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We study the inverse problem for a differential equation of 2b$$ 2b $$‐order with the Caputo fractional derivative over time and Schwartz‐type distribution in its right‐hand side. The generalized solution of the Cauchy problem for such an equation, space‐dependent part of a source, and a time‐dependent reaction coefficient in the equation are unknown. We find sufficient conditions for unique local in time solvability of the inverse problem under time‐ and space‐integral overdetermination conditions. [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, based on the differential quadrature method (DQM), matrix operators are derived for fractional integration and Caputo differentiation. These operators generalize the efficient DQM to fractional calculus. The proposed fractional differential/integral quadrature method (FDIQM) is used to solve various types of fractional ordinary and partial differential equations. FDIQM unifies the solution of multi‐integer fractional‐order differential equations leading to significant simplification in the implementation. Numerous examples are presented to demonstrate the accuracy of the operators. Other examples are presented to solve various fractional differential equations including time‐fractional sub‐diffusion equation, linear/nonlinear, and multiorder fractional differential equations. In addition, numerous boundary conditions are considered including mixed fractional derivatives. Further, a nonlinear fractional eigenvalue problem is solved efficiently, and its bifurcation diagrams are obtained. Comparisons between the proposed method and the existing ones are included, showing the ease of implementation, efficiency, and applicability of FDIQM. [ABSTRACT FROM AUTHOR]

Richards in [35] proposed a modification of the logistic model to model growth of biological populations. In this paper, we give a new representation (or characterization) of the solution to the Richards‐type fractional differential equation Dαy(t)=y(t)·(1+a(t)yβ(t))$$ {\mathcal{D}}^{\alpha }y(t)=y(t)\cdotp \left(1+a(t){y}^{\beta }(t)\right) $$ for t≥0$$ t\ge 0 $$, where a:[0,∞)→ℝ$$ a:\left[0,\infty \right)\to \mathrm{\mathbb{R}} $$ is a continuously differentiable function on [0,∞),α∈(0,1)$$ \left[0,\infty \right),\alpha \in \left(0,1\right) $$ and β$$ \beta $$ is a positive real constant. The obtained representation of the solution can be used effectively for computational and analytic purposes. This study improves and generalizes the results obtained on fractional logistic ordinary differential equation. [ABSTRACT FROM AUTHOR]

In the present paper, we construct the existence of nontrivial solutions to a new kind of Hadamard fractional boundary value problem on an unbounded domain. With the contribution of some fixed point theorems in cone and the corresponding Green function, we ensure sufficient conditions for the Hadamard fractional boundary value problem. Also, the paper concludes with two examples to demonstrate our results. [ABSTRACT FROM AUTHOR]

Summary Wireless sensor networks (WSNs) contain different sensors, which collect various data in the monitoring area. In general, one of the significant resources in WSNs is energy, which prolongs the network's lifetime. The energy‐efficient routing algorithms reduce energy consumption and enhance the survival cycle of WSNs. Thus, this work developed the optimization‐based WSN routing and deep learning (DL)–enabled energy prediction scheme for efficient routing in WSNs. Initially, the WSN simulation is carried out, and then, the node with minimum energy consumption is chosen as the cluster head (CH). Here, the proposed rat hawk optimization (RHO) algorithm is established for finding the best CH, and the RHO is the integration of rat swarm optimization (RSO) and fire hawk optimization (FHO). Furthermore, the routing is accomplished by the developed fractional rat hawk optimization (FRHO) using the fitness function includes delay, distance, link lifetime, and predicted energy of a network for predicting the finest route. Here, the fractional calculus (FC) is incorporated with the RHO to form the FRHO. The energy prediction is achieved by deep recurrent neural network (DRNN). The energy, delay, and throughput evaluation metrics are considered for revealing the efficiency of the proposed system, and the proposed system achieves the best results of 0.246 J, 0.190 s, and 67.13 Mbps, respectively. [ABSTRACT FROM AUTHOR]

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of n$$ n $$‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration. [ABSTRACT FROM AUTHOR]

