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This paper presents a new direct adaptive control (DAC) technique using Caputo's definition of the fractional-order derivative. This is the first time a fractional-order adaptive law is introduced to work together with an integer-order stable manifold for approximating the uncertainty of a class of nonlinear systems. The DAC approach uses universal function approximators such as multi-layer perceptrons with one hidden layer or fuzzy systems to approximate the controller. This paper presents a new lemma, which elucidates and clarifies the link between the Caputo and the Riemann–Liouville definitions. The introduced lemma is useful in developing a Lyapunov candidate to prove the stability of using the proposed fractional-order adaptive law. This is further explained by a numerical example, which is provided to elucidate the practicality of using the fractional-order derivative for updating the approximator parameters. The main novelty of the results in this paper is a rigorous stability proof of the fractional DAC approach for a class of nonlinear systems that is subjected to unstructured uncertainty and deals with the adaptation mechanism using a traditional integer-order stable manifold. This makes the control scheme easier to implement in practice. The fractional-order adaptation law provides greater degrees of freedom and a potentially larger functional control structure than the conventional adaptive control. Finally, the paper demonstrates that traditional integer-order DAC is a special case of the more general fractional-order DAC scheme introduced here. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of weak nonzero solutions for the following fractional Hamiltonian systems: t D ∞ α - ∞ D t α u (t) - L (t) u + λ u + ∇ W (t , u) = 0 , u ∈ H α (R , R N) , t ∈ R , where α ∈ (1 2 , 1 ] , λ ∈ R , - ∞ D t α and t D ∞ α are left and right Liouville–Weyl fractional derivatives of order α on real line R , the matrix L(t) is not necessarily coercive nor uniformly positive definite and W : R × R N → R satisfies some new general and weak conditions. Our results are proved using new symmetric mountain pass theorem established by Kajikia. Some recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the mean square locally and globally asymptotic stability problem of discrete-time fractional order stochastic neural networks(DFOSNNs) with multiple time-varying delays are investigated. Firstly, the DFOSNNs model with multiple time-varying delays are established. Then, the general framework for the mixed delays Halanay-type inequality has been established with no harsh assumption. Further, the locally and globally asymptotic stability of DFOSNNs with multiple time-varying delays in mean square sense are proved via Halanay-type inequality respectively. In addition, the sufficient condition for globally asymptotic stability of DFOSNNs with multiple time-varying delays in mean square sense is derived via Lyapunov–Krasovskii functional method. From the results obtained in this paper, the two theorems of global asymptotic stability are complement each other. Finally, three numerical examples are given to show the validity of the obtained 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 is devoted to studying a class of fuzzy fractional switched implicit differential equations (FFSIDEs) with non-instantaneous impulses that there are few papers considering this issue. Considering switching law and the memory property of fractional calculus, we first present a formula of solution for FFSIDEs with non-instantaneous impulses. Subsequently, based on a sequence of Picard functions, we explore the existence of solutions for the addressed equations by successive approximation. Furthermore, Ulam–Hyers (U–H) stability for this considered equations is derived. The main results are obtained using fuzzy-valued fractional calculus and nonlinear analysis. Finally, two numerical examples illustrating the theoretical result are given. [ABSTRACT FROM AUTHOR]

This paper investigates a technique that uses Riemann-Liouville fractional integrals to study several Euler-Maclaurin-type inequalities for various function classes. Afterwards, we provide our results by using special cases of obtained theorems and This paper is to derive examples. Moreover, we give some Euler-Maclaurin-type inequalities for bounded functions by fractional integrals. Furthermore, we construct some fractional Euler-Maclaurin-type inequalities for Lipschitzian functions. Finally, we offer some Euler-Maclaurin-type inequalities by fractional integrals of bounded variation. [ABSTRACT FROM AUTHOR]

