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This article is aimed at quantifying the health condition of the power transformer insulation system through its frequency-domain dielectric characteristic by using the concept of the resistor and fractional calculus. In this regard, the dielectric model of the fractional Poynting–Thomson (PT) is proposed, and a mathematical model for parameter identification of the proposed model is presented. The shuffled frog leaping algorithm (SFLA) is used to realize the parameter identification and the fractional PT model parameters of the oil-paper samples under different insulation health conditions. The influence of moisture and aging on the fractional PT dielectric model parameters is experimentally studied. Results show that, with the increase in the water content within the insulation system, the parameters of the fractional PT model gradually decrease, and there is a strong correlation between the relaxation time and the water content. This facilitates the employment of the fractional PT model parameters to quantify the aging and moisture content of the power transformer insulation system. [ABSTRACT FROM AUTHOR]

This paper explores the Apéry-like series and demonstrates the derivation of closed-form expressions using fractional calculus. We consider a variety of Apéry-like functions, which were categorized by their functional forms and coefficients by applying the Riemann–Liouville fractional integral and derivative to examine their properties across various domains. The study focuses on establishing rigorous mathematical frameworks that unveil new insights into the behaviors of these series, contributing to a deeper understanding of number theory and mathematical analysis. Key results include proofs of convergence and divergence within specified intervals and the derivation of closed-form solutions through fractional integration and differentiation. This paper also introduces a method aimed at conjecturing mathematical constants through continued fractions as an application of our results. Finally, we provide the proof of validation for three unproven conjectures of continued fractions obtained from the Ramanujan Machine. [ABSTRACT FROM AUTHOR]

The continuous Hopfield network (CHN) is a common recurrent neural network. The CHN tool can be used to solve a number of ranking and optimization problems, where the equilibrium states of the ordinary differential equation (ODE) related to the CHN give the solution to any given problem. Because of the non-local characteristic of the "infinite memory" effect, fractional-order (FO) systems have been proved to describe more accurately the behavior of real dynamical systems, compared to the model's ODE. In this paper, a fractional-order variant of a Hopfield neural network is introduced to solve a Quadratic Knap Sac Problem (QKSP), namely the fractional CHN (FRAC-CHN). Firstly, the system is integrated with the quadratic method for fractional-order equations whose trajectories have shown erratic paths and jumps to other basin attractions. To avoid these drawbacks, a new algorithm for obtaining an equilibrium point for a CHN is introduced in this paper, namely the optimal fractional CHN (OPT-FRAC-CHN). This is a variable time-step method that converges to a good local minima in just a few iterations. Compared with the non-variable time-stepping CHN method, the optimal time-stepping CHN method (OPT-CHN) and the FRAC-CHN method, the OPT-FRAC-CHN method, produce the best local minima for random CHN instances and for the optimal feeding problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider and study in detail the generalized Fox–Wright function Ψ ˜ q p introduced in our recent work as an extension of the Fox–Wright function Ψ q p . This special function can be seen as an important case of the so-called I-functions of Rathie and H ¯ -functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, Ψ ˜ q p is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of Ψ ˜ q p are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys "Going Next after 'A Guide to Special Functions in Fractional Calculus': A Discussion Survey", and "A Guide to Special Functions of Fractional Calculus", published recently in this journal. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain R + n with positive coordinates. Due to the generality of the Fox H-function, many special integral transforms have the form studied in this paper, including operators with such kernels as generalized hypergeometric functions, classical hypergeometric functions, Bessel and modified Bessel functions and so on. Moreover, most important fractional integral operators, such as the Riemann–Liouville type, are covered by the class under consideration. The mapping properties in Lebesgue-weighted spaces, such as the boundedness, the range and the representations of the considered transformation, are established. In special cases, it is applied to the specific integral transforms mentioned above. We use a modern technique based on the extensive use of the Mellin transform and its properties. Moreover, we generalize our own previous results from the one-dimensional case to the multi-dimensional one. The multi-dimensional case is more complex and needs more delicate techniques. [ABSTRACT FROM AUTHOR]

