Search

In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments. [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]

This paper considers the inverse problem of determining the time-dependent coefficient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag–Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved. [ABSTRACT FROM AUTHOR]

In this paper, an inverse problem of determining a kernel in a one-dimensional integro-differential time-fractional diffusion equation with initial-boundary and overdetermination conditions is investigated. An auxiliary problem equivalent to the problem is introduced first. By Fourier method this auxilary problem is reduced to equivalent integral equations. Then, using estimates of the Mittag–Leffler function and successive aproximation method, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel which will be used in study of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of positive solutions to a system of fractional differential equations that include the (r 1 , r 2 , r 3) -Laplacian operator, three-point boundary conditions, and various fractional derivatives. We use a combination of techniques, including cone expansion and compression of the functional type, and the Leggett–Williams fixed point theorem, to prove the existence of positive solutions. Finally, we provide two examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

Calcium signaling in nerve cells is a crucial activity for the human brain to execute a diversity of its functions. An alteration in the signaling process leads to cell death. To date, several attempts registered to study the calcium distribution in nerve cells like neurons, astrocytes, etc. in the form of the integer-order model. In this paper, a fractional-order mathematical model to study the spatiotemporal profile of calcium in nerve cells is assembled and analyzed. The proposed model is solved by the finite element method for space derivative and finite difference method for time derivative. The classical case of the calcium dynamics model is recovered by setting the fractional parameter and that validates the model for classical sense. The numerical computations have systematically presented the impact of a fractional parameter on nerve cells. It is observed that calbindin-D28k provides a significant effect on the spatiotemporal variation of calcium profile due to the amalgamation of the memory of nerve cells. The presence of excess amounts of calbindin-D28k controls the intracellular calcium level and prevents the nerve cell from toxicity. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of weak Wardowski contractive multivalued mappings and investigate the solvability of generalized '-Caputo snap boundary fractional differential inclusions. Our results utilize some existing results regarding snap boundary fractional differential inclusions. An example is given to illustrate the applicability of our main results. [ABSTRACT FROM AUTHOR]

Copyright of Boletin de Geologia is the property of Universidad Industrial de Santander and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we study the existence of solutions of a nonlocal Cauchy problem for nonlinear fractional integro-differential equations involving generalized Katugampola fractional derivative. By using fixed point theorems, the results are obtained in weighted space of continuous functions. In the last, results are illustrated with suitable examples. [ABSTRACT FROM AUTHOR]

A control scheme for finite-time stabilization of unstable orbits of the fractional difference logistic map is proposed in this paper. The presented technique is based on isolated perturbation impulses used to correct the evolution of the map's trajectory after it deviates too far from the neighborhood of the unstable orbit, and does not require any feedback control loops. The magnitude of the control impulses is determined by means of H-rank algorithm, which helps to reveal the pseudo-manifold of non-asymptotic convergence of the fractional difference logistic map. Numerical experiments are used to illustrate the effectiveness and the feasibility of the proposed approach, which is applicable beyond the studied fractional difference logistic map. [ABSTRACT FROM AUTHOR]

The stochastic transition behavior of tri-stable states in a fractional-order generalized Van der Pol (VDP) system under multiplicative Gaussian white noise (GWN) excitation is investigated. First, according to the minimal mean square error (MMSE) concept, the fractional derivative can be equivalent to a linear combination of damping and restoring forces, and the original system can be simplified into an equivalent integer-order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and based on singularity theory, the critical parameters for stochastic P -bifurcation of the system are found. Finally, the properties of stationary PDF curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each sub-region divided by the transition set curves. The consistency between numerical results obtained by Monte-Carlo simulation and analytical solutions verified the accuracy of the theoretical analysis process and the method used in this paper has a direct guidance in the design of fractional-order controller to adjust the system behavior. [ABSTRACT FROM AUTHOR]

In this paper, we derive the Shehu transform of the Caputo-Fabrizio fractional derivative without singular kernel. Moreover, we apply the proposed transform to solve Caputo-Fabrizio fractional differential equations and fractional integral equation. Furthermore, we present simulation examples to validate the result. [ABSTRACT FROM AUTHOR]