The Bat-inspired algorithm (BA), as one of the swarm intelligence algorithms, has a high potential for solving global optimization problems. This algorithm possesses an additional inherent capability compared to other swarm intelligence algorithms, which is the inclusion of a local search mechanism. Although this increases the convergence speed of the algorithm, excessive focus on exploitation, especially in the initial iterations, may lead to premature convergence and stagnation at the local optimum. In this paper, to address this drawback and strengthen the exploration capability while also achieving a balance between it and the exploitation capability, an improved version of BA, called fractional-order BA (FOBA), is introduced. The development of velocity and position vectors in FOBA, using the concept of fractional-order derivatives, extends the memory related to the previous behaviors of artificial bats and controls the convergence of the algorithm. To evaluate the proposed algorithm, ten well-known benchmark functions are used, and the results are compared with standard and state-of-the-art metaheuristic algorithms that have been introduced recently. Experimental results show that FOBA performs better than all compared algorithms. Furthermore, the proposed algorithm is used to optimize the weights and bias of the MLP neural network using six classification datasets. The results demonstrate the effectiveness and efficiency of FOBA. [ABSTRACT FROM AUTHOR]

In the paper, the affinity of quadratic fractional functions and the gradient of pseudolinear quadratic fractional functions are characterized. [ABSTRACT FROM AUTHOR]

In his first paper on the generalization of the tautochrone problem, that was published in 1823, Niels Henrik Abel presented a complete framework for fractional-order calculus, and used the clear and appropriate notation for fractional-order integration and differentiation. [ABSTRACT FROM AUTHOR]

In this paper, we prove an interpolation inequality on Riemann–Liouville fractional integrals and then use it to study the strong stability and semi-uniform stability of fractional resolvent families of order 0 < α < 2 . Let A denote the generator of a bounded fractional resolvent family. We show that if σ (A) ∩ (i R) α is countable and σ r (A) ∩ (i R) α = ∅ , then the bounded fractional resolvent family is strongly stable. And the semi-uniform stability of the fractional resolvent family is equivalent to σ (A) ∩ (i R) α = ∅ . Moreover, the relation between decay rates of semi-uniform stability and growth of the resolvent of A along (i R) α is given. [ABSTRACT FROM AUTHOR]

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson's second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. [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]

Multi-frame image super-resolution reconstruction focuses on obtaining a high-resolution (HR) image from a low-resolution image. Since the super-resolution problem is ill-posed, it is frequent to use regularization techniques. However, the choice of these regularization terms is not always direct. Generally, every term should contain some prior knowledge over the image, that is why it plays an important role in increasing the quality of the restored HR image. In this paper, we propose a total variable-order variation based-regularization to provide a natural-looking and an effective reconstruction of the desired HR image. We also present some existence and uniqueness results of our proposed model. The alternating direction method of multipliers algorithm is employed to implement the numerical simulations. Experimental results reveal that the proposed approach can reconstruct and recover high-quality results visually and quantitatively. [ABSTRACT FROM AUTHOR]

Considering the individual differences between neurons, the fractional-order framework is introduced, and the neurons with various orders denote the individual differences during the cell differentiation. In this paper, the fractional-order FithzHugh–Nagumo (FHN) neural circuit is used to reproduce the firing patterns. In addition, an energy balance strategy is applied to determine the inter-neuronal communication. The neurons with energy imbalance exchange the information whereas the synaptic channels are blocked when energy balance is achieved. Two neurons coupled by this strategy achieve the phase synchronization and phase lock, and it indicates the two neurons generate spiking at the same time or with an interval. Similarly, the synchronization results are also obtained in the chain neuronal network, and the neurons exhibit the same firing patterns since the synchronization factor is closed to 1. Particularly, the neurons with order diversities lead to the heterogeneity and gradient field in the regular network, and the target wave is developed over time. With the wave spreading in the network, the silent states and exciting states appear in the whole network. The formation and diffusion of the target wave reveals the information transmission in neuronal network, and it indicates the individual differences paly an essential role in the collective behavior of neurons. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of mild solution for impulsive fractional differential inclusions in Banach spaces involving the Hilfer derivative. Our study is based on the nonlinear alternative of Leray–Schauder type for multivalued maps due to Martelli. An example will be added to illustrate the main result. [ABSTRACT FROM AUTHOR]

