Topic Editors

Dr. António Lopes
Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200–465 Porto, Portugal
Prof. Dr. Alireza Alfi
Department of Electrical and Robotic Engineering, Shahrood University of Technology, Shahrood 95161-36199, Iran
Prof. Dr. Liping Chen
School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China
Dr. Sergio Adriani David
Faculdade de Zootecnia e Engenharia de Alimentos da USP, University of São Paulo, Av. Duque de Caxias-Norte, 225, Jardim Elite, 13635-900 Pirassununga, SP, Brazil

Fractional Calculus: Theory and Applications

Abstract submission deadline
closed (30 September 2022)
Manuscript submission deadline
30 January 2023
Viewed by
29835

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.

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

Keywords

  • fractional differential and integral equations
  • fractional dynamics and control
  • fractional calculus of variations
  • symmetry
  • applications of fractional calculus to real-world problems

Participating Journals

Journal Name Impact Factor CiteScore Launched Year First Decision (median) APC
Axioms
axioms
1.824 2.6 2012 19 Days 1600 CHF Submit
Fractal and Fractional
fractalfract
3.577 2.8 2017 17 Days 1800 CHF Submit
Mathematical and Computational Applications
mca
- - 1996 23.5 Days 1400 CHF Submit
Mathematics
mathematics
2.592 2.9 2013 17.8 Days 1800 CHF Submit
Symmetry
symmetry
2.940 4.3 2009 13.8 Days 1800 CHF Submit

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Published Papers (32 papers)