ABSTRACT Wireless power transfer (WPT) is an emerging technology that enables the wireless transfer of electrical energy from power supplies to electrical equipment. It has been widely used in electric vehicles, mobile phones, household appliances, medical devices, and other fields. However, the frequency splitting and bifurcation phenomena existing in WPT systems are the fundamental obstacles and challenges that affect the effective operation of WPT systems. In this paper, a method based on the fractional‐order circuit is proposed to simultaneously suppress the frequency splitting and bifurcation phenomena by changing the order of fractional‐order capacitor. By replacing the compensation capacitor in the transmitter of the traditional WPT system with a fractional‐order capacitor, a fractional‐order WPT system is formed. Then, using fractional calculus and circuit theory, the mathematical model of the proposed WPT system containing a fractional‐order capacitor is established, and the frequency splitting and bifurcation phenomena are analyzed. The theoretical results show that the frequency splitting and bifurcation phenomena are suppressed only by adjusting the order of fractional‐order capacitor, and the output power of the original resonant frequency is improved. Finally, the experimental prototype is implemented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

The main innovation of this article is to establish the existence and approximate controllability of Hilfer fractional differential inclusions of order 1<μ<2$$ 1<\mu <2 $$ using Clarke's subdifferential type in Hilbert spaces. By applying the fixed‐point theorem for multivalued maps, fractional calculus, generalized Clarke's subdifferential, and cosine families, we derive a novel existence result for the mild solutions of the Hilfer fractional differential inclusion system. Furthermore, we develop and demonstrate a new set of sufficient conditions for the approximate controllability of Hilfer fractional differential systems. These criteria are based on the assumption that the associated linear part of the system is approximately controllable. Finally, an example is provided to show the theoretical findings. [ABSTRACT FROM AUTHOR]

Summary In this paper, computational analysis of a Caputo fractional variable‐order system with inductor‐capacitor (LC) and resistor‐capacitor (RC) electrical circuit models is presented. The existence and uniqueness of solutions to the given problem are determined using Schaefer's fixed point theorem and the Banach contraction principle, respectively. The proposed problem's computational consequences are addressed and analyzed using modified Euler and Runge–Kutta fourth‐order techniques. Furthermore, the suggested model compares several orders, including integer, fractional, and variable orders. To demonstrate the utility of the proposed approach, computational simulations are carried out on LC and RC circuit models of various orders. Furthermore, a comparative analysis with previous investigations has been carried. For the given problem, the numerical solution results in high‐precision approximations. [ABSTRACT FROM AUTHOR]

Fractional calculus is a method to predict processes mathematically. This study uses fractional order models to determine whether starch hydration is governed by Fickian or anomalous diffusion. Native and modified starches are compared and classified based on their diffusive characteristics and the type of diffusion observed. The study aims to adjust the equation of the analytical solution of the diffusion model to study the hydration of both native and modified starches. The fractional order diffusion model is generalized to compare the two models and identify whether anomalous mechanisms exist in native and modified starches. The results show that water absorption by native and modified starch granules is characterized by anomalous diffusion. This is due to the temperature conditions and differences in the chemical potential of the starches. It is verified that the diffusive characteristics of native and modified starches differ under the same hydration conditions. [ABSTRACT FROM AUTHOR]

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

In this article, we generalized Mittag‐Leffler‐type functions F¯A(3),F¯B(3),F¯C(3)$$ {\overline{F}}_A&#x0005E;{(3)},{\overline{F}}_B&#x0005E;{(3)},{\overline{F}}_C&#x0005E;{(3)} $$, and F¯D(3)$$ {\overline{F}}_D&#x0005E;{(3)} $$, which correspond, respectively, to the familiar Lauricella hypergeometric functions FA(3),FB(3),FC(3)$$ {F}_A&#x0005E;{(3)},{F}_B&#x0005E;{(3)},{F}_C&#x0005E;{(3)} $$, and FD(3)$$ {F}_D&#x0005E;{(3)} $$ of three variables. Initially, from the Mittag‐Leffler type function in the simplest form to the functions we are studying, necessary information about the development history, study, and importance of this and hypergeometric type functions will be introduced. Among the various properties and characteristics of these three‐variable Mittag‐Leffler‐type function F¯D(3)$$ {\overline{F}}_D&#x0005E;{(3)} $$, which we investigate in the article, include their relationships with other extensions and generalizations of the classical Mittag‐Leffler functions, their three‐dimensional convergence regions, their Euler‐type integral representations, their Laplace transforms, and their connections with the Riemann‐Liouville operators of fractional calculus. The link of three‐variable Mittag‐Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics.Therefore, we provide the systems of partial differential equations which are associated with them. [ABSTRACT FROM AUTHOR]

