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The study of this article deals with the controllability results for time‐varying fractional dynamical systems in terms of Caputo‐type fractional derivatives having a prescribed or predetermined control. We demonstrate the controllability results for time‐varying linear fractional dynamical systems using Gramian technique and fractional calculus. Additionally, we explore the controllability results for semi‐linear and nonlinear fractional dynamical systems through the fixed point techniques. Several numerical examples are illustrated to validate the theoretical results. [ABSTRACT FROM AUTHOR]

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]

Zika is a flavivirus that is transmitted to humans either through the bites of infected Aedes mosquitoes or through sexual transmission. Zika has been associated with congenital anomalies, like microcephalus. We developed and analyzed the fractional‐order Zika virus model in this paper, considering the vector transmission route with human influence. The model consists of four compartments: susceptible individuals are x1(t), infected individuals are x2(t), x3(t) shows susceptible mosquitos, and x4(t) shows the infected mosquitos. The fractional parameter is used to develop the system of complex nonlinear differential equations by using Caputo and Atangana–Baleanu derivative. The stability analysis as well as qualitative analysis of the fractional‐order model has been made and verify the non‐negative unique solution. Finally, numerical simulations of the model with Caputo and Atangana Baleanu are discussed to present the graphical results for different fractional‐order values as well as for the classical model. A comparison has been made to check the accuracy and effectiveness of the developed technique for our obtained results. This investigative research leads to the latest information sector included in the evolution of the Zika virus with the application of fractional analysis in population dynamics. [ABSTRACT FROM AUTHOR]

The dynamic electromechanical coupling behavior of composite materials is highly dependent on external excitation frequency. While degradable biomolecular materials typically exhibit lower piezoelectric coefficients compared to ceramics, neglecting their frequency‐dependent performance in the design of piezoelectric devices further leads to less efficient utilization of their piezoelectric properties. This oversight greatly hinders the practical application of these materials. To address this, a novel fractional derivation (FD) theory‐assisted model is introduced to reversely design the glycine‐polyvinyl alcohol (PVA) thin films for versatile enhanced bio‐applications. An electromechanical coupling model incorporating FD theory is developed to learn the relationships between FD parameters, film dimensions, and dynamic electromechanical properties. This model accurately predicts the electromechanical performance of the films across a wide frequency range, validated by both finite element simulations and experimental results. This therefore allows to establish key design principles for piezoelectric thin film in bioenergy harvesting and sensing, by tailoring thin film parameters to enhance the piezoelectric performance at specific stimuli frequencies. Demonstrations of glycine‐PVA film devices guided by this model reveal excellent performance in ultrasonic energy harvesting and carotid artery bio‐signal sensing. This study provides a robust theoretical framework for designing and optimizing biodegradable piezoelectric materials for various practical applications. [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 this paper, we propose an anisotropic diffusion equation based on fractional‐order tensor diffusion applied on noisy images. We start by investigating the existence and uniqueness results of the proposed model. We also provide a discretization scheme based on the finite difference method. In order to prove the efficacy of our proposed approach, we deliver some simulations, which show robustness and remarkable results against noise while preserving edges and corners, compared to some well‐known models in literature. For a fair comparison, we manage to obtain the optimal parameters of every model thanks to PSNR and SSIM. We demonstrate the robustness efficacy of our model against high level of noise. [ABSTRACT FROM AUTHOR]

We reconstruct the initial functions from the trace of the solution of an initial value problem for the wave equation on the light cone. A method to recover the initial function from the solution of the wave equation on the light cone is already known for odd spatial dimensions. We generalize their work to the fractional wave equation and all dimensions. In other words, we present a method to reconstruct the initial functions from the solution of the fractional wave equation on the light cone. [ABSTRACT FROM AUTHOR]