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Article
The Power Fractional Calculus: First Definitions and Properties with Applications to Power Fractional Differential Equations
Mathematics 2022, 10(19), 3594; https://doi.org/10.3390/math10193594 - 01 Oct 2022
Cited by 1 | Viewed by 511
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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Article
Some Results on a New Refinable Class Suitable for Fractional Differential Problems
Fractal Fract. 2022, 6(9), 521; https://doi.org/10.3390/fractalfract6090521 - 15 Sep 2022
Viewed by 434
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 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 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)
Article
Analytical Investigations into Anomalous Diffusion Driven by Stress Redistribution Events: Consequences of Lévy Flights
Mathematics 2022, 10(18), 3235; https://doi.org/10.3390/math10183235 - 06 Sep 2022
Viewed by 485
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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Article
Hermite–Hadamard-Type Inequalities Involving Harmonically Convex Function via the Atangana–Baleanu Fractional Integral Operator
Symmetry 2022, 14(9), 1774; https://doi.org/10.3390/sym14091774 - 25 Aug 2022
Viewed by 458
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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Article
Generalized Space-Time Fractional Stochastic Kinetic Equation
Fractal Fract. 2022, 6(8), 450; https://doi.org/10.3390/fractalfract6080450 - 18 Aug 2022
Viewed by 415
Abstract
In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in Rd 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 Rd 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)
Article
Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators
Fractal Fract. 2022, 6(8), 436; https://doi.org/10.3390/fractalfract6080436 - 11 Aug 2022
Viewed by 577
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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Article
On Hilfer Generalized Proportional Nabla Fractional Difference Operators
Mathematics 2022, 10(15), 2654; https://doi.org/10.3390/math10152654 - 28 Jul 2022
Viewed by 355
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)
Article
Noether and Lie Symmetry for Singular Systems Involving Mixed Derivatives
Symmetry 2022, 14(6), 1225; https://doi.org/10.3390/sym14061225 - 13 Jun 2022
Viewed by 568
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)
Article
Regularization for a Sideways Problem of the Non-Homogeneous Fractional Diffusion Equation
Fractal Fract. 2022, 6(6), 312; https://doi.org/10.3390/fractalfract6060312 - 02 Jun 2022
Viewed by 697
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)
Article
Hermite–Hadamard-Type Inequalities for h-Convex Functions Involving New Fractional Integral Operators with Exponential Kernel
Fractal Fract. 2022, 6(6), 309; https://doi.org/10.3390/fractalfract6060309 - 01 Jun 2022
Viewed by 772
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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Article
Complex Dynamic Behaviour of Food Web Model with Generalized Fractional Operator
Mathematics 2022, 10(10), 1702; https://doi.org/10.3390/math10101702 - 16 May 2022
Viewed by 695
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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Article
Left Riemann–Liouville Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales
Fractal Fract. 2022, 6(5), 268; https://doi.org/10.3390/fractalfract6050268 - 15 May 2022
Viewed by 839
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)
Article
Hadamard-Type Fractional Integro-Differential Problem: A Note on Some Asymptotic Behavior of Solutions
Fractal Fract. 2022, 6(5), 267; https://doi.org/10.3390/fractalfract6050267 - 15 May 2022
Viewed by 815
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)
Article
Laplace Transform for Solving System of Integro-Fractional Differential Equations of Volterra Type with Variable Coefficients and Multi-Time Delay
Symmetry 2022, 14(5), 984; https://doi.org/10.3390/sym14050984 - 11 May 2022
Cited by 1 | Viewed by 699
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)
Article
Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method
Fractal Fract. 2022, 6(5), 252; https://doi.org/10.3390/fractalfract6050252 - 05 May 2022
Cited by 2 | Viewed by 883
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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Article
Investigating a Generalized Fractional Quadratic Integral Equation
Fractal Fract. 2022, 6(5), 251; https://doi.org/10.3390/fractalfract6050251 - 01 May 2022
Cited by 1 | Viewed by 850
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 ζ 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)
Article
Epidemic Dynamics of a Fractional-Order SIR Weighted Network Model and Its Targeted Immunity Control
Fractal Fract. 2022, 6(5), 232; https://doi.org/10.3390/fractalfract6050232 - 22 Apr 2022
Viewed by 859
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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Article
Continuum Damage Dynamic Model Combined with Transient Elastic Equation and Heat Conduction Equation to Solve RPV Stress
Fractal Fract. 2022, 6(4), 215; https://doi.org/10.3390/fractalfract6040215 - 11 Apr 2022
Cited by 2 | Viewed by 1000
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 μ^,d^), the Young’s modulus decreases point by point, and the Poisson’s ratio increases with the increase of temperature (damage factor μ^,Et). 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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Article
Impact of Al2O3 in Electrically Conducting Mineral Oil-Based Maxwell Nanofluid: Application to the Petroleum Industry
Fractal Fract. 2022, 6(4), 180; https://doi.org/10.3390/fractalfract6040180 - 24 Mar 2022
Cited by 3 | Viewed by 1060
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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Article
Existence of Mild Solutions for Hilfer Fractional Evolution Equations with Almost Sectorial Operators
Axioms 2022, 11(4), 144; https://doi.org/10.3390/axioms11040144 - 22 Mar 2022
Cited by 6 | Viewed by 978
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)
Article
Existence and Hyers–Ulam Stability for a Multi-Term Fractional Differential Equation with Infinite Delay
Mathematics 2022, 10(7), 1013; https://doi.org/10.3390/math10071013 - 22 Mar 2022
Cited by 2 | Viewed by 755
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)
Article
Multi-Point Boundary Value Problems for (k, ϕ)-Hilfer Fractional Differential Equations and Inclusions
Axioms 2022, 11(3), 110; https://doi.org/10.3390/axioms11030110 - 02 Mar 2022
Cited by 6 | Viewed by 1399
Abstract
In this paper we initiate the study of boundary value problems for fractional differential equations and inclusions involving (k,ϕ)-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,ϕ)-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)
Article
A Numerical Study of the Fractional Order Dynamical Nonlinear Susceptible Infected and Quarantine Differential Model Using the Stochastic Numerical Approach
Fractal Fract. 2022, 6(3), 139; https://doi.org/10.3390/fractalfract6030139 - 01 Mar 2022
Cited by 14 | Viewed by 1250
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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Article
Parameter Identification and the Finite-Time Combination–Combination Synchronization of Fractional-Order Chaotic Systems with Different Structures under Multiple Stochastic Disturbances
Mathematics 2022, 10(5), 712; https://doi.org/10.3390/math10050712 - 24 Feb 2022
Cited by 2 | Viewed by 673
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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Article
Equity Warrants Pricing Formula for Uncertain Financial Market
Math. Comput. Appl. 2022, 27(2), 18; https://doi.org/10.3390/mca27020018 - 22 Feb 2022
Viewed by 1282
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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Article
Switched Fractional Order Multiagent Systems Containment Control with Event-Triggered Mechanism and Input Quantization
Fractal Fract. 2022, 6(2), 77; https://doi.org/10.3390/fractalfract6020077 - 31 Jan 2022
Cited by 1 | Viewed by 1222
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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Article
On Transformation Involving Basic Analogue to the Aleph-Function of Two Variables
Fractal Fract. 2022, 6(2), 71; https://doi.org/10.3390/fractalfract6020071 - 28 Jan 2022
Viewed by 1404
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)
Article
Overview of One-Dimensional Continuous Functions with Fractional Integral and Applications in Reinforcement Learning
Fractal Fract. 2022, 6(2), 69; https://doi.org/10.3390/fractalfract6020069 - 27 Jan 2022
Cited by 1 | Viewed by 1261
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 ho¨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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Article
Existence and Uniqueness of Mild Solution Where α ∈ (1,2) for Fuzzy Fractional Evolution Equations with Uncertainty
Fractal Fract. 2022, 6(2), 65; https://doi.org/10.3390/fractalfract6020065 - 26 Jan 2022
Cited by 9 | Viewed by 1345
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 cDαx(t)=f(t,x(t),Dβx(t)),Iαx(0)=x0,x(0)=x1, where 1<α<2,0<β<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)
Article
Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion
Mathematics 2022, 10(3), 340; https://doi.org/10.3390/math10030340 - 23 Jan 2022
Viewed by 1440
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 {BtH,t0} [...] 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 {BtH,t0} and sub-fractional Brownian motions {ξtH,t0} with Hurst parameter H(12,1). 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)
Article
Non-Instantaneous Impulsive BVPs Involving Generalized Liouville–Caputo Derivative
Mathematics 2022, 10(3), 291; https://doi.org/10.3390/math10030291 - 18 Jan 2022
Viewed by 589
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)
Article
Recursive Identification for MIMO Fractional-Order Hammerstein Model Based on AIAGS
Mathematics 2022, 10(2), 212; https://doi.org/10.3390/math10020212 - 11 Jan 2022
Viewed by 628
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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