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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]

This article examines a new family of (p,q)‐Laplacian type nonlinear fractional impulsive differential coupled equations involving both the ψ$$ \psi $$‐Caputo fractional derivative and ψ$$ \psi $$–Riemann–Liouville fractional integral. With the help of Nehari manifold in critical point theory and fractional calculus properties, we obtain the existence of at least one nontrivial ground state solution for the coupled system with some natural and easily verifiable superlinear conditions on the nonlinearity. [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]

Analytical solution of initial value problem for the fractional oscillation equation with two Caputo fractional derivatives x′′(t)+aDtαx(t)+cx′(t)+bDtβx(t)+kx(t)=q(t)$$ {x}&#x0005E;{\prime \prime }(t)&#x0002B;a{D}_t&#x0005E;{\alpha }x(t)&#x0002B;c{x}&#x0005E;{\prime }(t)&#x0002B;b{D}_t&#x0005E;{\beta }x(t)&#x0002B; kx(t)&#x0003D;q(t) $$, where the coefficients and orders satisfy a,b,c,k>0,1<α≤2$$ a,b,c,k>0,1<\alpha \le 2 $$ and 0<β≤1$$ 0<\beta \le 1 $$, is investigated by using the Laplace transform and complex inverse integral method on the principal Riemann surface. It is proved by using the argument principle that the characteristic equation has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface under the assumption that α$$ \alpha $$ and β$$ \beta $$ are not both integers. Then three fundamental solutions, the unit impulse response, the unit initial displacement response, and the unit initial rate response, are derived analytically. Each of these solutions is expressed into a superposition of a classical damped oscillation decaying exponentially and a real Laplace integration decaying in a negative power law. Finally, the asymptotic behaviors of these analytical solutions for sufficiently large t$$ t $$ are determined as monotonous decays in a power of negative exponent. [ABSTRACT FROM AUTHOR]

Malicious traffic segregation and attack detection caused major financial loss and became one of the most serious security hazards. Moreover, cyber security attack is the major issue, which impacts network security. The network attack methods are constantly being upgraded by the technology development and it remains a major issue for detection and protection against network attacks. For this, it is required to present an effective strategy for detecting and maintaining network security. The work provides timely and accurate congestion attack detection and identification. In the Internet of Things (IoT) cloud system malicious traffic segregation and attack detection based on a hybrid optimization-enabled deep learning (DL) network is developed in this research. At first, the input log files are gathered from the simulation of IoT sensors and the superior route is selected by the proposed Fractional Hunger Jellyfish Search Optimization (FHGJO) algorithm. The FHGJO is the integration of Hunger Game Jelly Fish Optimization (HGJO) and Fractional Calculus (FC). Furthermore, the HGJO is the combination of Hunger Game Search Optimization (HGS) with Jellyfish Optimization (JSO). Then, the segregation is done based on the fitness measures and for preprocessing; the input data is fed using quantile normalization. The feature selection process is employed using the weighted Euclidian distance (WED). With the SpinalNet, the malicious segregation is categorized as malicious and non-malicious and the proposed FHJGO is used to tune the SpinalNet. Furthermore, the proposed FHGJO-trained Deep Quantum Neural Network (DQNN) is utilized to detect the attack and classifies it into a Denial-of-Service (DOS) attack, Distributed Denial of Service (DDoS) attack, and buffer overflow attack. Moreover, the proposed model is evaluated using the NSL-KDD dataset and BoT-IoT dataset. The proposed method ensures network security with 0.931 accuracy, 0.923 sensitivity, and 0.936 specificity. [ABSTRACT FROM AUTHOR]

Summary Electric vehicles (EVs) are the emerging environmentally friendly approach that is used to minimize greenhouse gases and carbon dioxide (CO2) emissions in the atmosphere. A clear strategy is required for scheduling charging stations (CS) to EVs based on their applications. In this research work, a robust routing and effective charge scheduling approach are devised using a cloud‐assisted Vehicular Ad Hoc NETwork (VANET) for charging EVs. Here, the multi‐objectives, like predicted traffic density, battery power, and distance, are used to identify the optimal routing of EV to CS. The predicted traffic density is evaluated using Deep Long Short Term Memory (DLSTM) and is trained using a developed Jaya Election‐Based Optimization Algorithm (JEBOA), which is the incorporation of Jaya Optimization (Jaya) and Election‐Based Optimization Algorithm (EBOA). Next to optimal routing, the charge scheduling process is carried out using the Fractional Jaya Election‐Based Optimization Algorithm (Fractional JEBOA) by considering the priority, response time, and latency of the EV. The designed Fractional JEBOA is the integration of Fractional Calculus (FC) and the developed JEBOA. Moreover, the various evaluation metrics are considered to calculate the performance of the designed method, which attained a delay of 0.243 ms, distance of 35 km, power of 95 W, response time of 0.441 s and traffic density of 0.664. [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]