The application of fractional calculus for solving nonlinear equations through iterative techniques is an emerging area of research. In recent times, several Newton-type methods have been proposed which particularly utilize the concept of fractional order derivatives. However, convergence of such methods essentially require the existence of at least first order derivative. Accordingly, in the case where derivative is not feasible to obtain, the derivative-free methods are of much significance. In this paper, a fractional Traub-Steffensen-type method is developed, the formulation of which is based on the idea of conformable fractional derivatives of order α ∈ (0 , 1 ] . In addition, the scheme is extended to its multi-dimensional case to solve the systems of equations. The proposed derivative-free scheme is further investigated for its dynamical aspects and convergence characteristics by varying the value of α in given range. In this concern, the convergence planes are presented in a well-defined region which in general provide the fundamental information about the stability of given method. Furthermore, the numerical performance is analyzed by locating the solutions of variety of nonlinear equations, including some applied problems. [ABSTRACT FROM AUTHOR]

This paper deals with the fractional-order modeling, stability analysis and control of robotic manipulators, namely a single flexible link robotic manipulator (SFLRM) and 2DOF Serial Flexible Joint Robotic Manipulator (2DSFJ). The control law is derived using Pole Placement (PP) method. This paper uses Mittag–Leffler function for the analysis of SFLRM in the time domain. The stability analysis of the fractional model is carried in a transformed Ω -Domain, and from the analysis, it is observed that the response of the fractional model of SFLRM robotic manipulator is stable. The main motive behind this analysis is to understand the fractional behavior of robotic manipulators, and it is well known from literature that most of the real-world systems have their own fractional behavior. Furthermore, a real-time SFLRM and 2DSFJ setups are considered to validate the results obtained and it is found that the control law suggested by PP method improves the settling time of the robotic manipulators. [ABSTRACT FROM AUTHOR]

In this paper, a predefined-time fractional-order time-varying sliding mode controller is proposed for a class of second-order systems. The state errors are converged to zero at a predefined time, which can be set in advance by an explicit parameter. The stability is proved by Lyapunov second method and the predefined-time convergence is directly proved by solving the analytical formula for the state errors with squeeze theorem and mean value theorem. The sliding mode arrival phase caused by the initial state errors is eliminated in the designed controller, which has stronger robustness. Compared with the integer-order ones, the predefined-time fractional-order time-varying sliding mode controller proposed in this paper has stronger anti-interference ability. The controller is applied to the design of terminal guidance law for hypersonic vehicles. The simulation results show that the designed guidance law can ensure high precision in terms of impact angle error and miss distance under severe interference. [ABSTRACT FROM AUTHOR]

A new lumped capacitance model that employs fractional order operators is proposed for use on transient heat conduction problems. Details and implications of the fractional lumped capacitance model's development and application are discussed. The model is shown to agree with observed heating and cooling temperature profiles of laser aiming paper being heated by a laser under various conditions. [ABSTRACT FROM AUTHOR]

In this paper we provide a way to handle some symmetric fractional boundary-value problems. Indeed, first, we consider some system of fractional equations. We introduce the existence and uniqueness of solutions of the systems of equations and we show that they are entire functions of the spectral parameter. In particular, we show that the solutions are at most of order 1/2. Moreover we share the integration by parts rule for vector-valued functions that enables us to obtain some symmetric equations. These symmetries allow us to handle 2 - sequential and 4 - sequential fractional boundary-value problems. We provide some expansion formulas for the bilinear forms of the solutions of 2 - sequential and 4 - sequential fractional equations which admit us to impose some unusual boundary conditions for the solutions of fractional differential equations. We show that the systems of eigenfunctions of 2 - sequential and 4 - sequential fractional boundary value problems are complete in both energy and mean. Furthermore, we study on the zeros of solutions of 2 - sequential fractional differential equations. At the end of the paper we show that 6 - sequential fractional differential equation can also be handled as a system of equations and hence almost all the results obtained in the paper can be carried for such boundary-value problems. [ABSTRACT FROM AUTHOR]

