launching innovative analytical techniques and applications. These Hermite–Hadamard inequalities are discovered in this [...] Read more.

Fractional integrals and inequalities have recently become quite popular and have been the prime consideration for many studies. The results of many different types of inequalities have been studied by launching innovative analytical techniques and applications. These Hermite–Hadamard inequalities are discovered in this study using Atangana–Baleanu integral operators, which provide both practical and powerful results. In this paper, a symmetric study of integral inequalities of Hermite–Hadamard type is provided based on an identity proved for Atangana–Baleanu integral operators and using functions whose absolute value of the second derivative is harmonic convex. The proven Hermite–Hadamard-type inequalities have been observed to be valid for a choice of any harmonic convex function with the help of examples. Moreover, fractional inequalities and their solutions are applied in many symmetrical domains. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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24 pages, 406 KiB  

Open AccessArticle

Generalized Space-Time Fractional Stochastic Kinetic Equation

by Junfeng Liu, Zhigang Yao and Bin Zhang

Fractal Fract. 2022, 6(8), 450; https://doi.org/10.3390/fractalfract6080450 - 18 Aug 2022

Viewed by 1225

Abstract

In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in ${\mathbb{R}}^d$ with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the nonlinear stochastic heat equation involving [...] Read more.

In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in ${\mathbb{R}}^d$ with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the nonlinear stochastic heat equation involving fractional derivatives in time and fractional Laplacian in space. We firstly give a necessary condition on the spatial covariance for the existence and uniqueness of the solution. Furthermore, we also study various properties of the solution, such as Hölder regularity, the upper bound of second moment, and the stationarity with respect to the spatial variable in the case of linear additive noise. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

16 pages, 392 KiB  

Open AccessArticle

Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators

by Christopher N. Angstmann, Stuart-James M. Burney, Bruce I. Henry and Byron A. Jacobs

Fractal Fract. 2022, 6(8), 436; https://doi.org/10.3390/fractalfract6080436 - 11 Aug 2022

Viewed by 1418

Abstract

Motivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solutions to [...] Read more.

Motivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solutions to these initial value problems possess discontinuities at the origin. We also show how these initial value problems can be re-formulated to provide solutions that are continuous at the origin but this imposes further constraints on the system. Consideration of the intrinsic discontinuities, or constraints, in these initial value problems is important if they are to be employed in mathematical modelling applications. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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16 pages, 309 KiB  

Open AccessArticle

On Hilfer Generalized Proportional Nabla Fractional Difference Operators

by Qiushuang Wang and Run Xu

Mathematics 2022, 10(15), 2654; https://doi.org/10.3390/math10152654 - 28 Jul 2022

Viewed by 1040

Abstract

In this paper, the Hilfer type generalized proportional nabla fractional differences are defined. A few important properties in the left case are derived and the properties in the right case are proved by Q-operator. The discrete Laplace transform in the sense of [...] Read more.

In this paper, the Hilfer type generalized proportional nabla fractional differences are defined. A few important properties in the left case are derived and the properties in the right case are proved by Q-operator. The discrete Laplace transform in the sense of the left Hilfer generalized proportional fractional difference is explored. Furthermore, An initial value problem with the new operator and its generalized solution are considered. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

16 pages, 364 KiB  

Open AccessArticle

Noether and Lie Symmetry for Singular Systems Involving Mixed Derivatives

by Chuan-Jing Song

Symmetry 2022, 14(6), 1225; https://doi.org/10.3390/sym14061225 - 13 Jun 2022

Cited by 1 | Viewed by 1160

Abstract

Singular systems play an important role in many fields, and some new fractional operators, which are general, have been proposed recently. Therefore, singular systems on the basis of the mixed derivatives including the integer order derivative and the generalized fractional operators are studied. [...] Read more.

Singular systems play an important role in many fields, and some new fractional operators, which are general, have been proposed recently. Therefore, singular systems on the basis of the mixed derivatives including the integer order derivative and the generalized fractional operators are studied. Firstly, Lagrange equations within mixed derivatives are established, and the primary constraints are presented for the singular systems. Then the constrained Hamilton equations are constructed by introducing the Lagrange multipliers. Thirdly, Noether symmetry, Lie symmetry and the corresponding conserved quantities for the constrained Hamiltonian systems are investigated. And finally, an example is given to illustrate the methods and results. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

22 pages, 345 KiB  

Open AccessArticle

Regularization for a Sideways Problem of the Non-Homogeneous Fractional Diffusion Equation

by Yonggang Chen, Yu Qiao and Xiangtuan Xiong

Fractal Fract. 2022, 6(6), 312; https://doi.org/10.3390/fractalfract6060312 - 02 Jun 2022

Viewed by 1560

Abstract

In this article, we investigate a sideways problem of the non-homogeneous time-fractional diffusion equation, which is highly ill-posed. Such a model is obtained from the classical non-homogeneous sideways heat equation by replacing the first-order time derivative by the Caputo fractional derivative. We achieve [...] Read more.