In the present paper, we study an initial-boundary problem for the parabolic equation ∂/∂t μ (x, t) + A (x, D) μ (x, t) = 0 with general elliptic operator and with the fractional initial conditions. Here A (x, D) non negative elliptic operator which defined in any N dimensional domain Ω with the smooth boundary. We prove the existence and uniqueness of the solution of the problem. In particular, initial conditions can be given through the one-sided Marchaud, Grunwald-Letnikov or Liouville-Weyl fractional derivatives. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence and uniqueness of solutions to uncertain fractional differential equations proposed by Canonical Liu's process. To this end, we provide and prove a a novel existence and uniqueness theorem for uncertain fractional differential equations under Local Lipschitz and monotone conditions is provided and proved. This result helps us to consider and analyze solutions to a wide range of nonlinear uncertain fractional differential equations driven by Canonical's process to be considered and analyzed. [ABSTRACT FROM AUTHOR]

In this paper, stochastic bifurcations of a fractional-order smooth and discontinuous (SD) oscillator composed of different viscoelastic materials are studied. As a widely applicable algorithm for various fractional-orders cases, an extended fast algorithm is introduced to obtain the statistics of the response, where the fractional derivative is separated into a history part and a local part with a predetermined memory length. The local part is approximated by a highly accurate algorithm while the history part is computed by an efficient convolution algorithm. Through this accurate and fast method, effects of the system parameters on the dynamic behaviors, such as the fractional order, smoothness parameter, and frequency of harmonic force, are thus successfully investigated. Abundant stochastic P-bifurcation phenomena are discussed in detail. Further, it is found that only when the damping material shows nearly elastic behaviors, the probability density functions of the system exhibit the crater shape. Experiments show that the fast algorithm is accurate for different fractional orders. [ABSTRACT FROM AUTHOR]

In economics, depreciation functions (operator kernels) are certain decreasing functions, which are assumed to be equal to unity at zero. Usually, an exponential function is used as a depreciation function. However, exponential functions in operator kernels do not allow simultaneous consideration of memory effects and depreciation effects. In this paper, it is proposed to consider depreciation of a non-exponential type, and simultaneously take into account memory effects by using the Prabhakar fractional derivatives and integrals. Integro-differential operators with the Prabhakar (generalized Mittag-Leffler) function in the kernels are considered. The important distinguishing features of the Prabhakar function in operator kernels, which allow us to take into account non-exponential depreciation and fading memory in economics, are described. In this paper, equations with the following operators are considered: (a) the Prabhakar fractional integral, which contains the Prabhakar function as the kernels; (b) the Prabhakar fractional derivative of Riemann–Liouville type proposed by Kilbas, Saigo, and Saxena in 2004, which is left inverse for the Prabhakar fractional integral; and (c) the Prabhakar operator of Caputo type proposed by D'Ovidio and Polito, which is also called the regularized Prabhakar fractional derivative. The solutions of fractional differential equations with the Prabhakar operator and its special cases are suggested. The asymptotic behavior of these solutions is discussed. [ABSTRACT FROM AUTHOR]

Recently, different extensions of the fractional derivative operator are found in many research papers. The main aim of this paper is to establish an extension of the extended Caputo fractional derivative operator. The extension of an extended fractional derivative of some elementary functions derives by considering an extension of beta function which includes the Mittag-Leffler function in the kernel. Further, an extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and the generating relations for extension of extended hypergeometric functions also presented in this study. [ABSTRACT FROM AUTHOR]

The fractional derivative method is considered as a possible formalism for the non-destructive insulation diagnosis by polarization and depolarization current measurements. The underlying motive is based on non-Debye relaxation processes clearly observed in long-time relaxation experiments with oil–paper, electrolytic and ultra-capacitors, transformers and other systems. Modes of relaxation at different time scales depend on charging prehistory and are sensitive to the state of a system. Fractional formalism provides useful parameters characterizing the state of the insulation system, which can be extracted from polarization–depolarization current measurements. Adequacy of the technique is examined by comparison of theoretical and experimental results for oil–paper, and ultra-capacitors. [ABSTRACT FROM AUTHOR]