This paper addresses some interesting results of optimal feedback control for neutral stochastic fractional systems in Hilbert spaces. We study the primary results by utilizing the theoretical concepts related to fractional calculus, fixed point theory of multivalued maps, and properties of generalized Clarke’s subdifferential type. Initially, we prove the existence of a mild solution by applying a fixed point technique. Then, we calculate the existence of feasible pair by employing the Filippov theorem and the Cesari property. Also, optimal feedback control results is developed under sufficient conditions. In addition, involving nonlocal conditions for the given control systems is discussed in a separate section. Further, we extend the given control systems with Sobolev-type and the existence of mild solution for the obtained system is evaluated by using the same fixed point theorem. In the end, a simple example is given for the effectiveness of the discussion. [ABSTRACT FROM AUTHOR]

This paper proposes an adaptive fractional-order sliding mode controller to control and stabilize a nonlinear uncertain disturbed robotic manipulator in fixed-time. Fractional calculus is used to construct a fractional-order sliding mode controller (FtNTSM) that suppresses chattering to help the robotic manipulator converge to equilibrium in a fixed-settling time based on fixed-time stability theory. Then, adaptive control is introduced and combined with FtNTSM to overcome the unknown system dynamics. The convergence time of the proposed fixed-time fractional-order sliding mode controller (AFtNTSM) is independent of beginning circumstances and can be precisely assessed, unlike the finite-time control approach. Finally, numerical simulations show that the adaptive fractional-order sliding mode controller outperforms finite-time sliding mode controller. [ABSTRACT FROM AUTHOR]

In this paper, by applying fractional quantum calculus, we study a nonlinear Duffing-type equation with three sequential fractional q-derivatives. We prove the existence and uniqueness results by using standard fixed-point theorems (Banach and Schaefer fixed-point theorems). We also discuss the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of the mentioned Duffing problem. Finally, we present an illustrative example and nice application; a Duffing-type oscillator equation with regard to our outcomes. [ABSTRACT FROM AUTHOR]

The paper is essentially motivated by the demonstrated potential for applications of the presented results in numerous widespread research areas, such as the mathematical, physical, engineering, and statistical sciences. The main object is to introduce and investigate a class of fractional integral operators involving a certain general family of multiindex Mittag-Leffler functions in their kernel. Among other results obtained in the paper, we establish several interesting expressions for the composition of well-known fractional integral and fractional derivative operators, such as (e.g.) the Riemann–Liouville fractional integral and fractional derivative operators, the Hilfer fractional derivative operator, and the above-mentioned fractional integral operator involving the general family of multiindex Mittag-Leffler functions in its kernel. Our main result is a generalization of the results obtained in earlier investigations in this field. We also present some potentially useful integral representations for the product of two members of the general family of multiindex Mittag-Leffler functions in terms of the well-known Fox–Wright hypergeometric function pΨ q with p numerator and q denominator parameters. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the optimal feedback control for a system of semilinear neutral stochastic integrodifferential equations in Hilbert space. Using the fixed-point technique, fractional calculus, and stochastic theory we obtained the existence result for the given feedback control system. The existence of feasible pairs were employed by involving Filippov theorem and the Cesari property. Moreover, optimal feedback control pairs are presented under sufficient conditions and also, we extend the discussion to the given system with nonlocal conditions. Finally, an example is given to validate our findings. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear fractional p-Laplacian boundary value problem containing both left Riemann–Liouville and right Caputo fractional derivatives with initial and integral conditions. Some new results on the existence and uniqueness of a solution for the model are obtained as well as the Ulam stability of the solutions. Two examples are provided to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a weighted fractional stochastic integro-differential equation with infinite delay and nonzero initial values involving a Riemann–Liouville fractional derivative of order 1 / 2 < α < 1 . The existence of a mild solution is investigated using fractional calculus, stochastic analysis, and the fixed point theorem. An example is also provided to illustrate the obtained result. [ABSTRACT FROM AUTHOR]