In this article, we investigate a sideways problem of the non-homogeneous time-fractional diffusion equation, which is highly ill-posed. Such a model is obtained from the classical non-homogeneous sideways heat equation by replacing the first-order time derivative by the Caputo fractional derivative. We achieve the result of conditional stability under an a priori assumption. Two regularization strategies, based on the truncation of high frequency components, are constructed for solving the inverse problem in the presence of noisy data, and the corresponding error estimates are proved. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

16 pages, 369 KiB  

Open AccessArticle

Hermite–Hadamard-Type Inequalities for h-Convex Functions Involving New Fractional Integral Operators with Exponential Kernel

by Yaoqun Wu

Fractal Fract. 2022, 6(6), 309; https://doi.org/10.3390/fractalfract6060309 - 01 Jun 2022

Viewed by 1526

Abstract

In this paper, we use two new fractional integral operators with exponential kernel about the midpoint of the interval to construct some Hermite–Hadamard type fractional integral inequalities for h-convex functions. Taking two integral identities about the first and second derivatives of the [...] Read more.

In this paper, we use two new fractional integral operators with exponential kernel about the midpoint of the interval to construct some Hermite–Hadamard type fractional integral inequalities for h-convex functions. Taking two integral identities about the first and second derivatives of the function as auxiliary functions, the main results are obtained by using the properties of h-convexity and the module. In order to illustrate the application of the results, we propose four examples and plot function images to intuitively present the meaning of the inequalities in the main results, and we verify the correctness of the conclusion. This study further expands the generalization of Hermite–Hadamard-type inequalities and provides some research references for the study of Hermite–Hadamard-type inequalities. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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23 pages, 3753 KiB  

Open AccessArticle

Complex Dynamic Behaviour of Food Web Model with Generalized Fractional Operator

by Ajay Kumar, Sara Salem Alzaid, Badr Saad T. Alkahtani and Sunil Kumar

Mathematics 2022, 10(10), 1702; https://doi.org/10.3390/math10101702 - 16 May 2022

Cited by 4 | Viewed by 1595

Abstract

We apply a new generalized Caputo operator to investigate the dynamical behaviour of the non-integer food web model (FWM). This dynamical model has three population species and is nonlinear. Three types of species are considered in this population: prey species, intermediate predators, and [...] Read more.

We apply a new generalized Caputo operator to investigate the dynamical behaviour of the non-integer food web model (FWM). This dynamical model has three population species and is nonlinear. Three types of species are considered in this population: prey species, intermediate predators, and top predators, and the top predators are also divided into mature and immature predators. We calculated the uniqueness and existence of the solutions applying the fixed-point hypothesis. Our study examines the possibility of obtaining new dynamical phase portraits with the new generalized Caputo operator and demonstrates the portraits for several values of fractional order. A generalized predictor–corrector (P-C) approach is utilized in numerically solving this food web model. In the case of the nonlinear equations system, the effectiveness of the used scheme is highly evident and easy to implement. In addition, stability analysis was conducted for this numerical scheme. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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34 pages, 481 KiB  

Open AccessArticle

Left Riemann–Liouville Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

by Xing Hu and Yongkun Li

Fractal Fract. 2022, 6(5), 268; https://doi.org/10.3390/fractalfract6050268 - 15 May 2022

Cited by 2 | Viewed by 1697

Abstract

First, we show the equivalence of two definitions of the left Riemann–Liouville fractional integral on time scales. Then, we establish and characterize fractional Sobolev space with the help of the notion of left Riemann–Liouville fractional derivative on time scales. At the same time, [...] Read more.