A new insight is presented into the solution of the erythrocyte sedimentation rate (ESR) model, based on fractional derivative with respect to time, with non‐zero uniform velocity of blood by incorporating the concentration gradient of the blood nutrients. An analytical solution is acquired for the modified ESR fractional model in addition to presenting some new interesting results. The best possible suitable range for the concentration gradient is found for the model whose use will be helpful in approximating the clinical data from laboratory tests in a profound and accurate manner and also in diagnosing the ESR rate more accurately. Further, an appropriate range is proposed for the uniform velocity of blood as well as the fractional order of the time derivative to construct the feasible model. In addition, it is also shown what value of the fractional order gives a closer resemblance to the clinical data. Validation and verification of the obtained results against earlier results demonstrate the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

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

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]

BACKGROUND: This article aims to fit kinetic models of the passion fruit seed drying process, in addition to mathematical modeling using traditional models and fractional calculus of the extraction of lipids from the seeds. RESULTS: Regarding the drying process, it was found that the model that best suited the two drying temperatures employed was the fractional order one. CONCLUSION: Analysis showed that for both drying conditions the value of parameter α was greater than 1, indicating that the profile that occurs for the seeds is superdiffusion. Regarding the mathematical fits of oil extraction, it was found that the two models that presented the best fits for the lipid extraction kinetics were hyperbolic and fractional order. Regarding the fractional order model, as the value of parameter α was lower than 1, it can be seen that the process again does not follow Fick's law, and in this case a subdiffusion process occurs. © 2024 Society of Chemical Industry (SCI). [ABSTRACT FROM AUTHOR]

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

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

This paper 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]

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]

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]

We propose a novel dynamical model for blood alcohol concentration that incorporates ψ$$ \psi $$‐Caputo fractional derivatives. Using the generalized Laplace transform technique, we successfully derive an analytic solution for both the alcohol concentration in the stomach and the alcohol concentration in the blood of an individual. These analytical formulas provide us a straightforward numerical scheme, which demonstrates the efficacy of the ψ$$ \psi $$‐Caputo derivative operator in achieving a better fit to real experimental data on blood alcohol levels available in the literature. In comparison with existing classical and fractional models found in the literature, our model outperforms them significantly. Indeed, by employing a simple yet nonstandard kernel function ψ(t)$$ \psi (t) $$, we are able to reduce the error by more than half, resulting in an impressive gain improvement of 59%. [ABSTRACT FROM AUTHOR]

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

This paper is devoted to analyzing a kind of fractional stochastic neutral system (FSNS). Firstly, by introducing the notion of newly defined two-parameter Mittag-Leffler matrix function, we derive the solution of the corresponding linear stochastic system. Subsequently, for the linear case, by virtue of the Grammian matrix, we give a suffcient and necessary condition to guarantee the relatively exact controllability for the addressed case. Furthermore, for the nonlinear one, the relatively exact controllability is obtained by fixed point and explore it via Banach contraction principle. Finally, two examples are provided to intensify our theoretical conclusions. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE) is a fundamental equation in the field of nonlinear optics and plays an important role in the study of many physical phenomena. The present study introduces a new model that demonstrates the novelty of the paper and provides the advancement of knowledge in the area of nonlinear optics by solving a challenging problem known as the extended (3 + 1)-dimensional nonlinear conformable Kudryashov's equation (CKE) with generalized anti-cubic nonlinearity, which is a generalization of the NLSE to three spatial dimension and one temporal dimension for the first time. This work is significant because it advances our understanding of nonlinear optics and its applications to solve complex equations in physics and related disciplines. The extended hyperbolic function method (EHFM) and Nucci's reduction method are applied to the extended (3 + 1)-dimensional nonlinear CKE with generalized anti-cubic nonlinearity. The equation is solved by using the concept of conformable derivative, a recently developed operator in fractional calculus, which has advantages over other fractional derivatives in terms of accuracy and flexibility. The attained solutions include periodic singular, dark 1-soliton, singular 1-soliton, and bright 1-soliton which are visualized using 3D and contour plots. This study highlights the potential of using conformable derivative and the applied techniques to solve complex nonlinear differential equations in various fields. The obtained solutions and analysis will be useful in the design and analysis of optical communication systems and other related fields. Overall, this study contributes for the understanding of the dynamics of the extended (3+1)-dimensional nonlinear CKE and offers new insights into the use of mathematical techniques to tackle complex problems in physics and related fields. [ABSTRACT FROM AUTHOR]