This paper considers a nonlinear fractional-order boundary value problem   H D a , g α 1 , β , μ x (t) + f (t , x (t) , H D a , g α 2 , β , μ x (t)) = 0 , for t ∈ [ a , b ] , α 1 ∈ (1 , 2 ] , α 2 ∈ (0 , 1 ] , β ∈ [ 0 , 1 ] with appropriate integral boundary conditions on the Hölder spaces. Here, f is a real-valued function that satisfies the Hölder condition, and   H D a , g α , β , μ represents the tempered-Hilfer fractional derivative of order α > 0 with parameter μ ∈ R + and type β ∈ [ 0 , 1 ] . The corresponding integral problem is introduced in the study of this issue. This paper addresses a fundamental issue in the field, namely the circumstances under which differential and integral problems are equivalent. This approach enables the study of differential problems using integral operators. In order to achieve this, tempered fractional calculus and the equivalence problem of the studied problems are introduced and studied. The selection of an appropriate function space is of fundamental importance. This paper investigates the applicability of these operators on Hölder spaces and provides a comprehensive rationale for this choice. [ABSTRACT FROM AUTHOR]

The document is an editorial conclusion for a special issue on "New Theory and Applications of Nonlinear Analysis, Fractional Calculus, and Optimization." It highlights the widespread and significant applications of nonlinear analysis in various fields of mathematics and science. The special issue includes 10 research papers that cover topics such as optimality conditions, fixed point results, determinantal expressions, pricing of credit risk derivatives, pseudo-asymptotically periodic solutions, common fixed points, local unique solvability, finite-time stability, summation and integral representations, and the existence of solutions. The authors express their appreciation to the contributors and reviewers and hope that the papers inspire future research in these areas. [Extracted from the article]

The oil‐paper compound insulation plays a vital role in the insulation structure of power transformer. To obtain the characteristics of the transformer's insulation system, it is of great importance to study the dielectric response of oil‐paper compound insulation. In this study, fractional calculus is applied to model the oil‐paper compound insulation. Both low‐frequency and broadband high‐frequency models are proposed and then they are verified by fitting the measured data of different insulation papers. Finally they are applied to two different cases: (i) the low‐frequency model for fitting polarity reversal property; (ii) the broadband high‐frequency model for studying the influence of the parameters variations on the frequency domain spectroscopy (FDS). The results demonstrate the advantages of the proposed models when compared with traditional ones. [ABSTRACT FROM AUTHOR]

Several real-life problems that involve a time delay are modeled using fractional time-delay systems. However, most studies related to these systems assume that the delay is already known, which is not the case in practical scenarios where the delay is often uncertain or unknown. To address this issue, this paper proposes an algebraic and robust method to estimate the input delay for a class of fractional time-delay systems in a noisy environment, by applying a cross-convolution approach. Besides, a filtering methodology is incorporated with the proposed approach to enhance its efficacy. In addition, this paper presents novel theories on convolution in the field of fractional calculus. Finally, the performance of the proposed approach is demonstrated by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper. [ABSTRACT FROM AUTHOR]

This paper establishes a linear relationship between the order of the Riemann–Liouville fractional calculus and the exponent of the Hölder condition, whether the Hölder condition is global, local, or at a single point. We propose and prove a control inequality between the Hölder derivative ( H f (x , α) as defined in Proposition 12) of a continuous function and the Hölder derivative of the Riemann–Liouville fractional calculus of this function. In addition, this paper provides a more accurate estimation of the Box dimension of the graph of the Riemann–Liouville fractional integral of an arbitrary continuous function. More specifically, it establishes the result that whenever there is a continuous function whose graph has the upper Box dimension s with 1 < s ≤ 2 , the graph of its Riemann–Liouville fractional integral of order ν , with 0 < ν < 1 , has the upper Box dimension not greater than s − (s − 1) ν. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach to obtaining analytical solutions for fuzzy fractional differential equations in the context of weighted Caputo-type generalized Hukuhara derivatives. The paper proposes the use of non-singular kernels to improve the accuracy of fractional calculus in fuzzy space and establishes the uniqueness of solutions for fuzzy fractional differential equations. The paper also introduces the concept of fuzzy Laplace transforms to facilitate the solution of these equations. Practical examples, such as the fuzzy fractional Newton’s law of heating and cooling, are provided to demonstrate the effectiveness of the proposed method. Overall, this paper contributes to the development of practical solutions for real-world problems in fuzzy space and enhances the accuracy of fractional calculus in this context. [ABSTRACT FROM AUTHOR]