The aim of this article is to find a geometric and physical interpretation of fractional order derivatives for a general class of functions defined over a bounded or unbounded domain. We show theoretically and geometrically that the absolute value of the fractional derivative value of a function is inversely proportional to the area of the triangle. Further, we prove geometrically that the fractional derivatives are inversely proportional to the classical integration in some sense. The established results are verified numerically for non‐monotonic, trigonometric, and power functions. Further, this article establishes a significant connection between the area of the projected fence and the area of triangles. As the area of triangles decreases, the area of the projected fence increases, and vice versa. We calculate the turning points of the fractional derivative values of different functions with respect to order α$$ \alpha $$, including non‐monotonic, trigonometric, and power functions. In particular, we demonstrate that for the power function xβ$$ {x}^{\beta } $$, with β$$ \beta $$ being a positive real number, the value α=0.5$$ \alpha =0.5 $$ is a turning point when x=β$$ x=\beta $$. However, for x>β$$ x>\beta $$, the turning point shifts to the left of point (0.5,0)$$ \left(0.5,0\right) $$ and shifts to the right of point (0.5,0)$$ \left(0.5,0\right) $$ for x<β.$$ x<\beta. $$ We discuss the physical interpretation of fractional order derivatives in terms of fractional divergence. We present some applications of fractional tangent lines in the field of numerical analysis. [ABSTRACT FROM AUTHOR]

In this study, we investigate the effect of Clarke's subdifferential type on the optimal control results for fractional differential systems of order 1

Summary: Time history analysis is sometimes used in an estimation of the wind‐induced response of a tall building. However, time history analysis for the wind‐induced behavior by the ensemble‐averaging wind force costs much computation time. This paper provides a reliable prediction method for the wind‐induced response of the viscoelastic (VE)‐damped system considering its frequency dependency coupling with frame damping effect with frequency spectral method. VE damper used in high‐rise buildings can dissipate energy from excessive vibration induced by seismic or wind excitation. Fractional derivative (FD) model of VE dampers can express VE frequency dependency clearly, but it is computationally complicated. The herein proposed prediction method is based on frequency spectral method and evaluated wind‐induced responses of the single degree of freedom (SDOF) VE‐damped system with a FD VE damper subjected to the respective 1st modal along‐ and across‐wind force. The maximum error of wind‐induced responses of the VE‐damped system, such as the root mean square value of responses, the total input energy, and the total energy dissipation, is within 5%. In summary, the proposed method had high accuracy in the prediction of wind‐induced responses of the VE‐damped system considering its frequency dependency with the coupling effect of frame damping. [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]

This paper focuses on revisiting and improving the results regarding the existence of a solution to the implicit fractional differential equations (FDEs) given in the following form: DCρω(ϰ)=fϰ,ω(ϰ),DCρω(ϰ)$$ {\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)&#x0003D;f\left(\mathrm{\varkappa},\omega \left(\mathrm{\varkappa}\right),{\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)\right) $$for all ϰ∈[0,T]$$ \mathrm{\varkappa}\in \left[0,T\right] $$ with the initial condition ω(0)=ω0$$ \omega (0)&#x0003D;{\omega}_0 $$, where 0<ρ<1,DCρ$$ 0<\rho <1,{\mathcal{D}}_C&#x0005E;{\rho } $$ is the Caputo fractional derivative (CFD). Mainly, we aim to weaken the necessary conditions often found in the literature for guaranteeing the existence of solution. [ABSTRACT FROM AUTHOR]

The impulsive effect on the fractional stochastic integrodifferential system of order ϱ∈(1,2)$$ \varrho \in \left(1,2\right) $$ with infinite delay via sectorial operator is investigated in this work. The primary findings of this study are tested by referring to multivalued functions, sectorial operators, fractional derivatives, and the fixed point approach. Initially, the presence of a mild solution for the systems class was introduced. The existence outcomes of semilinear control problems were then established. Eventually, an application is given for deriving the theory of the key findings. [ABSTRACT FROM AUTHOR]

In this paper, we study a time‐fractional initial‐boundary value problem of Kirchhoff type involving memory term for non‐homogeneous materials. As a consequence of energy argument, we derive L∞0,T;H01(Ω)$$ {L}^{\infty}\left(0,T;{H}_0^1\left(\Omega \right)\right) $$ bound as well as L2(0,T;H2(Ω))$$ {L}^2\left(0,T;{H}^2\left(\Omega \right)\right) $$ bound on the solution of the considered problem by defining two new discrete Laplacian operators. Using these a priori bounds, existence and uniqueness of the weak solution to the considered problem are established. Further, we study semi discrete formulation of the problem by discretizing the space domain using a conforming finite element method (FEM) and keeping the time variable continuous. The semi discrete error analysis is carried out by modifying the standard Ritz‐Volterra projection operator in such a way that it reduces the complexities arising from the Kirchhoff type nonlinearity. Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical solution of the problem under consideration. This method has a convergence rate of O(h+k2−α)$$ O\left(h+{k}^{2-\alpha}\right) $$, where α(0<α<1)$$ \alpha \kern0.1em \left(0<\alpha <1\right) $$ is the fractional derivative exponent and h$$ h $$ and k$$ k $$ are the discretization parameters in the space and time directions, respectively. This convergence rate is further improved to second order in the time direction by proposing a novel linearized L2‐1 σ$$ {}_{\sigma } $$ Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims. [ABSTRACT FROM AUTHOR]