Based on the fractional calculus and sliding mode control (SMC) techniques, this paper presents a predefined-time synchronization scheme for fractionalorder chaotic systems (FOCSs). Firstly, a predefined-time control method is proposed for fractional-order systems. Subsequently, a novel sliding surface is presented to ensure predefined-time convergence of the synchronization error. Then, a controller is designed by combining the predefined-time stability and the SMC method to ensure that the synchronization error converges to zero within the predefined time. Finally, the feasibility and robustness of the scheme are illustrated with two numerical simulation examples. [ABSTRACT FROM AUTHOR]

We consider the relationship between the sign of the fractional difference (Δαu) (n) and the positivity or monotonicity of u. Our focus is on the case in which the fractional difference can be negative, and we show that surprisingly, (Δαu) (n) > -C, where C > 0 is a constant, can still imply that u is increasing or is positive. We also consider the setting of sequential difference operators such as Δβ. Δα for suitable choices of the parameters α and β. As is demonstrated by explicit examples, our results substantially improve some recent results in the literature and, moreover, shed light on some previous observations that heretofore were only able to be investigated by numerical simulations. We also provide applications of our results to an analysis of fractional-order initial value problems. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the approximate controllability of fractional stochastic Sobolev-type hemivariational differential systems of order r ∈ (1, 2) with sectorial operators. Firstly, by using stochastic analysis, fractional calculus, sine and cosine family operators, sectorial operators, and the fixed-point theorems of multivalued maps, we show the existence of mild solutions for the fractional stochastic evolution systems. Then, we provide a sufficient condition to guarantee the approximate controllability of the stochastic evolution systems. Next, our results cover problems involving nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study. [ABSTRACT FROM AUTHOR]

In this paper, we use the generalized notions of Riemann–Liouville (fractional calculus with respect to a regular function) to extend the definitions of fractional integration and derivative from the functional sense to the distributional sense. First, we give some properties of fractional integral and derivative for the functions infinitely differentiable with compact support. Then, we define the weak derivative, as well as the integral and derivative of a distribution with compact support, and the integral and derivative of a distribution using the convolution product. Then, we generalize those concepts from the unidimensional to the multidimensional case. Finally, we propose the definitions of some usual differential operators. [ABSTRACT FROM AUTHOR]

A European option is one of the common types of options in financial markets, which can be modeled by a time‐fractional parabolic PDE, known as the time‐fractional Black–Scholes equation (BSE). In this article, we propose an effective numerical scheme by applying Müntz–Legendre wavelets (MLW) for the solution of the given BSE. Different from classical wavelets (such as Legendre and Chebyshev), the MLW have an extra parameter representing the fractional order. Therefore, they provide more reliable results for certain fractional calculus problems. The convergence analysis of the method is provided in detail. Several test examples are given to illustrate the advantages of MLW over other classical wavelets and the high accuracy of this technique compared to existing methods in the literature. [ABSTRACT FROM AUTHOR]

In the paper, the maiden application of Tilted Integral Fractional Derivative with Filter plus Fractional Derivative controller (TIDμ1NDμ2$TI{D^{{\mu _1}}}N{D^{{\mu _2}}}$), called TIFDNFD controller, is proposed to achieve load frequency control (LFC) of two area‐multi source power system. In the study, both areas include thermal power, hydropower, and gas turbine power systems. In this study, generation rate constraint and governor dead‐band are integrated into the power system to evaluate the performance of the controller in a non‐linear environment. The wild horse optimizer (WHO) is used for controller parameter tuning in the LFC problem for the first time. The performance of the proposed controller is demonstrated by comparing it with the recently published studies including different controllers. In this study, to obtain more realistic results, solar photovoltaic (PV) power, wind power, and load demand with stochastic form have been added to the power system. In addition, the dynamic effect of the hydrogen storage system, consisting of an electrolyser and a fuel cell, is considered a grid supporter. The superiority of the WHO algorithm in optimization has been verified by comparing it with three optimization algorithms recently proposed. Finally, the resilience and robustness of the power system are evaluated in perturbed system parameters. [ABSTRACT FROM AUTHOR]