One of the modern, recently developed mathematical approaches for modeling various complex chaotic processes (the bacteria migration is apparently one of them), is the application of fractional differential equations. Introduction of fractional derivatives is also a very effective approach for investigation of the reactive processes (growth of bacteria in our case). Our recent advances in application of fractional differential equations for modeling the anomalous transport of reactive and non-reactive contaminants (see our recent publications in the References) allow us to expect that the anomalous transport of growing bacteria can also be effectively described by the models with fractional derivatives. Based on these modern approaches, utilizing fractional differential equations, in this paper we developed a reliable mathematical model that could be properly calibrated and, consequently, provide an adequate description of the growing bacteria transport. This model accounts for the memory effects in the bacteria transport due to the random character of bacteria trapping and release by the porous matrix. Two types of bacteria in the saturated porous medium are considered: mobile and immobile bacteria. Bacteria in the mobile phase are migrating in the fluid and have the velocity of the bulk flow, whereas bacteria in the immobile phase are the bacteria that are captured by the porous matrix. These bacteria have zero velocity and can cause clogging of some pores (therefore, porosity is possibly not constant). Examining different conventional models and comparing computations based on these models, we show that this extremely complex character of bacteria transport cannot be described by the traditional approach based on classical partial differential equations. In this paper we suggest fractional differential equations as a simple but very effective tool that can be used for constructing the proper model capable of simulating all the above-mentioned effects associated with migration of alive bacteria. Using this approach, a reliable model of the growing bacteria transport in the porous medium is developed and validated by comparison with experimental laboratory results. We proved that this novel model can be properly linearized and calibrated, so that an excellent agreement with available experimental results can be achieved. This simple model can be used in many applications, for example, as a part of more general mathematical models for predicting the outcomes of the bioremediation of contaminated soils. [ABSTRACT FROM AUTHOR]

Asymmetry plays a significant role in the transmission dynamics in novel discrete fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed discrete models that incorporate different symmetry developmental stages. This paper introduces a Taylor monomial falling function and presents some properties of this function in a delta fractional model with Green's function kernel. In the deterministic case, Green's function will be non-negative, and this shows that the function has an upper bound for its maximum point. More precisely, in this paper, based on the properties of the Taylor monomial falling function, we investigate Lyapunov-type inequalities for a delta fractional boundary value problem of Riemann–Liouville type. [ABSTRACT FROM AUTHOR]

The normalized fractional derivatives (NFDs) are a normalization of the existing fractional derivatives (FDs) which admit geometrical meanings. They have finite ordinary derivatives at the initial point, which cause smoothness of solutions for related fractional differential equations (FDEs). For FDs with non-singular kernels, the NFDs do not vanish at the starting point, in general, and therefore related FDEs admit solutions without the need to impose extra conditions. In this paper, we consider FDEs with the normalized Caputo–Fabrizio derivative of Caputo type. We show in closed forms the solutions of related FDEs and show that the Cauchy problem with the NFD admits a nontrivial solution. We also define the higher-order NFDs and show that related FDEs can be solved by transforming them to integro-differential equations with integer orders. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of multiplicity of positive solutions to a new generalized Riemann–Liouville type fractional Fisher‐like equation on a semi‐infinite interval equipped with nonlocal multipoint boundary conditions involving Riemann–Liouville fractional derivative and integral operators. The existence of at least two positive solutions for the given problem is established by using the concept of complete continuity and iterative positive solutions. We show the existence of at least three positive solutions to the problem at hand by applying the generalized Leggett–Williams fixed‐point theorem due to Bai and Ge [Z. Bai, B. Ge, Existence of three positive solutions for some second‐order boundary value problems, Comput. Math. Appl. 48 (2014) 699‐70]. Illustrative examples are constructed to demonstrate the effectiveness of the main results. It has also been indicated in Section 5 that some new results appear as special cases by choosing the parameters involved in the given problem appropriately. [ABSTRACT FROM AUTHOR]

The human eye is a fascinating optical system, with the crystalline lens playing a significant role in focusing light onto the retina of the eye. The ray tracing through the crystalline lens problem is a challenging problem in optics. In this paper, the case of a non-homogeneous optical medium is investigated, and the ray equation is numerically solved to get the ray paths. The governing equation is an ODE with a fractional derivative given in the Caputo sense. A novel numerical scheme is based on the Jacobi polynomial collocation technique to tackle this problem. A fast and accurate Broyden's Quasi-Newton algorithm is applied to solve the nonlinear system of equations obtained from the collocation process. Numerical results are stated in detail to show the efficiency of our technique and are compared with other analytical and numerical methods using tables and illustrated figures, which will be useful to corroborate the clinical and physical data. Ray tracing through the crystalline lens is not only fascinating from a scientific perspective but also has practical implications across various domains, and the proposed scheme is considered a promising and practically reliable method to address such types of applications. [ABSTRACT FROM AUTHOR]