Summary: In this work, the analysis of negative group delay circuits in fractional domain has been conducted. The low pass and the high pass negative group delay circuits constructed based on passive network have been chosen as our candidate circuits. For performing the analysis in fractional domain, the novel Caputo–Fabrizio derivative‐based fractional impedance have been introduced to both circuits. The crucial parameters, the necessary conditions, and the existence conditions of both candidate negative group delay circuits have been formulated. The simulations have been conducted based on the formulated results where the comparisons with the conventional prototypes have been made. The verifications by the proof‐of‐concept circuits have also been performed. In addition, the effects of variation in the order of fractional impedances have been investigated. In summary, it has been found that considering these negative group delay circuits in the fractional domain, which effectively taking unavoidable nonidealities that cannot be modeled in conventional domain into account, significantly alters their characteristics especially the low pass circuit. However they can retain their low pass and high pass negative group delay functions. [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]

The primary objective of this research is to generalize the concept of controllability with constraints to cases where the output function is a Riemann--Liouville fractional derivative of order α. More precisely, if α = 0, we obtain the enlarged controllability of the system state and enlarged controllability of the gradient state of the system is obtained with α = 1. Moreover, we aim to characterize the optimal control that enables us to guide a linear parabolic system to a fractional final state within the evolution system's domain. We adopt two approaches to solve the aforementioned problem: The first is based on the Lagrangian technique, and the second one employs the subdifferential theory. Subsequently, we develop an algorithm to compute the minimum energy control. Finally, we provide numerical simulations to illustrate the validity of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we discuss the existence and approximate controllability results for impulsive fractional differential systems of order 1

In our article, we are primarily concentrating on existence and controllability of nonlocal mixed Volterra‐Fredholm type fractional delay integro‐differential equations of order 1 < r < 2. By applying the results and facts belongs to the cosine function of operators, fractional calculus, the measure of noncompactness and fixed point approach, the main results are established. Initially, we focus the existence of mild solution, and later we establish the controllability of the considered fractional system. Finally, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the theory of a generalized Fourier transform in order to solve differential equations with a generalized fractional derivative, and we state its main properties. In particular, we obtain the new corresponding convolution, inverse and Plancherel formulas, and Hausdorff–Young type inequality. We show that this generalized Fourier transform is useful in the study of several fractional differential equations (both ordinary and partial differential equations). [ABSTRACT FROM AUTHOR]

Monkeypox virus is one of the major causes of both smallpox and cowpox infection in our society. It is typically located next to tropical rain forests in remote villages in Central and West Africa. The disease is brought on by the monkeypox virus, a member of the Orthopoxvirus genus and the Poxviridae family. For analysis and the dynamical behaviour of the monkeypox virus infection, we developed a fractional order model with the Mittag‐Leffler kernel. The uniqueness, positivity, and boundedness of the model are treated with fixed point theory results. A Lyapunov function is used to construct both local and global asymptotic stability of the system for both endemic and disease‐free equilibrium points. Finally, numerical simulations are carried out using the effective numerical scheme with an extended Mittag‐Leffler function to demonstrate the accuracy of the suggested approaches. [ABSTRACT FROM AUTHOR]

All signals obtained as instrumental response of analytical apparatus are affected by noise, as in Raman spectroscopy. Whereas Raman scattering is an inherently weak process, the noise background may lead to misinterpretations. Although surface amplification of the Raman signal using metallic nanoparticles has been a strategy employed to partially solve the signal‐to‐noise problem, the preprocessing of Raman spectral data through the use of mathematical filters has become an integral part of Raman spectroscopy analysis. In this paper, a Tikhonov modified method to remove random noise in experimental data is presented. In order to refine and improve the Tikhonov method as a filter, the proposed method includes Euclidean norm of the fractional‐order derivative of the solution as an additional criterion in Tikhonov function. In the strategy used here, the solution depends on the regularization parameter, λ, and on the fractional derivative order, α. As will be demonstrated, with the algorithm presented here, it is possible to obtain a noise‐free spectrum without affecting the fidelity of the molecular signal. In this alternative, the fractional derivative works as a fine control parameter for the usual Tikhonov method. The proposed method was applied to simulated data and to surface‐enhanced Raman scattering (SERS) spectra of crystal violet dye in Ag nanoparticles colloidal dispersion. [ABSTRACT FROM AUTHOR]