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 work, we introduce a novel idea of generalized (k , ψ) -Hilfer proportional fractional operators. The proposed operator combines the (k , ψ) -Riemann–Liouville and (k , ψ) -Caputo proportional fractional operators. Some properties and auxiliary results of the proposed operators are investigated. The ψ-Laplace transform and its properties of the proposed operators are established and utilized to solve Cauchy-type problems. Furthermore, the uniqueness result for a higher-order initial value problem under (k , ψ) -Hilfer proportional fractional operators is proved by using Picard's iterative technique. At the end, examples are provided to present the theoretical results. This new type of proposed operator can help other researchers who are still working on real-world problems. [ABSTRACT FROM AUTHOR]

To accurately investigate the nonlinear dynamic characteristics of a flyback converter, a fractional‐order state‐space averaged model of a flyback converter in continuous conduction mode (CCM) is established based on fractional calculus theory. And nonlinear dynamical bifurcation maps which use reference current as bifurcation parameters are obtained. The period‐doubling bifurcation is analyzed and compared with that of an integral‐order flyback converter. The results show that under certain operating conditions, the fractional‐order flyback converter exhibits period‐doubling bifurcation as certain circuit and control parameters change. Under the same circuit conditions, there is a difference in the stable parameter region between the fractional‐ and integral‐order models of the flyback converter. The stable zone of the fractional‐order flyback converter is larger than that of the integral‐order one. Therefore, the circuit is more difficult to enter the state of the period‐doubling bifurcation and chaos. The stability domain of period‐doubling bifurcation can be accurately predicted by using the predictive correction model of the fractional‐order flyback converter. Finally, by performing circuit simulations and hardware‐in‐the‐loop experiments, the rationality and correctness of the theoretical analysis are verified. [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, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equation where fractional derivatives in the sense of Caputo and Riemann–Liouville are applied. Our method extends the recently proposed fractional reduced differential transform method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady‐state solutions or spatial blow‐ups of the fractional Helmholtz equation in a better way. More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady‐solutions and when spatial blow‐ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting. Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann–Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel–Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems. [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 work, we proposed some generalized Darbo type fixed point theorem by employing the concept of generalized operators and measure of non-compactness within the framework of partial order Banach spaces. Some fixed point theorems and their consequences are presented as part of our exploration. These theorems encompass the presentation of various fixed point theorems and the subsequent implications they entail. Further, we demonstrate the practical application of our findings by establishing the existence of solutions for delay differential equations. To substantiate our conclusions, we provide numerical estimations based on a real-world example. This research contributes to the advancement of fixed point theory in the context of partial order Banach spaces and showcases its practical relevance through a concrete application and empirical support. [ABSTRACT FROM AUTHOR]

A new fractional strain tensor ϵ α (u) of order α (0 < α < 1 ) is introduced for a displacement u of a body occupying the entire three-dimensional space. For α ↑ 1 , the fractional strain tensor approaches the classical infinitesimal strain tensor of the linear elasticity. It is shown that ϵ α (u) satisfies Korn's inequality (in a general L p version, 1 < p < ∞ ) and the fractional analog of Saint-Venant's compatibility condition. The strain ϵ α (u) is then used to formulate a three-dimensional fractional linear elasticity theory. The equilibrium of the body in an external force f is determined by the Euler-Lagrange equation of the total energy functional. The solution u is given by Green's function G α : u (x) = ∫ R n G α (x − y) f (y) d y , x ∈ R 3. For an isotropic body the equilibrium equation reads − μ (− Δ) α u + (λ + μ) ∇ α div α u + f = 0 where λ , μ are the Lamé moduli of the material and (− Δ) α , ∇ α and div α are the fractional laplacean, gradient and divergence. Green's function can be determined explicitly in this case: G α (x) = c α μ | x | 3 − 2 α (I − λ + μ 2 α (λ + 2 μ) (I + (2 α − 3) | x | − 2 x ⊗ x)) , x ∈ R 3 , x ≠ 0 , where I is the identity tensor (matrix), and c α a normalization factor (determined below). For α ↑ 1 the function G α approaches Green's function of the standard linear elasticity. Similar approach applies to the equilibrium solution. [ABSTRACT FROM AUTHOR]