This article discusses changes in heat transfer resulting from laser pulses and magnetic fields produced in thermoelastic materials under initial stress. This paper introduces a novel approach to modeling generalized thermoelastic materials by including fractional time derivatives with Eringen's non‐local thermoelastic theory. This model incorporates both the Caputo–Fabrizio and the Atangana–Baleanu derivatives, which are novel forms of fractional derivatives in the domain of fractional calculus. Analytical formulations for system variables, such as temperature and thermal stress, were derived using the Laplace transform method. This was done considering the effects of laser pulses, non‐local actuators, and fractional actuators. The findings of these investigations are showcased through numerical illustrations and visual representations. The research also included comparisons between the acquired results and those derived from earlier theories, which may be regarded as a specific instance. Validating the suggested model and showing its correctness and applicability is seen as a crucial step. [ABSTRACT FROM AUTHOR]

In this paper, an analytical approach has been developed for electromagnetic field enhancement in fractal nanostructure under the quasi-static limit. A metallic fractal embedded in an unbounded, isotropic, and lossless chiral medium has been treated under this approach. In addition, the field enhancement is also noted by imposing surface plasmon resonance condition. To achieve the desired objective, Maxwell divergence equations are considered and written in an appropriate form by mean of fractional calculus. [ABSTRACT FROM AUTHOR]

Unstable first order time delay systems are frequently encountered in industrial applications, such as chemical plants, hydraulic processes, in satellite communications or economic systems, to name just a few. The control of such processes is a challenging issue. In this paper a filtered Smith Predictor control structure is used to compensate for the process time delays and to ensure the stability of the overall closed loop system. A simplified type of a fractional order PI controller is then designed to meet zero steady state error and an overshoot requirement. The tuning is based on the root locus analysis of the closed loop fractional order system. Simulation examples are provided to validate the proposed method and to demonstrate the efficiency of the proposed control method. Comparisons with two existing methods are included to highlight the possibility of using the proposed method as an alternative solution for controlling these types of processes. [ABSTRACT FROM AUTHOR]

Summary: Fractional‐order calculus is an extension of the integer‐order calculus in that its order is not limited to integers but can be noninteger as well, thus making the application of fractional‐order calculus more flexible. It has been shown that actual reactive elements such as capacitors and inductors have fractional‐order properties and that the theory of fractional‐order calculus can be utilized to more accurately characterize circuits and systems containing reactive elements. However, there is no uniform definition of fractional order calculus. Different definitions will yield different results in solving the same fractional‐order circuits and systems, and it is worth exploring whether these differences have an impact on the accuracy of solving fractional‐order systems. Taking a fractional‐order boost converter running in discontinuous conduction mode (DCM) as an example, this paper focuses on the differences between the Riemann–Liouville (R–L) and the Caputo fractional calculus definitions. Effects of the two definitions on the division of the converter's CCM (continuous conduction mode) and DCM operating regions are analyzed, as well as their impact on the characterization of a fractional‐order boost converter in DCM operation, including the DC (direct current) operating point, current ripple, and voltage gain. Simulations and experiments show that the analytical results based on the R–L definition can more accurately describe the operation of a real fractional‐order converter than those based on the Caputo definition. [ABSTRACT FROM AUTHOR]

In this paper, optimal estimates on the decaying rates of Jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the range of the L p -space under the Riemann-Liouville fractional integral. The connection between these new spaces and classical fractional-order Sobolev spaces is then elucidated. Under this framework, the optimal decaying rate of Jacobi expansion coefficients is obtained, based on which the projection errors under different norms are given. This work is expected to introduce fractional calculus into traditional fields in approximation theory and to explore the possibility in solving classical problems by this ‘new’ tool. [ABSTRACT FROM AUTHOR]

This article explores integral inequalities within the framework of local fractional calculus, focusing on the class of generalized α$$ \alpha $$‐convex functions. It introduces a novel extension of the Hermite‐Hadamard inequality and derives numerous fractal inequalities through a novel multiparameterized identity. The primary aim is to generalize existing inequalities, highlighting that previously established results can be obtained by setting specific parameters within the main inequalities. The validity of the derived results is demonstrated through an illustrative example, accompanied by 2D and 3D graphical representations. Lastly, the paper discusses potential practical applications of these findings. [ABSTRACT FROM AUTHOR]