We study the inverse problem for a differential equation of order 2b$$ 2b $$ with the time Caputo fractional derivative and a source with values in Schwartz‐type distributions. The generalized solution of the Cauchy problem for such an equation with initial values and a time‐dependent reaction coefficient are unknown. We find sufficient conditions for the unique solvability of the inverse problem under a time‐ and space‐integral overdetermination conditions. [ABSTRACT FROM AUTHOR]

The dynamic characteristics of the fractional Duffing system with nonlinear time delay term are studied according to the average method and the harmonic balance method. First, the steady‐state solutions of the fractional Duffing system with a nonlinear time delay term are obtained by applying the average method and the harmonic balance method. The analytical expressions of the amplitude–frequency characteristic curves under the average method and the harmonic balance method are also established. Second, the stability criterion of the system is obtained according to the indirect method of studying the motion stability problem, and the concept of the stability condition parameter is proposed in this process. Third, the transient solution of the fractional Duffing system with a nonlinear time delay term is obtained utilizing the average method, and the approximate analytical solution of the fractional Duffing system with nonlinear time delay term is obtained. Finally, the amplitude–frequency characteristic curves of the different fractional differential terms, coefficients of different fractional differential terms, magnitude of different time delay quantities, and feedback strength of different delay are analyzed by numerical simulation. Furthermore, the amplitude–frequency characteristic curves obtained by the average method and the harmonic balance method under different order of fractional differential term are compared and analyzed by numerical simulation. The time sequence and phase diagrams of the system under different order of fractional differential term and different time delay quantities are analyzed by numerical simulation. In addition, the correctness of the analytical analysis is verified by comparing the analytical results with the numerical simulation results under the average method and the harmonic balance method, respectively, by numerical simulation. [ABSTRACT FROM AUTHOR]

The study of the spread of epidemics has gained significant attention in recent years, due to ongoing and recurring outbreaks of diseases such as COVID‐19, dengue, Ebola, and West Nile virus. In particular, modeling the spatial spread of these epidemics is crucial. This article explores the use of fractional diffusion as a means of describing non‐local infection spread. The Grünwald–Letnikov formulation of fractional diffusion is presented, along with several mass‐conserving boundary conditions, that is, we aim to design the boundary conditions in a mass‐conserving way, by not allowing gain or loss of the total population. The stationary points of the model for both sticky and reflecting boundary conditions are discussed, with numerical examples provided to illustrate the results. It is shown that reflecting boundary conditions are more reasonable, as the stationary point for sticky boundary conditions is infinite at the boundaries, while reflecting boundary conditions only have the trivial stationary point, given sufficiently fine discretization. The numerical results were applied to an SI model with fractional diffusion, highlighting the dependence of the system on the value of the fractional derivative. Results indicate that as the order of the derivative increases, the diffusivity also increases, accompanied by a slight increase in the average number of infected individuals. These models have the potential to provide valuable insights into the dynamics of disease spread and aid in the development of effective control strategies. [ABSTRACT FROM AUTHOR]

Mathematical simulation of drug diffusion is a significant tool for predicting the bio‐transport process. Moreover, the reported models in the literature are based on Fick's approach, which leads to an infinite propagation speed. Consequently, it is essential to construct a mathematical model to represent the diffusion processes for estimating drug concentrations at different sites and throughout the circulation. Thus, in this article, the diffusion process is employed to propose three models for estimating the drug release from multi‐layer cylindrical tablets. A fractional model is presented based on Fick's approach, while classical and fractional Cattaneo models are presented using the relaxed principle. Various numerical methods are used to solve the specified problem. The numerical scheme's stability and convergence are demonstrated. Drug concentration and mass profiles are presented for the tablet and the external medium and compared with the in vivo plasma profiles. The results show the efficiency and precision of the proposed fractional models based on the fourth‐order weighted‐shifted Grünwald–Letnikov difference operator approximation. These models are compatible with the in vivo data compared with the classical Fick's one. [ABSTRACT FROM AUTHOR]