First, we show the equivalence of two definitions of the left Riemann–Liouville fractional integral on time scales. Then, we establish and characterize fractional Sobolev space with the help of the notion of left Riemann–Liouville fractional derivative on time scales. At the same time, we define weak left fractional derivatives and demonstrate that they coincide with the left Riemann–Liouville ones on time scales. Next, we prove the equivalence of two kinds of norms in the introduced space and derive its completeness, reflexivity, separability, and some embedding. Finally, as an application, by constructing an appropriate variational setting, using the mountain pass theorem and the genus properties, the existence of weak solutions for a class of Kirchhoff-type fractional p-Laplacian systems on time scales with boundary conditions is studied, and three results of the existence of weak solutions for this problem is obtained. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

19 pages, 358 KiB  

Open AccessArticle

Hadamard-Type Fractional Integro-Differential Problem: A Note on Some Asymptotic Behavior of Solutions

by Ahmad Mugbil and Nasser-Eddine Tatar

Fractal Fract. 2022, 6(5), 267; https://doi.org/10.3390/fractalfract6050267 - 15 May 2022

Cited by 1 | Viewed by 1503

Abstract

As a follow-up to the inherent nature of Hadamard-Type Fractional Integro-differential problem, little is known about some asymptotic behaviors of solutions. In this paper, an integro-differential problem involving Hadamard fractional derivatives is investigated. The leading derivative is of an order between one and [...] Read more.

As a follow-up to the inherent nature of Hadamard-Type Fractional Integro-differential problem, little is known about some asymptotic behaviors of solutions. In this paper, an integro-differential problem involving Hadamard fractional derivatives is investigated. The leading derivative is of an order between one and two whereas the nonlinearities may contain fractional derivatives of an order between zero and one as well as some non-local terms. Under some reasonable conditions, we prove that solutions are asymptotic to logarithmic functions. Our approach is based on a generalized version of Bihari–LaSalle inequality, which we prove. In addition, several manipulations and crucial estimates have been used. An example supporting our findings is provided. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

11 pages, 303 KiB  

Open AccessArticle

Laplace Transform for Solving System of Integro-Fractional Differential Equations of Volterra Type with Variable Coefficients and Multi-Time Delay

by Miran B. M. Amin and Shazad Shawki Ahmad

Symmetry 2022, 14(5), 984; https://doi.org/10.3390/sym14050984 - 11 May 2022

Cited by 1 | Viewed by 1491

Abstract

This study is the first to use Laplace transform methods to solve a system of Caputo fractional Volterra integro-differential equations with variable coefficients and a constant multi-time delay. This technique is based on different types of kernels, which we will explain in this [...] Read more.

This study is the first to use Laplace transform methods to solve a system of Caputo fractional Volterra integro-differential equations with variable coefficients and a constant multi-time delay. This technique is based on different types of kernels, which we will explain in this paper. Symmetry kernels, which have properties of difference kernels or simple degenerate kernels, are able to compute analytical work. These are demonstrated by solving certain examples and analyzing the effectiveness and precision of cause techniques. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

16 pages, 8066 KiB  

Open AccessArticle

Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method

by Mohammed Shqair, Mohammed Alabedalhadi, Shrideh Al-Omari and Mohammed Al-Smadi

Fractal Fract. 2022, 6(5), 252; https://doi.org/10.3390/fractalfract6050252 - 05 May 2022

Cited by 17 | Viewed by 1808

Abstract

The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel [...] Read more.

The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex fractional systems by identifying suitable options for arbitrary elliptic functions. To understand the physical characteristics of NFMT, the 3D graphical representations of the obtained propagation wave solutions for some free physical parameters are randomly drawn for a different order of the fractional derivatives. The results indicate that the proposed method is reliable, simple, and powerful enough to handle more complicated nonlinear fractional partial differential equations in quantum mechanics. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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12 pages, 814 KiB  

Open AccessArticle

Investigating a Generalized Fractional Quadratic Integral Equation

by Basim N. Abood, Saleh S. Redhwan, Omar Bazighifan and Kamsing Nonlaopon

Fractal Fract. 2022, 6(5), 251; https://doi.org/10.3390/fractalfract6050251 - 01 May 2022

Cited by 6 | Viewed by 1531

Abstract

In this article, we investigate the analytical and approximate solutions for a fractional quadratic integral equation in the frame of the generalized Riemann–Liouville fractional integral operator with respect to another function. The existence and uniqueness results obtained. Moreover, some new special results corresponding [...] Read more.