Our research is about the Sturm-Liouville equation which contains conformable fractional derivatives of order α - (0,1) in lieu of the ordinary derivatives. First, we present the eigenvalues, eigenfunctions, and nodal points, and the properties of nodal points are used for the reconstruction of an integral equation. Then, the Bernstein technique was utilized to solve the inverse problem, and the approximation of solving this problem was calculated. Finally, the numerical examples were introduced to explain the results. Moreover, the analogy of this technique is shown in a numerical example with the Chebyshev interpolation technique. [ABSTRACT FROM AUTHOR]

This paper presents the development of a series of fractional multi-step linear finite difference methods (FLMMs) designed to address fractional multi-delay pantograph differential equations of order 0 < α ≤ 1. These p-FLMMs are constructed using fractional backward differentiation formulas of first and second orders, thereby facilitating the numerical solution of fractional differential equations. Notably, we employ accurate approximations for the delayed components of the equation, guaranteeing the retention of stability and convergence characteristics in the proposed p-FLMMs. To substantiate our theoretical findings, we offer numerical examples that corroborate the efficacy and reliability of our approach. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalized delta q-Mittag-Leffler function. Also, we solve some Caputo delta q-fractional dynamic equations and these solutions are expressed by means of the newly introduced delta q-Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

The objectives of this study are to develop the SEIR model for COVID-19 and evaluate its main parameters such as therapeutic vaccines, vaccination rate, and effectiveness of prophylactic. Global and local stability of the model and numerical simulation are examined. The local stability of equilibrium points was classified. A Lyapunov function is constructed to analyze the global stability of the disease-free equilibrium. The simulation part is based on two situations, including the USA and Iran. Our results provide a good contribution to the current research on this topic. [ABSTRACT FROM AUTHOR]

This paper explored the finite-time stability (FTS) of fractional-order fuzzy inertial neural network with mixed delays. First, the dimension of the model was reduced by the order reduction method. Second, by leveraging the fractional-order finite-time stability theorem, fractional calculus and inequality methods, we established some sufficient conditions to guarantee the FTS of the model under feasible delay-dependent feedback controller and delay-dependent adaptive controller, respectively. Additionally, we derived the settling times (STs) for each control strategy. Finally, we provided two examples to substantiate our findings. [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]

In this study, we employ the Laplace Variational Iterational Method (LVIM) as a sophisticated mathematical tool to investigate into the complex dynamics of cancer cells under the influence of chemotherapy. The LVIM, a method combining Laplace transformations and variational iteration techniques, is specifically adapted to address a system of time fractional differential equations (FDEs) that characterizes the temporal behavior of cancer cells. To enhance the efficacy of our approach, we introduce a semi-analytic version of LVIM, which proves to be a powerful and versatile tool for solving mathematical problems involving fractional derivatives. The focus of our analysis centers on elucidating the impact of chemotherapy, with a particular emphasis on drug diffusion within cancer cells and the fractality of DNA walks. Through numerical exploration encompassing varying fractional order derivatives, our study exposes nonlinear behaviors that remain secret in systems featuring only integer order derivatives. Notably, the methodology we propose is not only applicable to the specific cases examined in this research but also exhibits broad versatility, making it suitable for exploring the effects of different drugs and types of cancers. This research contributes valuable insights into the dynamics of cancer cells, aiding in the understanding of the implications for therapeutic strategies in the context of cancer treatment. [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]

We consider the initial value problem for an ordinary differential equation with distributed order differentiation operator regarded as the integral of the Riemann–Liouville fractional derivative with respect to the order of differentiation over a Borel measure with compact support. We construct the fundamental solution to the equation in terms of the Wright function with distributed parameters and obtain the general representation for solutions to this equation. We prove the existence and uniqueness theorem for the Cauchy problem and express the solution in an explicit form. [ABSTRACT FROM AUTHOR]