In this article, we develop a meshfree numerical solver for fractional‐order governing differential equations. More specifically, we develop a mesh‐free interpolation‐based element‐free Galerkin numerical model for the fractional‐order governing differential equations. The proposed fractional element‐free Garlekin (f‐EFG) numerical model is a lighter and more accurate alternative to existing mesh‐based finite element solvers for the fractional‐order governing differential equations. We demonstrate here that the f‐EFG with moving least squares (MLS) interpolants are naturally suitable for the approximation of fractional‐order derivatives in terms of the corresponding nodal values, thereby alleviating several issues with FE solvers for such integro‐differential governing equations. We demonstrate the efficacy of the proposed numerical model for numerical solutions with benchmark problems on the linear and nonlinear elastic response of nonlocal elastic solid modeled via fractional‐order governing differential equations. However, it must be noted that the proposed f‐EFG algorithm can be extended to fractional‐order governing differential equations in diverse applications, including multiscale and multiphysics studies. [ABSTRACT FROM AUTHOR]

In this paper, we study fractional reaction–diffusion equation using Jacobi spectral method. Reaction–diffusion equation is used as model for spatial effects in ecology. In this method, the reaction–diffusion equation is changed to the systems of equations. Convergence analysis for the proposed spectral method is established from theoretical as well as numerical point of view. Some numerical examples are solved using proposed numerical method showing the feasibility of the method. The efficiency of the spectral method is shown by listing CPU time of computation. [ABSTRACT FROM AUTHOR]

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

In this paper, a free boundary model of tumor growth with drug application including the heterogeneity or different types of tumor cells (caused by mutations and different values of drug and nutrient concentrations inside the tumor) is studied. Heterogeneity is included in the model by a variable −1≤y≤1$$ -1\le y\le 1 $$. It is assumed that converting from mutation y1$$ {y}_1 $$ to mutation y2$$ {y}_2 $$ happens with probability P(y1,y2)$$ P\left({y}_1,{y}_2\right) $$. A Caputo time fractional‐order hyperbolic equation describes the evolution of tumor cells depending on y$$ y $$. It also includes two Caputo time fractional‐order parabolic equations describing the diffusions of nutrients (e.g., oxygen and glucose) and drug concentrations. Instead of integer‐order time derivatives, the fractional ones are considered. In this study, it is aimed to employ the least squares curve fitting method to fit the order of fractional derivatives, coefficient, and rates of the model. For this purpose, using the mathematical model, we have considered the boundary densities of the tumor cells of different types and near‐boundary concentrations of drug and nutrient as the functions of the unknown orders, coefficients and rates (unknown variables). Due to the complexity of the problem, we have obtained the functions numerically. For this, using a change of variable, we have changed the free boundary problem to a problem with fixed domain. Thus, Riemann–Liouville fractional‐order integrals are added to the problem. In the spatial domain, the problem is discretized using the collocation method. In the temporal domain, the fractional derivatives and integrals (with order α$$ \alpha $$) are approximated in mesh points (with step size t∗$$ {t}&#x0005E;{\ast } $$) applying a method with error O((t∗)2−α)$$ O\left({\left({t}&#x0005E;{\ast}\right)}&#x0005E;{2-\alpha}\right) $$. Then, the unknown variables are obtained by fitting the functions for unknown variables to the data. In order to obtain the variables, a quadratic objective function is considered. The discretized function is substituted into the objective function, then the objective function is minimized using the trust‐region reflective algorithm. Finally, some numerical examples are presented to verify the efficiency of the method. In the examples, we have added noises generated from Gaussian distributions to the data and the effects of noise on the fitted coefficients, and rates are illustrated using some figures and tables. The noisy data are also plotted to have a clear vision of the effects of noises on the data. It is shown that despite the noise, the prediction of the radius of the tumor is acceptable. [ABSTRACT FROM AUTHOR]