The main objective is to apply the concept of newly developed idea of the fractional‐order derivative of the Rabotnov fractional–exponential function in fluid dynamics. In this article, a newly developed idea of the fractional‐order derivative of the Rabotnov fractional–exponential function and the nonsingular kernel has been applied to study viscous fluid flow in the presence of an applied magnetic field. The flow is considered over an infinite vertical plate moving with arbitrary velocity. The modeled problem is transformed into a nondimensional form via dimensionless analysis, and then the Laplace transform method is applied for the solution of the problem. Due to the complexity in Laplace inversion, a strong numerical inversion procedure, namely, Zakian's algorithm, has been used, and the results are computed in various plots and tables. The corresponding discussion of results is included in detail. It is concluded that the generalized fractional‐order derivative is accurate and efficient for describing general fractional‐order dynamics in complex and power law phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the qualitative behavior of a class of fractional SEIR epidemic models with a more general incidence rate function and time delay to incorporate latent infected individuals. We first prove positivity and boundedness of solutions of the system. The basic reproduction number R0$$ {\mathcal{R}}_0 $$ of the model is computed using the method of next generation matrix, and we prove that if R0<1$$ {\mathcal{R}}_0<1 $$, the healthy equilibrium is locally asymptotically stable, and when R0>1$$ {\mathcal{R}}_0>1 $$, the system admits a unique endemic equilibrium which is locally asymptotically stable. Moreover, using a suitable Lyapunov function and some results about the theory of stability of differential equations of delayed fractional‐order type, we give a complete study of global stability for both healthy and endemic steady states. The model is used to describe the COVID‐19 outbreak in Algeria at its beginning in February 2020. A numerical scheme, based on Adams–Bashforth–Moulton method, is used to run the numerical simulations and shows that the number of new infected individuals will peak around late July 2020. Further, numerical simulations show that around 90% of the population in Algeria will be infected. Compared with the WHO data, our results are much more close to real data. Our model with fractional derivative and delay can then better fit the data of Algeria at the beginning of infection and before the lock and isolation measures. The model we propose is a generalization of several SEIR other models with fractional derivative and delay in literature. [ABSTRACT FROM AUTHOR]

An inverse source problem (ISP) of recovering space‐dependent source term along with diffusion concentration for a two‐dimensional diffusion equation involving integral convolution of arbitrary memory kernel in time is considered. The unique existence of the solution is proved using a bi‐orthogonal system of functions obtained from the associated non‐self‐adjoint spectral problem and its adjoint problem which form Riesz basis in L2[(0,1)×(0,1)]$$ {L}&#x0005E;2\left[\left(0,1\right)\times \left(0,1\right)\right] $$. In addition, some particular cases of ISP are described as special cases of our results. [ABSTRACT FROM AUTHOR]

The goal of this study is to analyze and obtain a new numerical approach to an important mathematical model called polio, which is one of the highly infectious and dangerous diseases challenging many lives in most developing nations, most especially in Africa, Latin America, and Asia. A number of research outputs have proven beyond doubt that modeling with noninteger‐order derivative is much more accurate and reliable when compared with the integer‐order counterparts. In the present case, an extension is given to the polio model by replacing the classical time derivative with the newly defined operator known as the Atangana–Baleanu fractional derivative which has its formulation based on the noble Mittag–Leffler kernel. This derivative has been tested and applied in number of ways to model a range of physical phenomena in science and engineering. The Picard–Lindelöf theorem is applied to determine the condition under which the proposed model has a solution, also to show that the solution exists and unique. Local stability analysis of the disease‐free equilibrium and endemic state is also discussed. A novel approximation based on the Adams–Bashforth method is formulated to numerically approximate the fractional derivative operator. To justify the theoretical findings, some numerical results obtained for different instances of fractional order are presented. [ABSTRACT FROM AUTHOR]

In this paper, impulsive fractional differential equations with Hilfer fractional derivatives of order 0<μ<1$$ 0<\mu <1 $$ and type 0≤ν≤1$$ 0\le \nu \le 1 $$ is considered. Convergence analysis of P$$ P $$‐type and PIμ$$ P{I}^{\mu } $$‐type open‐loop iterative learning scheme is studied in the sense of λ$$ \lambda $$‐norm. Examples are provided to explain the theory developed. [ABSTRACT FROM AUTHOR]