In this article, we investigate the analytical and approximate solutions for a fractional quadratic integral equation in the frame of the generalized Riemann–Liouville fractional integral operator with respect to another function. The existence and uniqueness results obtained. Moreover, some new special results corresponding to suitable values of the parameters $\zeta$ and q are given. The main results are proved by applying Banach’s fixed point theorem, the Adomian decomposition method, and Picard’s method. In the end, we present a numerical example to justify our results. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

17 pages, 5544 KiB  

Open AccessArticle

Epidemic Dynamics of a Fractional-Order SIR Weighted Network Model and Its Targeted Immunity Control

by Na Liu, Jie Fang, Junwei Sun and Sanyi Li

Fractal Fract. 2022, 6(5), 232; https://doi.org/10.3390/fractalfract6050232 - 22 Apr 2022

Cited by 3 | Viewed by 1641

Abstract

With outbreaks of epidemics, an enormous loss of life and property has been caused. Based on the influence of disease transmission and information propagation on the transmission characteristics of infectious diseases, in this paper, a fractional-order SIR epidemic model is put forward on [...] Read more.

With outbreaks of epidemics, an enormous loss of life and property has been caused. Based on the influence of disease transmission and information propagation on the transmission characteristics of infectious diseases, in this paper, a fractional-order SIR epidemic model is put forward on a two-layer weighted network. The local stability of the disease-free equilibrium is investigated. Moreover, a conclusion is obtained that there is no endemic equilibrium. Since the elderly and the children have fewer social tiers, a targeted immunity control that is based on age structure is proposed. Finally, an example is presented to demonstrate the effectiveness of the theoretical results. These studies contribute to a more comprehensive understanding of the epidemic transmission mechanism and play a positive guiding role in the prevention and control of some epidemics. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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37 pages, 4275 KiB  

Open AccessArticle

Continuum Damage Dynamic Model Combined with Transient Elastic Equation and Heat Conduction Equation to Solve RPV Stress

by Wenxing Chen, Shuyang Dai and Baojuan Zheng

Fractal Fract. 2022, 6(4), 215; https://doi.org/10.3390/fractalfract6040215 - 11 Apr 2022

Cited by 3 | Viewed by 2081

Abstract

The development of the world cannot be separated from energy: the energy crisis has become a major challenge in this era, and nuclear energy has been applied to many fields. This paper mainly studies the stress change of reaction pressure vessels (RPV). We [...] Read more.

The development of the world cannot be separated from energy: the energy crisis has become a major challenge in this era, and nuclear energy has been applied to many fields. This paper mainly studies the stress change of reaction pressure vessels (RPV). We established several different physical models to solve the same mechanical problem. Numerical methods range from 1D to 3D; the 1D model is mainly based on the mechanical equilibrium equations established by the internal pressure of RPV, the hoop stress, and the axial stress. We found that the hoop stress is twice the axial stress; this model is a rough estimate. For 2D RPV mechanical simulation, we proposed a new method, which combined the continuum damage dynamic model with the transient cross-section finite element method (CDDM-TCFEM). The advantage is that the temperature and shear strain can be linked by the damage factor effect on the elastic model and Poission ratio. The results show that with the increase of temperature (damage factor $\widehat{\mu },\widehat{d}$), the Young’s modulus decreases point by point, and the Poisson’s ratio increases with the increase of temperature (damage factor $\widehat{\mu },{\mathcal{E}}_{\mathrm{t}}$). The advantage of the CDDM-TCFEM is that the calculation efficiency is high. However, it is unable to obtain the overall mechanical cloud map. In order to solve this problem, we established the axisymmetric finite element model, and the results show that the stress value at both ends of RPV is significantly greater than that in the middle of the container. Meanwhile, the shape changes of 2D and 3D RPV are calculated and visualized. Finally, a 3D thermal–mechanical coupling model is established, and the cloud map of strain and displacement are also visualized. We found that the stress of the vessel wall near the nozzle decreases gradually from the inside surface to the outside, and the hoop stress is slightly larger than the axial stress. The main contribution of this paper is to establish a CDDM-TCFEM model considering the influence of temperature on elastic modulus and Poission ratio. It can dynamically describe the stress change of RPV; we have given the fitting formula of the internal temperature and pressure of RPV changing with time. We also establish a 3D coupling model and use the adaptive mesh to discretize the pipe. The numerical discrete theory of FDM-FEM is given, and the numerical results are visualized well. In addition, we have given error estimation for h-type and p-type adaptive meshes. So, our research can provide mechanical theoretical support for nuclear energy safety applications and RPV design. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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21 pages, 2776 KiB  

Open AccessArticle

Impact of Al₂O₃ in Electrically Conducting Mineral Oil-Based Maxwell Nanofluid: Application to the Petroleum Industry

by Hanifa Hanif and Sharidan Shafie

Fractal Fract. 2022, 6(4), 180; https://doi.org/10.3390/fractalfract6040180 - 24 Mar 2022

Cited by 14 | Viewed by 1996

Abstract

Alumina nanoparticles (Al₂O₃) are one of the essential metal oxides and have a wide range of applications and unique physio-chemical features. Most notably, alumina has been shown to have thermal properties such as high thermal conductivity and a convective [...] Read more.