The main focus of this article is to investigate the existence of feedback optimal control for the neutral fractional evolution systems in Hilbert spaces in the sense of the Caputo fractional derivatives. In order to establish the necessary conditions for the proposed problem, we apply the semigroup property, the fixed point theorem of multivalued maps, and the properties of generalized Clarke's subdifferentials. Then, by using the Filippov theorem and the Cesari property, a set of sufficient conditions is formulated to ensure the existence of a feasible pair for the feedback control systems. Finally, we apply our main results to obtain the optimal feedback control pair. In the end, an example is given to illustrate our theory. [ABSTRACT FROM AUTHOR]

This paper concerns the fractional derivative of the Lerch zeta function. The author already dealt with its functional equation. He reduced its computational cost and proved an approximate functional equation for this fractional derivative. Here, we study the mean square of this fractional derivative. Moreover, we deal with the distribution of zeros, showing some zero‐free regions. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to approximate the fractional calculus (FC) operator with general analytic kernel by using auxiliary newly defined linear positive operators. For this purpose, we introduce the Stancu variant of modified Bernstein–Kantorovich operators and investigate their simultaneous approximation properties. Then we construct new operators by means of these auxiliary operators, and based on the obtained results, we prove the main theorems on the approximation of the general FC operators. We also obtain some quantitative estimates for this approximation in terms of modulus of continuity and Lipschitz class functions. Additionally, we exhibit our approximation results for the well‐known FC operators such as Riemann–Liouville integral, Caputo derivative, Prabhakar integral, and Caputo–Prabhakar derivative. [ABSTRACT FROM AUTHOR]

In this paper, we construct a novel family of fractional‐type integral operators of a function f$$ f $$ by replacing sample values (f(k/n))k=0n$$ {\left(f\left(k/n\right)\right)}_{k=0}^n $$ with the fractional mean values of that function. We give some explicit formulas for higher order moments of the proposed operators and investigate some approximation properties. We also define the fractional variants of Mirakyan–Favard–Szász and Baskakov‐type operators and calculate the higher order moments of these operators. We give an explicit formula for fractional derivatives of proposed operators with the help of the Caputo‐type fractional derivative Furthermore, several graphical and numerical results are presented in detail to demonstrate the accuracy, applicability, and validity of the proposed operators. Finally, an illustrative real‐world example associated with the recent trend of Covid‐19 has been investigated to demonstrate the modeling capabilities of fractional‐type integral operators. [ABSTRACT FROM AUTHOR]

This work studies a mechanical system composed of two‐spring coupled masses arrangement without damping using fractional derivatives under the Caputo sense. We determine and implement an explicit model based on the Caputo‐Fabrizio operator. To achieve this model, we detail a systematic methodology that avoids dimensional inconsistencies. Then, we use the proposed model to explain the system dynamic acquired from an experimental (quite rudimentary) setup. We also compare our results to those achieved from the ordinary model and the numerical implementation using the Caputo derivative definition. Results prove that the proposed model describes the dynamic of this mechanical system better than the ordinary and numerical models. We also show that the fractional order allows the model (obtained from an idealized undamped system) to describe dynamics from fully dissipative to wholly oscillatory. Moreover, we discuss other exciting characteristics of the obtained responses. [ABSTRACT FROM AUTHOR]

In this paper, we systematically present some main results of the fractional calculus, almost periodic functions, fuzzy functions, fuzzy fractional calculus, and almost periodic generalized fuzzy dynamic equations on timescales. Moreover, the potential future research of almost periodic fractional fuzzy dynamic equations on timescales is discussed. The results presented in this survey can be applied to study the qualitative theory of almost periodic fractional fuzzy dynamic equations and fuzzy fractional calculus on timescales. [ABSTRACT FROM AUTHOR]