In this work, we study the stability of a one‐dimensional Timoshenko system with localized internal fractional Kelvin–Voigt damping in a bounded domain. First, we reformulate the system into an augmented model and using a general criteria of Arendt–Batty we prove the strong stability. Next, we investigate three cases: The first one when the damping is localized in the bending moment, the second case when the damping is localized in the shear stress, we prove that the energy of the system decays polynomially with rate t−1$$ {t}^{-1} $$ in both cases. In the third case, the fractional Kelvin–Voigt is acting on the shear stress and the bending moment simultaneously. We show that the system is polynomially stable with energy decay rate of type t−42−α$$ {t}^{\frac{-4}{2-\alpha }} $$, provided that the two dampings are acting in the same subinterval. The method is based on the frequency domain approach combined with multiplier technique. [ABSTRACT FROM AUTHOR]

Fractional calculus tools have been exploited to effectively model variety of engineering, physics and applied sciences problems. The concept of fractional derivative has been incorporated in the optimization process of least mean square (LMS) iterative adaptive method. This study exploits the recently introduced enhanced fractional derivative based LMS (EFDLMS) for parameter estimation of power signal formed by the combination of different sinusoids. The EFDLMS addresses the issue of fractional extreme points and provides faster convergence speed. The performance of EFDLMS is evaluated in detail by taking different levels of noise in the composite sinusoidal signal as well as considering various fractional orders in the EFDLMS. Simulation results reveal that the EDFLMS is faster in convergence speed than the conventional LMS (i.e., EFDLMS for unity fractional order). [ABSTRACT FROM AUTHOR]

In this article, we formulate fractional differential inclusion of pantograph type (IFDIP), incorporating impulsive behavior of the solution. The boundary conditions taken into account are nonlocal in nature. We will consider the convex problem and prove the Filippov–Wazewski‐type theorem. Moreover, existence of solution, uniqueness of a solution, and the topological properties of the solution's set will be examined for the problem under consideration. In the second part, the study will be confined to the second‐order impulsive fractional differential equation of pantograph type. For certain geometric characteristics of the solution's set, Aronszajn–Browder–Gupta‐type results will be explored for the newly introduced differential equation. Also, it will prove the existence of solution for the first‐order fractional differential equation of pantograph type having impulsive behavior of the solution. [ABSTRACT FROM AUTHOR]

This article deals with the approximate controllability problem for fractional evolution equations involving noninstantaneous impulses and state‐dependent delay. In order to derive sufficient conditions for the approximate controllability of our problem, we first consider the linear‐regulator problem and find the optimal control in the feedback form. By using this optimal control, we develop the approximate controllability of the linear fractional control system. Further, we obtain sufficient conditions for the approximate controllability of the nonlinear problem. In the end, we provide a concrete example to support the applicability of the derived results. [ABSTRACT FROM AUTHOR]

A backward problem for composite fractional relaxation equations is considered with Caputo's fractional derivative, which covers as particular case of Basset problem that concerns the unsteady motion of a particle accelerating in a viscous fluid in fluid dynamics. Based on a spectral problem, the representation of solutions is established. Next, we show the maximal regularity for the corresponding initial value problem. Due to the mildly ill‐posedness of current backward problem, the fractional Landweber regularization method will be applied to discuss convergence analysis and error estimates. [ABSTRACT FROM AUTHOR]

In this work, we consider a system of multidimensional wave equations coupled by velocities with one localized fractional boundary damping. First, using a general criteria of Arendt–Batty, by assuming that the boundary control region satisfy some geometric conditions, under the equality speed propagation and the coupling parameter of the two equations is small enough, we show the strong stability of our system in the absence of the compactness of the resolvent. Our system is not uniformly stable, in general, since it is the case of the interval. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach combining with a multiplier method. Indeed, by assuming that the boundary control region satisfy some geometric conditions, the waves propagate with equal speed, and the coupling parameter term is small enough, we establish a polynomial energy decay rate for smooth solutions, which depends on the order of the fractional derivative. [ABSTRACT FROM AUTHOR]