Alumina nanoparticles (Al₂O₃) are one of the essential metal oxides and have a wide range of applications and unique physio-chemical features. Most notably, alumina has been shown to have thermal properties such as high thermal conductivity and a convective heat transfer coefficient. Therefore, this study is conducted to integrate the adsorption of Al₂O₃ in mineral oil-based Maxwell fluid. The ambitious goal of this study is to intensify the mechanical and thermal properties of a Maxwell fluid under heat flux boundary conditions. The novelty of the research is increased by introducing fractional derivatives to the Maxwell model. There are various distinct types of fractional derivative definitions, with the Caputo fractional derivative being one of the most predominantly applied. Therefore, the fractoinal-order derivatives are evaluated using the fractional Caputo derivative, and the integer-order derivatives are evaluated using the Crank–Nicolson method. The obtained results are graphically displayed to demonstrate how all governing parameters, such as nanoparticle volume fraction, relaxation time, fractional derivative, magnetic field, thermal radiation, and viscous dissipation, have a significant impact on fluid flow and temperature distribution. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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13 pages, 294 KiB  

Open AccessArticle

Existence of Mild Solutions for Hilfer Fractional Evolution Equations with Almost Sectorial Operators

by Mian Zhou, Chengfu Li and Yong Zhou

Axioms 2022, 11(4), 144; https://doi.org/10.3390/axioms11040144 - 22 Mar 2022

Cited by 19 | Viewed by 2035

Abstract

In this paper, we obtain new sufficient conditions of the existence of mild solutions for Hilfer fractional evolution equations in the cases that the semigroup associated with an almost sectorial operator is compact as well as noncompact. Our results improve and extend some [...] Read more.

In this paper, we obtain new sufficient conditions of the existence of mild solutions for Hilfer fractional evolution equations in the cases that the semigroup associated with an almost sectorial operator is compact as well as noncompact. Our results improve and extend some recent results in references. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

15 pages, 310 KiB  

Open AccessArticle

Existence and Hyers–Ulam Stability for a Multi-Term Fractional Differential Equation with Infinite Delay

by Chen Chen and Qixiang Dong

Mathematics 2022, 10(7), 1013; https://doi.org/10.3390/math10071013 - 22 Mar 2022

Cited by 7 | Viewed by 1694

Abstract

This paper is devoted to investigating one type of nonlinear two-term fractional order delayed differential equations involving Caputo fractional derivatives. The Leray–Schauder alternative fixed-point theorem and Banach contraction principle are applied to analyze the existence and uniqueness of solutions to the problem with [...] Read more.

This paper is devoted to investigating one type of nonlinear two-term fractional order delayed differential equations involving Caputo fractional derivatives. The Leray–Schauder alternative fixed-point theorem and Banach contraction principle are applied to analyze the existence and uniqueness of solutions to the problem with infinite delay. Additionally, the Hyers–Ulam stability of fractional differential equations is considered for the delay conditions. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

17 pages, 358 KiB  

Open AccessArticle

Multi-Point Boundary Value Problems for (k, ϕ)-Hilfer Fractional Differential Equations and Inclusions

by Jessada Tariboon, Ayub Samadi and Sotiris K. Ntouyas

Axioms 2022, 11(3), 110; https://doi.org/10.3390/axioms11030110 - 02 Mar 2022

Cited by 9 | Viewed by 2283

Abstract

In this paper we initiate the study of boundary value problems for fractional differential equations and inclusions involving $(k,\varphi )$-Hilfer fractional derivative of order in $(1,2]$. In the single-valued case the existence and uniqueness [...] Read more.