This paper investigates the cluster synchronization of fractional‐order complex networks. Considering that impulsive control can reduce the update of controller, and the appearance of impulse is always dependent on each node in the networks instead of appearing at fixed instant, thus we design a variable‐time impulsive controller to control the considered networks. Foremost, several assumptions are proposed to guarantee the every solution of coupled error networks intersect each discontinuous impulsive surface exactly once. In addition, by utilizing the B‐equivalence method and the theory of fractional calculus, the variable‐time impulsive fractional‐order system is reduced to a fixed‐time impulsive fractional‐order system, which can be regarded as the comparison system of the former. Next, under the framework of 1‐norm, some sufficient conditions are presented to ensure that fractional‐order system and target trajectory ultimately achieve cluster synchronization. In the end, a numerical example is designed to illustrate the validity and feasibility of theoretical results. [ABSTRACT FROM AUTHOR]

In designing control systems, it is known that fractional-order proportional integral derivative (FOPID) controllers often provide greater flexibility than conventional proportional integral derivative (PID) controllers. This higher level of flexibility has proven to be extremely valuable for various applications such as vibration suppression in structural engineering. In this paper, we study the optimization of FOPID controllers using twelve well-established algorithms to minimize structural responses under seismic excitations. The algorithms include crystal structure algorithm (CryStAl), stochastic paint optimizer, particle swarm optimization, krill herd, harmony search, ant colony optimization, genetic algorithm, grey wolf optimizer, Harris hawks optimization, sparrow search algorithm, hippopotamus optimization algorithm, and duck swarm algorithm. In addition to highlighting the benefits of fractional calculus in structural control, this study provides a detailed analysis of FOPID controllers as well as a brief description of the algorithms used to optimize them. To evaluate the efficiency of the proposed techniques, two building models with different numbers of stories are examined. FOPID controllers are designed based on oustaloup's approximation and the El Centro earthquake data. Using five well-known metrics, the performances of the developed methods are evaluated against five earthquake scenarios, including the recent earthquake in Turkey. A non-parametric (Friedman) test is also employed to compare the algorithms based on their corresponding vibration reduction. The findings of this analysis show that CryStAl consistently performs better than the other algorithms for both building models, thus resulting in superior vibration suppression. [ABSTRACT FROM AUTHOR]

This paper considers containment control (CC) of delayed nonlinear fractional‐order multi‐agent systems (FMASs). A reliable and effective method is proposed to deal with the difficulties brought from fractional calculus and delays. Utilizing improved Razumikhin method, algebraic graph theory, and matrix analysis technique, several simple algebraic conditions are provided to solve CC by choosing nondelayed or delayed communication protocols, respectively. Numerical results are also offered to clarify the efficiency of our theoretical method. [ABSTRACT FROM AUTHOR]

A type of general fractional calculus with respect to another function is introduced in this paper. The kernel function within the fractional derivative and integral is specified as the Exponential function in a typical way. Based on the excellent property of the newly defined general fractional calculus, we proposed two viscoelastic models in the fractional manner compared with the classical calculus. [ABSTRACT FROM AUTHOR]

Caputo Fractional Derivative is a base for the Riemann-Liouville Derivative, Katugampola fractional derivative, and other fractional derivatives. In this paper, we have illustrated several critical features of the Caputo Derivative via Hilbert Transformation that have not yet been discussed. We have also investigated the behavior of the Riemann-Liouville integral and studied the results of the k-Riemann-Liouville fractional integral with the applications of Hilbert transformation. [ABSTRACT FROM AUTHOR]

Fractional uncertain differential equations are used for forecast in this paper. It is crucial to provide an accurate estimation method since fractional uncertain differential equations hold memory effects and parameter estimation strongly depends on the accuracy of numerical schemes. A generalized moment estimation method is developed by the famous L 1 numerical scheme. Adam optimization algorithm is adopted in the minimum optimization problem and the obtained optimal parameters can pass the uncertain hypothesis test. Compared with the uncertain differential equations, the fractional one holds more parameter freedom degrees and captures long-term interactions of observed data. Finally, the applications to aftershock frequency and pharmacokinetics are provided as examples. It can be concluded that fractional uncertain differential equations lead to the higher forecast accuracy which also provides some potential works on "fractional deep learning" under uncertain environments. [ABSTRACT FROM AUTHOR]

In this paper, the time-resolved model of the photoacoustic signal for samples with a complex inner structure is derived including local non-equilibrium of structural elements with multiple degrees of freedom, i.e., structural entropy of the system. The local non-equilibrium is taken into account through the fractional operator. By analyzing the model for two types of time-dependent excitation, a very short pulse and a very long pulse, it is shown that the rates of non-equilibrium relaxations in complex samples can be measured by applying the derived model and time-domain measurements. Limitations of the model and further directions of its development are discussed. [ABSTRACT FROM AUTHOR]