In a previous paper Guo et al. (Fractional integration on mixed norm spaces. I. Preprint, 2022), we characterized the boundedness of fractional integration operators between mixed norm spaces over the unit disk. In this paper, we characterize the boundedness between X and Y, where X , Y ∈ { H p (0 < p < ∞) , H ∞ , BMOA , B , H (p , q , α) }. As in Guo et al. (Fractional integration on mixed norm spaces. I. Preprint, 2022), we cover three types of fractional integration: Flett, Hadamard, and Riemann-Liouville, and we consider complex orders t ∈ C instead of mere t ∈ R . Our findings provide, in particular, a complete answer to a problem started by Flett in 1972 (Theorem C). [ABSTRACT FROM AUTHOR]

This paper presents (1) a new and improved fractional-order chaotic system (FOCS); (2) design of controllers for suppression of chaos for commensurate and incommensurate orders in the regions: $$0<\alpha \le 1$$ and $$1<\alpha <2$$ ; (3) synchronisation between the new FOCS (as master) and a non-identical FOCS (as slave) when the master is in either chaotic mode or chaos-suppressed mode and finally (4) analog circuit implementation to validate the numerical simulation results. The rich variety of complex nonlinear dynamical behaviour is explored by varying the parameters as well as the fractional orders. Effective control strategies are designed to suppress chaos in the new system. In addition to this, the paper contributes the idea of synchronisation of the FOCS when it switches between chaotic and chaos-suppressed mode. The control scheme put forward in the paper is able to synchronise the slave with the switching master effectively in spite of change in dynamics. The paper compares the improved features of the newly proposed FO model with a class of well-known FOCSs and elaborates its capability to enhance the performance of secure communication. Circuit implementation results for all the above are in perfect agreement with those obtained through theoretical analyses and numerical simulations and thus reveal that the objectives of the paper are successfully achieved. [ABSTRACT FROM AUTHOR]

This paper reviews the recent research in the application of fractional calculus in the models of linear viscoelasticity utilized in dynamic problems of mechanics of solids. The brief historical survey reflecting the contribution of Soviet mechanicians in the development of hereditary mechanics is provided. Different fractional derivative models of viscoelastic materials have been analyzed, among them those considering the bulk relaxation. It is shown that the models with time-dependent Poisson's operators allow one to describe the properties of viscoelastic auxetics, i.e. materials with negative Poisson's ratios. A companion article will review boundary-value dynamic problems utilizing the rheological models considered in the given paper and will provide a critical estimation of the results obtained in the field during last decade in the light of new apprehensions of the role and place of the fractional calculus in engineering practice. [ABSTRACT FROM AUTHOR]

In this paper, a general nonlinear discrete fractional boundary value problem is considered, of order between one and two. The main result is an existence theorem, proving the existence of at least one solution to the boundary value problem, subject to validity of a certain key inequality that allows unrestricted growth in the problem. The proof of this existence theorem is accomplished by using Brouwer's fixed point theorem as well as two other main results of this paper, namely, first, a result showing that the solutions of the boundary value problem are exactly the solutions to a certain equivalent integral representation, and, second, the establishment of solutions satisfying certain a priori bounds provided the key inequality holds. In order to establish the latter result, several novel discrete fractional inequalities are developed, each of them interesting in itself and of potential future use in different contexts. We illustrate the usefulness of our existence results by presenting two examples. [ABSTRACT FROM AUTHOR]