In this work, we consider a system of multidimensional wave equations coupled by velocities with one localized fractional boundary damping. First, using a general criteria of Arendt–Batty, by assuming that the boundary control region satisfy some geometric conditions, under the equality speed propagation and the coupling parameter of the two equations is small enough, we show the strong stability of our system in the absence of the compactness of the resolvent. Our system is not uniformly stable, in general, since it is the case of the interval. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach combining with a multiplier method. Indeed, by assuming that the boundary control region satisfy some geometric conditions, the waves propagate with equal speed, and the coupling parameter term is small enough, we establish a polynomial energy decay rate for smooth solutions, which depends on the order of the fractional derivative. [ABSTRACT FROM AUTHOR]

Summary: For wide application of the synchronization properties of nonlinear oscillators in control and signal processing engineering field, a common design method for nonlinear oscillator models is required. The output response of a nonlinear oscillator excited by a periodic signal implicitly includes information on both the instantaneous phase and the amplitude of the input signal. The design of a nonlinear oscillator model that can synchronize with an arbitrary cyclic phenomenon can be enabled by explicitly expressing the dynamics of both phase and amplitude with the phase reduction method and organizing them by the number of events in one period. Assuming the cyclic phenomenon to be a single periodic signal, the properties of the generalized nonlinear oscillator model depend on the phase resolution of the signal, which is associated with the phase‐shift of fractional calculus. Thus, this study verifies the validity of fractional calculus through the input‐output characteristic of the generalized nonlinear oscillator model. Numerical simulation using a fractional differentiator with backward‐difference demonstrated that increasing the phase resolution of a single periodic signal improves the estimation accuracy of the phase and amplitude of the signal by employing a generalized nonlinear oscillator model. [ABSTRACT FROM AUTHOR]

The goal of this research is to investigate the issue of existence and optimal control results for fractional mixed Volterra–Fredholm type integrodifferential systems of order 1

The main motivation of our conversation is the approximate controllability of fractional stochastic Sobolev‐type Volterra–Fredholm integro‐differential equation of order 1 < r < 2. Using principles and ideas from stochastic analysis, the theory of cosine family, fractional calculus, and Banach fixed point techniques, the key findings are established. We begin by emphasizing the existence of mild solutions and then demonstrate the approximate controllability of the fractional stochastic control equation. We then apply our findings to the theory of nonlocal conditions. At last, an application is established for drawing the theory of the main results. [ABSTRACT FROM AUTHOR]

Summary: The present manuscript investigates the action of the Hilfer fractional operator on the dynamics of resistor–capacitor (RC) and certain resistor–inductor‐capacitor (RLC) electric circuits using the Laplace transform method alongside utilizing the negative binomial formula. The choice of Hilfer's operator here was basically based on its interpolating nature between the well‐known Caputo and Riemann–Liouville fractional derivatives. Meanwhile, this operator has not been adequately scrutinized with regard to its effects on electric circuits in the literature. Thus, respective approximate exact solutions are determined convolutionally in the presence of certain periodic voltage source functions. Lastly, the significance of the operator on the dynamics of these models was demonstrated graphically for different values of the fractional‐order α and type β. More so, the present study will be beneficial in designing modern electric devices involving temporal delays and memory record. [ABSTRACT FROM AUTHOR]

This article investigates the issue of optimal control and approximate controllability results for fractional integrodifferential evolution equations with infinite delay of r∈(1,2) in Banach space. In the beginning, we analyze approximate controllability results for fractional integrodifferential evolution equations using the fractional calculations, cosine families, and Banach fixed point theorem. After, we developed the continuous dependence of the fractional integrodifferential evolution equations by using the Henry–Gronwall inequality. Furthermore, we tested the existence of optimal controls for the Lagrange problem. Lastly, an application is presented to illustrate the theory of the main results. [ABSTRACT FROM AUTHOR]

In this study, the peridynamic Drucker‐Prager plastic model with fractional order derivative is proposed to investigate the plastic behavior of surrounding rocks around tunnels, in which the Caputo fractional derivative is employed due to its mathematical simplicity. Instead of utilizing just one parameter for the typical nonassociated flow rule, such as dilation angle, multiple parameters, such as fractional order and interval of fractional derivative, are used to specify the direction of plastic deformation, and the fractional derivative based‐typical flow rule is proposed. As a result, compared with the traditional peridynamic Drucker‐Prager plastic model, the proposed method is a more generalized model including the typical nonassociated flow rule. Besides, based on the PD force density, the strategy of exertion of initial stress is proposed to simulate in‐situ stress in rocks. Taking a block subjected to compression as an example, the impacts of various factors, such as fractional order and interval of fractional derivative, are investigated. A numerical simulation of tunnel excavation in rocks is carried out and the numerical results obtained by the proposed method are verified by comparing with FEM results. [ABSTRACT FROM AUTHOR]