This paper aims to examine an approach that studies many Euler–Maclaurin-type inequalities for various function classes applying Riemann–Liouville fractional integrals. Afterwards, our results are provided by using special cases of obtained theorems and examples. Moreover, several Euler–Maclaurin-type inequalities are presented for bounded functions by fractional integrals. Some fractional Euler–Maclaurin-type inequalities are established for Lipschitzian functions. Finally, several Euler–Maclaurin-type inequalities are constructed by fractional integrals of bounded variation. [ABSTRACT FROM AUTHOR]

Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the limitations of traditional tools. Fractional calculus is a valuable tool for these issues, with applications ranging from membrane diffusion to electrical response of complex fluids, particularly electrolytic cells like liquid crystal cells. This paper presents the main fractional tools to formulate a diffusive model regarding time-fractional derivatives and modify the continuity equations stating the conservation laws. We explore two possible ways to introduce time-fractional derivatives to extend the continuity equations to the field of arbitrary-order derivatives. This investigation is essential, because while the mathematical description of neutral particle diffusion has been extensively covered by various authors, a comprehensive treatment of the problem for electrically charged particles remains in its early stages. For this reason, after presenting the appropriate mathematical tools based on fractional calculus, we demonstrate that generalizing the diffusion equation leads to a generalized definition of the displacement current. This modification has strong implications in defining the electrical impedance of electrolytic cells but, more importantly, in the formulation of the Maxwell equations in material systems. [ABSTRACT FROM AUTHOR]

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the C 0 algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map's states in commensurate and incommensurate cases. [ABSTRACT FROM AUTHOR]

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie's modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots. [ABSTRACT FROM AUTHOR]

This paper is devoted to introducing a new q-fractional calculus in the framework of Atangana–Baleanu (풜ℬ) and Caputo–Fabrizio (풞ℱ) operators. First, an appropriate q-Mittag-Leffler function is defined, and then q-analogues of fractional derivatives of Atangana–Baleanu–Riemann (풜ℬℛ) and Atangana–Baleanu–Caputo (풜ℬ풞) are derived. Next, the q-analogues of proper fractional integrals in the 풜ℬ sense are proved. Several important properties of these definitions are investigated by using the q-Laplace transform. Additionally, a suitable q-exponential function is defined, and the q-analogues of 풞ℱ fractional derivatives with their inverse operators are introduced. The higher-order extension of the q-analogues of 풜ℬ and 풞ℱ fractional operators is discussed. Finally, a demonstrative example is enhanced to check the effectiveness of q-풜ℬ풞 calculus. We believe that these outcomes will be the care of many researchers in the field of fractional calculus. [ABSTRACT FROM AUTHOR]

The objective of this paper is investigating the numerical solution and optical phenomena of highly nonlinear fractional-order partial differential equation namely fractional order Lakshmanan–Porsezian–Daniel model, incorporating spatio-temporal dispersion effects. The fractional derivative is considered in Caputo sense because this is the most crucial tool for working with integer order models in a fractional sense under the right subsidiary conditions, it has many benefits. To achieve the objective of this study, the Improved Adomian decomposition method has been utilized. This method is known for effectively handling nonlinearities and fractional derivatives iteratively. The obtained soliton solutions are presented graphically. We conduct a comparative analysis at ρ = 1 , focusing on the bright optical soliton solutions. To the best of our knowledge, there have not been previously reported identical or related findings. By comparing the obtained results with the exact solution, the accuracy of the considered method is examined. The applied method is explicit, efficient, and user-friendly for managing a wider range of fractional order nonlinear models, as demonstrated by the numerical outcomes. [ABSTRACT FROM AUTHOR]

In this paper, the Caputo fractional derivative is assumed to be the prey–predator model. In order to create Caputo fractional differential equations for the prey–predator model, a discretization process is first used. The fixed points of the model are categorized topologically. We identify requirements for the fixed points of the suggested prey–predator model's local asymptotic stability. We demonstrate analytically that, under specific parametric conditions, a fractional order prey–predator model supports both a Neimark–Sacker (NS) bifurcation and a Flip bifurcation. We present evidence for NS and Flip bifurcations using central manifold and bifurcation theory. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order prey–predator model. As the bifurcation parameter is increased, the system displays chaotic behavior. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. The suggested prey–predator dynamical system's chaotic behavior will be controlled by the OGY and hybrid control methodology, which will also visualize the chaotic state for various biological parameters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions and analyze the large-time behavior for Gurtin–Maccamy population model involving conformable fractional derivatives. As a preliminary step, we construct a generic structure of the solution associated with our proposed model by utilizing some basic properties and tools of conformable fractional calculus. We establish the existence of a unique solution of the given model with the given initial conditions. At last, by using the upper and lower solutions for the characteristic equation, we define the upper and lower boundaries for the obtained solution and describe the large-time behavior of the total population. [ABSTRACT FROM AUTHOR]