This scheme's main goal is to examine the existence, uniqueness, and continuous dependence of solutions for a nonlinear coupled system of fractional q‐integro‐differential equations involving the derivation and integration of fractional Caputo–Fabrizio. The numerical technique methodology of the proposed problem will be introduced. Proving the existence theorem depends on Schauder's fixed‐point theorem. To drive the numerical method, we use the definitions of the fractional derivative and integral of Caputo–Fabrizio and the q‐integral of the Riemann–Liouville type. Then, the integral part will be treated using the trapezoidal method, and the derivative part will be treated using the forward finite difference method. And therefore, the coupled system will be converted into a system of algebraic equation that will be solved together to get the solutions. Finally, we give two examples to illustrate the effectiveness of the suggested approach. [ABSTRACT FROM AUTHOR]

In our article, we are primarily concentrating on approximate controllability results for fractional Sobolev type Volterra‐Fredholm integro‐differential inclusions of order 1 < r < 2. By applying the results and ideas belongs to the cosine function of operators, fractional calculus and fixed point approach, the main results are established. Initially, we establish the approximate controllability of the considered fractional system, then continue to examine the system with the concept of nonlocal conditions. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

A novel approach to address the asymptotic stability problem of the decentralized formation control of multi-agent robot systems considering a cyclic pursuit formation graph is derived by using the classical Lyapunov's stability method and LaSalle's invariance principle. Theoretical development of a local potential function-based control law and the formulation of the position and formation error dynamics, from the viewpoint of the multi-variable conformable fractional-order calculus, is given. In addition, a comparison between the integer-order case and conformable fractional-order case is provided. Finally, a numerical example illustrates the performance of the developed results. [ABSTRACT FROM AUTHOR]

Steady‐state visual evoked potentials (SSVEP), significant in brain‐computer interfaces (BCI) and medical diagnostics, benefit from enhanced signal processing for improved analysis and interpretation. This study introduces a novel enhancement algorithm for SSVEP electroencephalogram (EEG) signals, employing fractional‐order differentiation operators combined with image processing techniques. Utilizing fractional‐order differentiation within a Laplace pyramid framework, the algorithm achieves hierarchical signal enhancement, facilitating detailed feature extraction and emphasizing SSVEP signal characteristics. This innovative approach merges the precision of fractional calculus with the structural benefits of the Laplace pyramid, leading to enhanced signal clarity and feature discrimination. The efficacy of this method was validated using canonical correlation analysis (CCA), filter bank CCA (FBCCA), and task‐related component analysis (TRCA) on a public dataset. Compared to conventional methods, our algorithm not only mitigates trend components in SSVEP signals but also significantly boosts the recognition accuracy of CCA, FBCCA, and TRCA algorithms. Experimental results indicate a marked improvement in recognition precision, underscoring the algorithm's potential to advance SSVEP‐based BCI research. [ABSTRACT FROM AUTHOR]

Fractional order calculus, with its inheritance and infinite memory properties, is a promising research area in information processing and modeling of certain physical systems, system identification, and control. In this paper, a fractional‐order gradient descent method is proposed for backpropagation training of multilayer feedforward neural networks. In particular, the Caputo derivative is used to define the measurement function and consider the L1/2$L_{1/2}$ smooth regular term. In addition, the monotonicity of the error function and the strong (weak) convergence theorem of the algorithm are rigorously proved. Finally, numerical experiments prove the correctness and effectiveness of the algorithm theory. [ABSTRACT FROM AUTHOR]

The Caputo operator has recently gained popularity as a widely used operator in fractional calculus. The purpose of this current research is to develop a new operator by combining the Caputo and proportional derivatives, resulting in the constant proportional Caputo (CPC) fractional operator. To demonstrate the dynamic behavior of this newly defined operator, it was applied to the unsteady Oldroyd‐B fluid model. Additionally, the research considered an Oldroyd‐B fluid in a generalized Darcy medium, considering non‐uniform temperature, radiation, and heat generation. Analytical solutions for the proposed model were obtained and presented in graphical form using the computational software MATHCAD. The impact of various physical parameters was also examined through graphical analysis of velocity and temperature profiles, as well as a comparison between isothermal and non‐uniform temperature. In conclusion, this research found that the CPC fractional operator effectively explains the dynamics of the Oldroyd‐B fluid model with stable and strong memory effects, compared to the classical model. [ABSTRACT FROM AUTHOR]