This paper concerns with the existence, uniqueness, Ulam's Hyer (UH) stability and total controllability results for the Hilfer fractional switched impulsive systems in finite-dimensional spaces. Mainly, this paper can be divided into three parts. In the first part, we examine the existence of a unique solution. In the second part, we establish the UH stability results, and in the third part, we study the total controllability results. The main outcomes are acquired by utilising the nonlinear analysis, fractional calculus, Mittag–Leffler function and Banach contraction principle. Finally, we have given two examples to validate the obtained analytical results. [ABSTRACT FROM AUTHOR]

This paper focuses on the existence and dynamic behaviors of multiple equilibria of switched fractional-order recurrent neural networks (SFRNNs) with time-varying delay. By applying the characteristics of Caputo fractional calculus and an effective state space partition method, sufficient criteria are derived to ascertain that the n-neuron SFRNN has 5 n equilibria, among which 3 n ones are locally asymptotically stable and the rest are unstable. Moreover, the multistability of integer-order neural networks as a special case is also taken into consideration in this paper. Two illustrative examples are provided to substantiate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving particular values to the parameters involved in this special function, this leads to some known special functions (as the classical Wright function, the α-Mittag-Leffler function, the Tricomi function, etc.) that on their turn appear as cases of the so-called multi-index Mittag-Leffler functions. As an application, we mention that this new generalization Wright function nis an isochronous solution of a nonlinear fractional partial differential equation. [ABSTRACT FROM AUTHOR]

The two-dimensional (2D) Otsu algorithm used in image segmentation considers both the gray scale information of the image and the spatial information contained in the neighborhood of pixels. The algorithm is quite effective, but a number of variations have been proposed to improve its performance. In this paper, a new adaptive fractional-order (FO) genetic-particle swarm optimization (FOGPSO) version is proposed. The FOGPSO associates the particle selection operations of a genetic algorithm (GA) and particle swarm optimization (PSO). Crossover and genetic mutation are used to avoid PSO falling into a local optimum. Fractional calculus operators are adopted in the updating scheme of velocity and position, while the order of the derivative is adaptively changed according to the state of the particles. Compared with the original PSO, the integer PSO, the fractional PSO (FOPSO), other improved versions of the PSO for Otsu algorithms, and other existing methods for 2D Otsu algorithms, the proposed method shows great superiority. Indeed, experimental results reveal that both qualitatively and quantitatively, through suitable indices, as the regional contrast, the intersection over union (IOU) and peak signal to noise ratio (PSNR), the FOGPSO outperforms the other methods, thus verifying the effectiveness of the new algorithm. [ABSTRACT FROM AUTHOR]

Artificial neural network (ANN) is the backbone of machine learning, specifically deep learning. The interpolating and learning ability of an ANN makes it an ideal tool for modelling, control and various other complex tasks. Fractional calculus (FC) involving derivatives and integrals of arbitrary non-integer order has recently been popular for its capability to model memory-type systems. There have been many attempts to explore the possibilities of combining these two fields, the most popular combination being the use of fractional derivative in the learning algorithm. This paper reviews the use of fractional calculus in various artificial neural network architectures, such as radial basis functions, recurrent neural networks, backpropagation NNs, and convolutional neural networks. These ANNs are popularly known as fractional-order artificial neural networks (FANNs). A detailed review of the various concepts related to FANNs, including activation functions, training algorithms based on fractional derivative, stability, synchronization, hardware implementations of FANNs, and real-world applications of FANNs, is presented. The study also highlights the advantage of combining fractional derivatives with ANN, the impact of fractional derivative order on performance indices like mean square error, the time required for training and testing FANN, stability, and synchronization in FANN. The survey reports interesting observations: combining FC to an ANN endows it with the memory feature; Caputo definition of fractional derivative is the most commonly used in FANNs; fractional derivative-based activation functions in ANN provide additional adjustable hyperparameters to the networks; the FANN has more degree of freedom for adjusting parameters compared to an ordinary ANN; use of multiple types of activation functions can be employed in FANN, and many more. [ABSTRACT FROM AUTHOR]