Fractional calculus serves as a versatile and potent tool for the modeling and control of intricate systems. This discussion debates the system of DFDEs with two regimes; theoretically and numerically. For theoretical analysis, we have established the EUE by leveraging the definition of Hilfer (α,β)-framework. Our investigation involved the examination of the possessions of the FRD, FCD, and FHD, utilizing their forcefulness and qualifications to convert the concerning delay system into an equivalent one of fractional DVIEs. By employing the CMT, we have successfully demonstrated the prescribed requirements. For numerical analysis, the Galerkin algorithm was implemented by leveraging OSLPs as a base function. This algorithm allows us to estimate the solution to the concerning system by transforming it into a series of algebraic equations. By employing the software MATHEMATICA 11, we have effortlessly demonstrated the requirements estimation of the nodal values. One of the key advantages of the deployed algorithm is its ability to achieve accurate results with fewer iterations compared to alternative methods. To validate the effectiveness and precision of our analysis, we conducted a comprehensive evaluation through various linear and nonlinear numerical applications. The results of these tests, accompanied by figures and tables, further support the superiority of our algorithm. Finally, an analysis of the numerical algorithm employed was provided along with insightful suggestions for potential future research directions. [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 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]

Scaling fractional-order memristors (fracmemristors) are a new research topic. Although remarkable progress has been made in the study of such memristors, there are some problems that require further study, for example, the effect of input signals on the instantaneous valid frequency range, failure to obtain the corresponding time-domain electrical characteristic expression of the fracmemristor, and the requirement of circuit schematics for using numerous memristor emulators. Most memristors in string scaling fracmemristor (SSF) circuit configurations must be of the floating-type. Therefore, the design and implementation of the circuit schematic are challenging. Motivated by this need, a SSF with a fixed valid frequency range was implemented using a simple circuit in this study. First, the circuit configuration of the SSF with a fixed valid frequency range and high approximation benefit was proposed. Subsequently, a simple circuit schematic of an incremental and decremental SSF was designed, and the corresponding time-domain electrical characteristic expression of the fracmemristor was obtained. Finally, rich analysis results for the electrical characteristics of the SSF were obtained. A circuit implementation of the SSF was used to verify the applicability of the theory. The main contributions of this study are as follows: the SSF circuit configuration and its simple circuit schematic with a fixed valid frequency range and high approximation benefit were devised, and the corresponding expressions of the time-domain electrical characteristics and the analysis results for the electrical characteristics were obtained. [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]

Stroke is a disease that is caused due to the blockage and burst in the blood vessels of the brain, thus resulting in abrupt brain dysfunction, like sensory or motor disorders, unconsciousness, limb paralysis, and pronunciation disorders. The existing stroke prediction algorithms have some limitations because of the lengthy testing procedures and hefty testing expenses. The main goal of this study is to develop and implement the proposed fusion-based, optimized deep learning model for stroke disease prediction using multimodalities. For that, this research considers the Computed Tomography (CT) and electroencephalogram (EEG) signals as input, and all of these inputs are processed separately to predict the stroke disease. While predicting the stroke disease with a CT image, the bilateral filter performs the pre-processing and the disease prediction is done with the DenseNet model, which is tuned by the proposed Jaya Fractional Reptile Search Algorithm (Jaya FRSA). Similar to how the proposed FRSA does CNN-LSTM training, the Convolutional Neural Network-Long Short Term Memory (CNN-LSTM) predicts the stroke disease using the EEG data as an input after the Gaussian filter removes signal noise. Additionally, the CT image and the EEG signal are processed independently from the image and signal properties. Additionally, the CNN-LSTM model and DenseNet model results are combined using the overlap coefficient to get the final disease prediction. According to the experimental study, the suggested method achieved the maximum image accuracy, sensitivity, and specificity of 0.924, 0.930, and 0.935. [ABSTRACT FROM AUTHOR]

The Hadamard fractional derivative and integral are important parts of fractional calculus which have been widely used in engineering, biology, neural networks, control theory, and so on. In addition, the periodic boundary conditions are an important class of symmetric two-point boundary conditions for differential equations and have wide applications. Therefore, this article considers a class of nonlinear Hadamard fractional coupling (p 1 , p 2) -Laplacian systems with periodic boundary value conditions. Based on nonlinear analysis methods and the contraction mapping principle, we obtain some new and easily verifiable sufficient criteria for the existence and uniqueness of solutions to this system. Moreover, we further discuss the generalized Ulam–Hyers (GUH) stability of this problem by using some inequality techniques. Finally, three examples and simulations explain the correctness and availability of our main results. [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]