In this paper we initiate the study of boundary value problems for fractional differential equations and inclusions involving $(k,\varphi )$-Hilfer fractional derivative of order in $(1,2]$. In the single-valued case the existence and uniqueness results are established by using classical fixed-point theorems, such as Banach, Krasnoselskiĭ and Leray-Schauder. In the multivalued case we consider both cases, when the right-hand side has convex or non-convex values. In the first case, we apply the Leray–Schauder nonlinear alternative for multivalued maps, and in the second, the Covit–Nadler fixed-point theorem for multivalued contractions. All results are well illustrated by numerical examples. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

13 pages, 6093 KiB  

Open AccessArticle

A Numerical Study of the Fractional Order Dynamical Nonlinear Susceptible Infected and Quarantine Differential Model Using the Stochastic Numerical Approach

by Thongchai Botmart, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Wajaree Weera, Rahma Sadat and Mohamed R. Ali

Fractal Fract. 2022, 6(3), 139; https://doi.org/10.3390/fractalfract6030139 - 01 Mar 2022

Cited by 22 | Viewed by 2224

Abstract

The theme of this study is to present the impacts and importance of the fractional order derivatives of the susceptible, infected and quarantine (SIQ) model based on the coronavirus with the lockdown effects. The purpose of these investigations is to achieve more accuracy [...] Read more.

The theme of this study is to present the impacts and importance of the fractional order derivatives of the susceptible, infected and quarantine (SIQ) model based on the coronavirus with the lockdown effects. The purpose of these investigations is to achieve more accuracy with the use of fractional derivatives in the SIQ model. The integer, nonlinear mathematical SIQ system with the lockdown effects is also provided in this study. The lockdown effects are categorized into the dynamics of the susceptible, infective and quarantine, generally known as SIQ mathematical system. The fractional order SIQ mathematical system has never been presented before, nor solved by using the strength of the stochastic solvers. The stochastic solvers based on the Levenberg-Marquardt backpropagation scheme (LMBS) along with the neural networks (NNs), i.e., LMBS-NNs have been implemented to solve the fractional order SIQ mathematical system. Three cases using different values of the fractional order have been provided to solve the fractional order SIQ mathematical model. The data to present the numerical solutions of the fractional order SIQ mathematical model is selected as 80% for training and 10% for both testing and validation. For the correctness of the LMBS-NNs, the obtained numerical results have been compared with the reference solutions through the Adams–Bashforth–Moulton based numerical solver. In order to authenticate the competence, consistency, validity, capability and exactness of the LMB-NNs, the numerical performances using the state transitions (STs), regression, correlation, mean square error (MSE) and error histograms (EHs) are also provided. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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26 pages, 2310 KiB  

Open AccessArticle

Parameter Identification and the Finite-Time Combination–Combination Synchronization of Fractional-Order Chaotic Systems with Different Structures under Multiple Stochastic Disturbances

by Weiqiu Pan, Tianzeng Li, Muhammad Sajid, Safdar Ali and Lingping Pu

Mathematics 2022, 10(5), 712; https://doi.org/10.3390/math10050712 - 24 Feb 2022

Cited by 9 | Viewed by 1555

Abstract

This paper researches the issue of the finite-time combination-combination (C-C) synchronization (FTCCS) of fractional order (FO) chaotic systems under multiple stochastic disturbances (SD) utilizing the nonsingular terminal sliding mode control (NTSMC) technique. The systems we considered have different characteristics of the structures and [...] Read more.

This paper researches the issue of the finite-time combination-combination (C-C) synchronization (FTCCS) of fractional order (FO) chaotic systems under multiple stochastic disturbances (SD) utilizing the nonsingular terminal sliding mode control (NTSMC) technique. The systems we considered have different characteristics of the structures and the parameters are unknown. The stochastic disturbances are considered parameter uncertainties, nonlinear uncertainties and external disturbances. The bounds of the uncertainties and disturbances are unknown. Firstly, we are going to put forward a new FO sliding surface in terms of fractional calculus. Secondly, some suitable adaptive control laws (ACL) are found to assess the unknown parameters and examine the upper bound of stochastic disturbances. Finally, combining the finite-time Lyapunov stability theory and the sliding mode control (SMC) technique, we propose a fractional-order adaptive combination controller that can achieve the finite-time synchronization of drive-response (D-R) systems. In this paper, some of the synchronization methods, such as chaos control, complete synchronization, projection synchronization, anti-synchronization, and so forth, have become special cases of combination-combination synchronization. Examples are presented to verify the usefulness and validity of the proposed scheme via MATLAB. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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8 pages, 239 KiB  

Open AccessArticle

Equity Warrants Pricing Formula for Uncertain Financial Market

by Foad Shokrollahi

Math. Comput. Appl. 2022, 27(2), 18; https://doi.org/10.3390/mca27020018 - 22 Feb 2022

Viewed by 2439

Abstract

In this paper, inside the system of uncertainty theory, the valuation of equity warrants is explored. Different from the strategies of probability theory, the valuation problem of equity warrants is unraveled by utilizing the strategy of uncertain calculus. Based on the suspicion that [...] Read more.