This paper proves an equality for the case of differentiable convex functions including the conformable fractional integrals. By using this equality, we establish several parameterized inequalities with the help of the conformable fractional integrals. Several inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Furthermore, we present previously achieved results and new results by using special cases of the obtained theorems. [ABSTRACT FROM AUTHOR]

In this EPJ Special Topics issue selected contributions on some recent developments in the theory and applications of fractional dynamical systems are presented. Results on topics involving fundamental theory, qualitative theory, iterative methods and numerous applications of fractional-order equations are reported. [ABSTRACT FROM AUTHOR]

Fractional calculus has always been regarded as an ideal mathematical tool to describe the memory of complex systems and special materials. A fractional-order memristor-based chaotic system with hidden dynamics is studied in this paper. The system can exhibit excellent dynamic behavior by introducing a quadratic nonlinear memristor. Asymmetric coexistence occurs when both the order and parameter change. Considering the practical application of fractional-order system, the spectral entropy (SE) algorithm is used to investigate the complexity of the system. Besides, synchronous experiment between two fractional-order system is carried out and the synchronization circuit is also designed. To verify the numerical simulation results, the hardware circuit is constructed, and the hidden attractors are successfully captured on the oscilloscope by hardware electronic circuit. [ABSTRACT FROM AUTHOR]

The Bénard problem consists in a system that couples the well-known Navier–Stokes equations and an advection-diffusion equation. In thin varying domains this leads to the g-Bénard problem, which turns out to be the classical Bénard problem when g is constant. The main goal of this paper is to, first of all, introduce the g-Bénard problem with time-fractional derivative of order α ∈ (0 , 1) . This formulation is new even in the classical Bénard problem, that is with constant g. The second goal of this paper is to prove the existence and uniqueness of a weak solution by means of the Faedo–Galerkin approximation method. Some recent works on time-fractional Navier–Stokes equations have opened new perspectives in studying variational aspects in problems involving time-fractional derivatives. [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]

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]

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]

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]

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]

This paper investigates the event-triggered impulsive synchronization of a variable-order fractional chaotic Lur'e system with the application of fractal interpolation based image encryption. The variable-order fractional Lur'e system is modeled into fractional order Chua's circuit model with short memory. The synchronization is achieved using event-triggered impulsive control and short memory for chaotic Lur'e systems, employing linear matrix inequalities. The variable-order fractional Chua's circuit systems successfully accomplish their synchronization using both impulsive and event-triggered impulsive control. Furthermore, fractal interpolation function (with function scalings) is applied to reconstruct the chaotic attractors. The Riemann-Liouville variable-order fractional calculus of diverse fractal functions is also investigated in connection with the study on variable-order fractional systems. The interpolated sequence, along with the Arnold transform, modified Fisher Yates algorithm, and additive diffusion, is utilized to propose an image encryption algorithm. Various statistical analyses are conducted to assess the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Medical image segmentation technology, as a key work of modern medical such as intelligent medical diagnosis, has attracted a lot of attention. However, the gray heterogeneity of medical images makes accurate target segmentation of " focus " greatly difficult. In recent years, fractional calculus has attracted much attention in the field of image processing because of its good nonlinear processing characteristics. In this paper, a medical image segmentation model based on Caputo fractional differentia, Caputo CV model, is proposed. The Caputo fractional differential with weak singularity is introduced into the model, and the feasibility of the model is proved theoretically. At the same time, according to the characteristics of high-order Caputo fractional differential and low-order Caputo fractional differential, in the numerical calculation of the model, the fitting term of the model is processed by the high-order Caputo fractional differential mask operator of order in interval (2,3), and the gradient in the approximation term of the model is discretized by the low-order Caputo fractional differential operator of order in interval (0,1). By doing this, the model can effectively maintain the nonlinear characteristics of image low-frequency information while enhancing the high-frequency information in the iteration process, so that make the evolution process of the model more stable and effective. A large number of experiments show that the proposed model can effectively improve the accuracy of gray inhomogeneous target segmentation and greatly reduce the running time. [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]