In this article, a novel collocation method is developed based on Gegenbauer wavelets together with the quasi‐linearization technique to facilitate the solution of population growth model of fractional order in a closed system. The operational matrices of fractional order integration are obtained via block‐pulse functions. The obtained matrices are employed to transform the given time‐fractional population growth model into a non‐linear system of algebraic equations. Then, the quasi‐linearization technique is invoked to convert the underlying equations to a linear system of equations. The performance and accuracy of the proposed method is elucidated by a presenting a comparison with some numerical methods existing in the open literature. The numerical outcomes shows that the present method is more efficient than the existing ones. [ABSTRACT FROM AUTHOR]

In this article, we study a model problem for the advection–reaction–diffusion equation involving a new nonsingular time‐fractional derivative with Rabotnov fractional‐exponential (RFE) kernel. In order to solve this model numerically, we first obtain the numerical approximation of RFE fractional derivative for a simple polynomial function, which gives rise to an operational matrix of fractional differentiation. We illustrate the accuracy and validity of this operational matrix with the aid of an example. We use Legendre collocation technique together with the newly developed operational matrix to find the numerical solution of the given model. The numerical results depict the feasibility and efficacy of our method. The error estimates show that our method is valid with great accuracy and is applicable to a fractional ODE system and an integral equation with RFE kernel fractional derivative. [ABSTRACT FROM AUTHOR]

In the current paper, a space–time spectral‐Galerkin method is presented for the one‐ and two‐dimensional (1D & 2D) fourth‐order time‐fractional partial integro‐differential equation (TFPIDE) with a weakly singular kernel. In temporal direction, a Petrov‐Galerkin approach is used for discretization. Indeed, eigenfunctions of the first and second kind fractional Sturm‐Liouville problem (FSLP), which called Jacobi polyfractonomials, are used as temporal basis for the trial and test spaces, respectively. Also, spatial discretization is based on the Galerkin approximation with a combination of Legendre polynomials as basis. Fully discrete scheme, give rises to obtaining approximate solution of desired problem via solving a system of linear algebraic equations. Spectral accuracy and efficiency of the considered method are numerically demonstrated by some test problems with smooth and non‐smooth exact solutions in one‐ and two‐dimensional cases. [ABSTRACT FROM AUTHOR]

Mathematical modeling of biological systems is an important issue having significant effect on human beings. In this direction, the description of immune systems is an attractive topic as a result of its ability to detect and eradicate abnormal cells. Therefore, this manuscript aims to investigate the asymptotic behavior of immunogenic tumor dynamics based on a new fractional model constructed by the concept of general fractional operators. We discuss the stability and equilibrium points corresponding to the new model; then we modify the predictor–corrector method in general sense to implement the model and compare the associated numerical results with some real experimental data. As an achievement, the new model provides a degree of flexibility enabling us to adjust the complex dynamics of biological system under study. Consequently, the new general model and its solution method presented in this paper for the immunogenic tumor dynamics are new and comprise quite different information than the other kinds of classical and fractional equations. In addition to these, we implement a tracking control method in order to decrease the development of tumor‐cell population. The satisfaction of control purpose is confirmed by some simulation results since the controlled variables track the tumor‐free steady state in the whole realistic cases. [ABSTRACT FROM AUTHOR]

In this work, we ponder on a Cauchy problem for the Rayleigh–Stokes equation accompanied by polynomial and gradient nonlinearities. We particularly concern about the behavior of mild solutions for the different instances of the nonlinear source term. In the case of polynomial nonlinearities, we present the local‐in‐time existence and uniqueness of the mild solution. Moreover, we claim that either it is the global‐in‐time or it blows up at a finite time. With reference to the case that the source function is global Lipschitzian, we observe that the solution always uniquely exists for a finite time and is continuously dependent. Eventually, we establish some regularity results for the mild solution. [ABSTRACT FROM AUTHOR]

In this present paper, we discuss some properties of fractional operators with fixed memory length (Riemann–Liouville fractional integral, Riemann–Liouville and Caputo fractional derivatives). Some observations and examples are discussed during the article, in order to make the results well defined and clear. Furthermore, we consider the fundamental theorem of calculus for fractional operators with fixed memory length. [ABSTRACT FROM AUTHOR]