In this paper, inside the system of uncertainty theory, the valuation of equity warrants is explored. Different from the strategies of probability theory, the valuation problem of equity warrants is unraveled by utilizing the strategy of uncertain calculus. Based on the suspicion that the firm price follows an uncertain differential equation, a valuation formula of equity warrants is proposed for an uncertain stock model. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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29 pages, 8287 KiB  

Open AccessArticle

Switched Fractional Order Multiagent Systems Containment Control with Event-Triggered Mechanism and Input Quantization

by Jiaxin Yuan and Tao Chen

Fractal Fract. 2022, 6(2), 77; https://doi.org/10.3390/fractalfract6020077 - 31 Jan 2022

Cited by 8 | Viewed by 2141

Abstract

This paper studies the containment control problem for a class of fractional order nonlinear multiagent systems in the presence of arbitrary switchings, unmeasured states, and quantized input signals by a hysteresis quantizer. Under the framework of the Lyapunov function theory, this paper proposes [...] Read more.

This paper studies the containment control problem for a class of fractional order nonlinear multiagent systems in the presence of arbitrary switchings, unmeasured states, and quantized input signals by a hysteresis quantizer. Under the framework of the Lyapunov function theory, this paper proposes an event-triggered adaptive neural network dynamic surface quantized controller, in which dynamic surface control technology can avoid “explosion of complexity” and obtain fractional derivatives for virtual control functions continuously. Radial basis function neural networks (RBFNNs) are used to approximate the unknown nonlinear functions, and an observer is designed to obtain the unmeasured states. The proposed distributed protocol can ensure all the signals remain semi-global uniformly ultimately bounded in the closed-loop system, and all followers can converge to the convex hull spanned by the leaders’ trajectory. Utilizing the combination of an event-triggered scheme and quantized control technology, the controller is updated aperiodically only at the event-sampled instants such that transmitting and computational costs are greatly reduced. Simulations compare the event-triggered scheme without quantization control technology with the control method proposed in this paper, and the results show that the event-triggered scheme combined with the quantization mechanism reduces the number of control inputs by 7% to 20%. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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8 pages, 292 KiB  

Open AccessArticle

On Transformation Involving Basic Analogue to the Aleph-Function of Two Variables

by Dinesh Kumar, Dumitru Baleanu, Frédéric Ayant and Norbert Südland

Fractal Fract. 2022, 6(2), 71; https://doi.org/10.3390/fractalfract6020071 - 28 Jan 2022

Cited by 2 | Viewed by 2149

Abstract

In our work, we derived the fractional order q-integrals and q-derivatives concerning a basic analogue to the Aleph-function of two variables (AFTV). We discussed a related application and the q-extension of the corresponding Leibniz rule. Finally, we presented two corollaries [...] Read more.

In our work, we derived the fractional order q-integrals and q-derivatives concerning a basic analogue to the Aleph-function of two variables (AFTV). We discussed a related application and the q-extension of the corresponding Leibniz rule. Finally, we presented two corollaries concerning the basic analogue to the I-function of two variables and the basic analogue to the Aleph-function of one variable. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

22 pages, 734 KiB  

Open AccessArticle

Overview of One-Dimensional Continuous Functions with Fractional Integral and Applications in Reinforcement Learning

by Wang Jun, Cao Lei, Wang Bin, Gong Hongtao and Tang Wei

Fractal Fract. 2022, 6(2), 69; https://doi.org/10.3390/fractalfract6020069 - 27 Jan 2022

Cited by 2 | Viewed by 2403

Abstract

One-dimensional continuous functions are important fundament for studying other complex functions. Many theories and methods applied to study one-dimensional continuous functions can also be accustomed to investigating the properties of multi-dimensional functions. The properties of one-dimensional continuous functions, such as dimensionality, continuity, and [...] Read more.