In this paper, a fractional adaptive backstepping sliding mode control system based on the Caputo-Fabrizio derivative for a 2-DoF mobile robot manipulator is designed and described. The fractional dynamics of the system are obtained using the Atangana-Baleanu derivative and the Euler-LaGrange formalism, which include external disturbances, parametric uncertainties, and nonholonomic constraints. A fractional sliding mode control strategy is designed for trajectory tracking tasks. In order to compensate for the chattering phenomenon, the proposed controller is combined with a fractional backstepping strategy. Additionally, the Caputo-Fabrizio derivative is introduced to further reduce the chattering effects and improve the driver performance. A fractional adaptive control law is used to cope with the parametric uncertainties, while adding robustness to external disturbances. To further improve system performance, a tracking control strategy based on a fractional neural network is added. For a comparative analysis, the conventional adaptive neural network backstepping sliding mode control is implemented. Numerical simulations are described and discussed to validate the effectiveness of our control system for trajectory tracking tasks under different operating conditions such as trajectory changes and external disturbances. Our proposed control system and its traditional counterpart were tuned using the Cuckoo optimization approach. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the existence and stability of a solution to the time fractional parameters Sc,Ec,Ic,Ac,Tc,R$$ \left({S}_c,{E}_c,{I}_c,{A}_c,{T}_c,R\right) $$ of Atangana–Baleanu fractional ordered 2019‐nCoV model. For the existence of the solution, we derive Darbo‐type fixed‐point theorems and its corollaries for (α,β)$$ \left(\alpha, \beta \right) $$‐admissible and ψ$$ \psi $$‐condensing operator in the setting of modular space. Moreover, the stability analysis and behavior of solution are analyzed by the numerical technique governed by Toufik and Atangana [1]. [ABSTRACT FROM AUTHOR]

This article extends the celebrated Riemann–Hilbert (RH) method equipped with mixed spectrum to a new integrable system of three-component coupled time-varying coefficient complex mKdV equations (ccmKdVEs for short) generated by the mixed spectral equations (msEs). Firstly, the ccmKdVEs and the msEs for generating the ccmKdVEs are proposed. Then, based on the msEs, a solvable RH problem related to the ccmKdVEs is constructed. By using the constructed RH problem with mixed spectrum, scattering data for the recovery of potential formulae are further determined. In the case of reflectionless coefficients, explicit N-soliton solutions of the ccmKdVEs are ultimately obtained. Taking N equal to 1 and 2 as examples, this paper reveals that the spatiotemporal solution structures with time-varying nonlinear dynamic characteristics localized in the ccmKdVEs is attributed to the multiple selectivity of mixed spectrum and time-varying coefficients. In addition, to further highlight the application of our work in fractional calculus, by appropriately selecting these time-varying coefficients, the ccmKdVEs are transformed into a conformable time-fractional order system of three-component coupled complex mKdV equations. Based on the obtained one-soliton solutions, a set of initial values are assigned to the transformed fractional order system, and the N-th iteration formulae of approximate solutions for this fractional order system are derived through the variational iteration method (VIM). [ABSTRACT FROM AUTHOR]

For a class of fractional-order singular multi-agent systems (FOSMASs) with local Lipschitz nonlinearity, this paper proposes a closed-loop D α -type iterative learning formation control law via input sharing to achieve the stable formation of FOSMASs in a finite time. Firstly, the formation control issue of FOSMASs with local Lipschitz nonlinearity under the fixed communication topology (FCT) is transformed into the consensus tracking control scenario. Secondly, by virtue of utilizing the characteristics of fractional calculus and the generalized Gronwall inequality, sufficient conditions for the convergence of formation error are given. Then, drawing upon the FCT, the iteration-varying switching communication topology is considered and examined. Ultimately, the validity of the D α -type learning method is showcased through two numerical cases. [ABSTRACT FROM AUTHOR]