To accurately investigate the nonlinear dynamic characteristics of a forward converter, a fractional-order state-space averaged model of a forward converter in continuous conduction mode (CCM) is established based on the fractional calculus theory. And nonlinear dynamical bifurcation maps which use PI controller parameters and a reference current as bifurcation parameters are obtained. The nonlinear dynamic behavior is analyzed and compared with that of an integral-order forward converter. The results show that under certain operating conditions, the fractional-order forward converter exhibits bifurcations characterized by low-frequency oscillations and period-doubling as certain circuit and control parameters change. Under the same circuit conditions, there is a difference in the stable parameter region between the fractional and integral-order models of the forward converter. The stable zone of the fractional-order forward converter is larger than that of the integral-order one. Therefore, the circuit struggles to enter states of bifurcation and chaos. The stability domain for low-frequency oscillations and period-doubling bifurcations can be accurately predicted by using a small signal model and a predictive correction model of the fractional-order forward converter, respectively. Finally, by performing circuit simulations and hardware-in-the-loop experiments, the rationality and correctness of the theoretical analysis are verified. [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 work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo's fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The hybrid approach considered in this work combines the advantages of both the Caputo and Hadamard fractional derivatives, leading to a more comprehensive and versatile model for describing sequential processes. To address the problem of the existence and uniqueness of solutions for such hybrid fractional sequential differential equations, we turn to Darbo's fixed-point theorem, a powerful mathematical tool that establishes the existence of fixed points for certain types of mappings. By appropriately transforming the differential equation into an equivalent fixed-point formulation, we can exploit the properties of Darbo's theorem to analyze the solutions' existence and uniqueness. The outcomes of this research expand the understanding of HCHFSDEs and contribute to the growing body of knowledge in fractional calculus and fixed-point theory. These findings are expected to have significant implications in various scientific and engineering applications, where sequential processes are prevalent, such as in physics, biology, finance, and control theory. [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]

In this study, we investigate the application of fractional calculus to the mathematical modeling of biological systems, focusing on fractional-order-in-time partial differential equations (FTPDEs). Fractional derivatives, especially those defined in the Caputo sense, provide a useful tool for modeling memory and hereditary characteristics, which are problems that are frequently faced with integer-order models. We use the Chebyshev spectral approach for spatial derivatives, which is known for its faster convergence rate, in conjunction with the L 1 scheme for time-fractional derivatives because of its high accuracy and robustness in handling nonlocal effects. A detailed theoretical analysis, followed by a number of numerical experiments, is performed to confirmed the theoretical justification. Our simulation results show that our numerical technique significantly improves the convergence rates, effectively tackles computing difficulties, and provides a realistic simulation of biological population dynamics. [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]

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]

In this study, we consider model equations K(m,n) with fractional Caputo time derivatives. By applying the Lie group symmetry method, we determine all symmetries for these equations and present the reduced symmetric equations for the equation K(m,n) with fractional Caputo time derivatives. Furthermore, we obtain the exact solution for K(1,1) with the fractional Caputo time derivative and provide graphs depicting the behavior at different orders of the fractional time derivative. Additionally, by considering the symmetries of the equation, we establish the conservation laws for K(m,m) with the fractional Caputo time derivative. [ABSTRACT FROM AUTHOR]

A mathematical model of thermoelectric MHD theory has been developed in the context of a study of heat conduction using a new fractal form of Green–Naghdi theory without energy dissipation (FGN-II). Theories of coupled (CT) and Green–Naghdi of type II (GN-II) thermoelectric fluid follow as limit cases. This model is applied to different problems of thermoelectric fluid bounded by an infinite plane surface with constant temperature in the presence of a uniform magnetic field. State space approach utilized for one-dimensional heat source situations. Techniques based on the Laplace transform are employed. The resulting formulation is applied to a problem for the entire space with heat sources distributed in a plane. Reflection approach tackles semi-space problem with heat source distribution and entire space solution. The numerical technique is used to achieve the Laplace transform inversion. The comparisons in the figures help assess how the fractional order parameter effects on the behavior of the field quantities in the new theory. [ABSTRACT FROM AUTHOR]