One-dimensional continuous functions are important fundament for studying other complex functions. Many theories and methods applied to study one-dimensional continuous functions can also be accustomed to investigating the properties of multi-dimensional functions. The properties of one-dimensional continuous functions, such as dimensionality, continuity, and boundedness, have been discussed from multiple perspectives. Therefore, the existing conclusions will be systematically sorted out according to the bounded variation, unbounded variation and h$\ddot{o}$lder continuity. At the same time, unbounded variation points are used to analyze continuous functions and construct unbounded variation functions innovatively. Possible applications of fractal and fractal dimension in reinforcement learning are predicted. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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18 pages, 357 KiB  

Open AccessArticle

Existence and Uniqueness of Mild Solution Where α ∈ (1,2) for Fuzzy Fractional Evolution Equations with Uncertainty

by Ramsha Shafqat, Azmat Ullah Khan Niazi, Mdi Begum Jeelani and Nadiyah Hussain Alharthi

Fractal Fract. 2022, 6(2), 65; https://doi.org/10.3390/fractalfract6020065 - 26 Jan 2022

Cited by 18 | Viewed by 2359

Abstract

This paper concerns with the existence and uniqueness of fuzzy fractional evolution equation with uncertainty involves function of form [...] Read more.

This paper concerns with the existence and uniqueness of fuzzy fractional evolution equation with uncertainty involves function of form ${}^c{D}^{\alpha }x\left(t\right)=f(t,x\left(t\right),D^{\beta }x\left(t\right)),I^{\alpha }x\left(0\right)=x_0,x^{\prime }\left(0\right)=x_1,$ where $1<\alpha <2,0<\beta <1$. After determining the equivalent integral form of solution we establish existence and uniqueness by using Rogers conditions, Kooi type conditions and Krasnoselskii-Krein type conditions. In addition, various numerical solutions have been presented to ensure that the main result is true and effective. Finally, a few examples which express fuzzy fractional evolution equations are shown. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

13 pages, 294 KiB  

Open AccessArticle

Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion

by Bodo Herzog

Mathematics 2022, 10(3), 340; https://doi.org/10.3390/math10030340 - 23 Jan 2022

Cited by 2 | Viewed by 2648

Abstract

The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions $\{B_t^H,t\ge 0\}$ [...] Read more.

The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions $\{B_t^H,t\ge 0\}$ and sub-fractional Brownian motions $\{{\xi }_t^H,t\ge 0\}$ with Hurst parameter $H\in (\frac{1}{2},1)$. We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

18 pages, 317 KiB  

Open AccessArticle

Non-Instantaneous Impulsive BVPs Involving Generalized Liouville–Caputo Derivative

by Ahmed Salem and Sanaa Abdullah

Mathematics 2022, 10(3), 291; https://doi.org/10.3390/math10030291 - 18 Jan 2022

Cited by 10 | Viewed by 1398

Abstract

This manuscript investigates the existence, uniqueness and Ulam–Hyers stability (UH) of solution to fractional differential equations with non-instantaneous impulses on an arbitrary domain. Using the modern tools of functional analysis, we achieve the required conditions. Finally, we provide an example of how our [...] Read more.

This manuscript investigates the existence, uniqueness and Ulam–Hyers stability (UH) of solution to fractional differential equations with non-instantaneous impulses on an arbitrary domain. Using the modern tools of functional analysis, we achieve the required conditions. Finally, we provide an example of how our results can be applied. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

21 pages, 4883 KiB  

Open AccessArticle

Recursive Identification for MIMO Fractional-Order Hammerstein Model Based on AIAGS

by Qibing Jin, Bin Wang and Zeyu Wang

Mathematics 2022, 10(2), 212; https://doi.org/10.3390/math10020212 - 11 Jan 2022

Cited by 2 | Viewed by 1497

Abstract

In this paper, adaptive immune algorithm based on a global search strategy (AIAGS) and auxiliary model recursive least square method (AMRLS) are used to identify the multiple-input multiple-output fractional-order Hammerstein model. The model’s nonlinear parameters, linear parameters, and fractional order are unknown. The [...] Read more.

In this paper, adaptive immune algorithm based on a global search strategy (AIAGS) and auxiliary model recursive least square method (AMRLS) are used to identify the multiple-input multiple-output fractional-order Hammerstein model. The model’s nonlinear parameters, linear parameters, and fractional order are unknown. The identification step is to use AIAGS to find the initial values of model coefficients and order at first, then bring the initial values into AMRLS to identify the coefficients and order of the model in turn. The expression of the linear block is the transfer function of the differential equation. By changing the stimulation function of the original algorithm, adopting the global search strategy before the local search strategy in the mutation operation, and adopting the parallel mechanism, AIAGS further strengthens the original algorithm’s optimization ability. The experimental results show that the proposed method is effective. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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