This paper introduces and explores the dynamics of a novel three-dimensional (3D) fractional map with hidden dynamics. The map is constructed through the integration of a discrete sinusoidal memristive into a discrete Duffing map. Moreover, a mathematical operator, namely, a fractional variable-order Caputo-like difference operator, is employed to establish the fractional form of the map with short memory. The numerical simulation results highlight its excellent dynamical behavior, revealing that the addition of the piecewise fractional order makes the memristive-based Duffing map even more chaotic. It is characterized by distinct features, including the absence of an equilibrium point and the presence of multiple hidden chaotic attractors. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of the SIR infectious disease model, with a specific emphasis on utilizing a harmonic mean-type incidence rate. It thoroughly analyzes the model's equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. A sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V-L) matrices, is developed to examine the overall trajectory of the model globally. In addition to that, we describe the transmission of infectious disease through a mathematical model using fractal-fractional differential operators. We prove the existence and uniqueness of solutions in the SIR model framework with a harmonic mean-type incidence rate by using the Banach contraction approach. Functional analysis is used together with the Ulam–Hyers (UH) stability approach to perform stability analysis. We simulate the numerical results by using a computational scheme with the help of MATLAB. This study advances our knowledge of the dynamics of epidemic dissemination and facilitates the development of disease prevention and mitigation tactics. [ABSTRACT FROM AUTHOR]

The two-parameter Mittag–Leffler function E α , β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When α > 1 , the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for α ∈ (0 , 1) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for E α , β (− t) , t ≥ 0 , with α ∈ (1 , 2) . This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants. [ABSTRACT FROM AUTHOR]

This paper considers a nonlinear impulsive fractional boundary value problem, which involves a ψ -Caputo-type fractional derivative and integral. Combining critical point theory and fractional calculus properties, such as the semigroup laws, and relationships between the fractional integration and differentiation, new multiplicity results of infinitely many solutions are established depending on some simple algebraic conditions. Finally, examples are also presented, which show that Caputo-type fractional models can be more accurate by selecting different kernels for the fractional integral and derivative. [ABSTRACT FROM AUTHOR]

In this article, we focus on the application of fractional differential calculus in image processing, specifically in the context of image decoding techniques. The article presents different methods of image decoding based on Riemann–Liouville fractional integration and compares their performance with established methods in the denoising of noisy images. Here we create new fractional masks by using the Middle Point approach method, the Grumwald–Letnikov scheme, the Toufik–Riemann scheme, and the Euler method with the help of the Riemann–Liouville fractional integral operator. The results demonstrate that the suggested algorithms outperform the compared techniques. The paper also suggests that the concepts presented can be applied to other definitions of fractional calculus. The future research goal is to determine the optimal fractional order for a given set of algorithms. [ABSTRACT FROM AUTHOR]

Image enhancement is a vital process that serves as a tool for improving the quality of a lot of real-life applications. Fractional calculus can be utilized in enhancing images using fractional order kernels, adding more controllability to the system, due to the flexible choice of the fractional order parameter, which adds extra degrees of freedom. The proposed system merges two fractional order kernels which helps in image enhancement techniques, and the contribution of this work is based on the study of how to optimize this process. The optimization of the two fractional kernels was done using the neural network optimization algorithm (NNA) to utilize the best order for the two kernels. In this paper, three fractional kernels are studied to highlight the performance of image enhancement using fractional kernels against different metrics. Furthermore, three different combinations of two kernels are combined and studied to enhance the metrics score by utilizing two different fractional orders for each kernel. Various optimization algorithms are used to obtain the optimum fractional order for both single and combined kernels. Using the constrained NNA, the evaluation metrics of the image enhancement show a 33% increase in measure of enhancement metric (EME), 21% increase in contrast, and 4% increase in average gradient compared to the best-achieved metrics by the literature while keeping the similarity metric above 0.75. [ABSTRACT FROM AUTHOR]

Recently, many scholars have discovered that fractional calculus possess infinite memory and can better reflect the memory characteristics of neurons. Therefore, this paper studies the Hopf bifurcation of a fractional‐order network with short‐cut connections structure and self‐delay feedback. Firstly, we use the Laplace transform to obtain the characteristic equation of the model, which is the transcendental equation containing four times transcendental item. Secondly, by selecting the communication delay as the bifurcation parameter and the other delay as the constant in its stability interval, the conditions for the occurrence of Hopf bifurcation are established; the bifurcation diagrams are provided to ensure that the derived bifurcation findings are accurate. Thirdly, in the case of identical neurons, the crossing curves method is exploited to the fractional‐order functional function equation to extract the Hopf bifurcation curve. Finally, two numerical examples are employed to confirm the efficiency of the developed theoretical outcomes. [ABSTRACT FROM AUTHOR]