The stochastic P-bifurcation behavior of tri-stability in a generalized Van der Pol system with fractional derivative under additive Gaussian white noise excitation is investigated. Firstly, based on the minimal mean square error principle, the fractional derivative is found to be equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Secondly, the stationary probability density function of the system amplitude is obtained by stochastic averaging, and according to the singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the nature of stationary probability density function curves of the system amplitude is qualitatively analyzed by choosing the corresponding parameters in each region divided by the transition set curves. The consistency between the analytical solutions and Monte-Carlo simulation results verifies the theoretical results in this paper. [ABSTRACT FROM AUTHOR]

In the spectral analysis of operators associated with Sturm–Liouville-type boundary value problems for fractional differential equations, the problem of positive definiteness or the problem of Hermitian nonnegativity of the corresponding kernels plays an important role. The present paper is mainly devoted to this problem. It should be noted that the operators under study are non-self-adjoint, their spectral structure is not well investigated. In this paper we use various methods to prove the Hermitian non-negativity of the studied kernels; in particular, a study of matrices that approximate the Green's function of the boundary value problem for a differential equation of fractional order is carried out. Using the well-known Livshits theorem, it is shown that the system of eigenfunctions of considered operator is complete in the space L 2 (0 , 1) . Generally speaking, it should be noted that this very important problem turned out to be very difficult. [ABSTRACT FROM AUTHOR]

This paper focuses on synchronization control of the fractional-order unified chaotic system. First, the stability of equilibria is analyzed by the Routh–Hurwitz criteria. Then, three different controllers are designed to achieve chaos synchronization of the drive-response unified system. Further, the impact of the fractional order on the synchronization performance is illustrated. Numerical simulations are carried out to verify the feasibility of theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper investigates finite-time synchronization of fractional-order quaternion-valued neural networks (F-QV-NNs), involving uncertainties in the system parameters. We adopt a combined approach of sliding mode control (SMC) and a non-separation strategy to achieve finite-time synchronization. Based on SMC theory, we construct a specific sliding surface and design a controller to ensure the occurrence of sliding motion. To achieve the desired sliding motion, we apply the fractional Lyapunov direct method to guide the system's states to the designed sliding surface. Moreover, the derived sufficient conditions ensure finite-time synchronization within the models. Lastly, we give a numerical example to validate the effectiveness of the acquired results. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of optical pulses of the fractional extended nonlinear Schrödinger equation and the effect of the third-order dispersion parameter are studied. The equation physically defines the propagation of femtosecond in nonlinear optical fiber and plasma physics. These types of equations are applicable in many real-world scenarios, such as physical sciences and optical engineering. One of the most important applications of such equations is the study of solitons, which are self-sustaining solitary waves that retain their shape and momentum even after collision with other solitons. Solitons are employed in many different fields, including fluid dynamics (where they simulate long-lasting ocean waves) and optical communications (where they transmit information over large distances without distortion). Different types of optical pulses, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted by using the recently integration method known as modified Sardar subequation method. The technique used in this analysis is effective and robust, as demonstrated by the computational results. Different sets and values of the physical parameters are used to study the optical soliton solutions of the resulting system. Moreover, different graphs with the associated parameter values are sketched to visualize the solutions behaviour. [ABSTRACT FROM AUTHOR]

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

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]

This paper proposes a new operational numerical method based on Touchard wavelets for solving fractional pantograph differential equations. First, we present an operational matrix of fractional integration as well as the fractional derivative of the Touchard wavelets. Then, by approximating the fractional derivative of the unknown function in terms of the Touchard wavelets and also by using collocation method, the original problem is reduced to a system of algebraic equations. Finally, to show the accuracy and the validity of the proposed technique, we provide some numerical examples. [ABSTRACT FROM AUTHOR]

Intensity inhomogeneity (IIH) and noise are ubiquitous in images acquired by a variety of imaging modalities, which pose an ongoing challenge for segmentation methods used in many applications. This paper proposes a level set evolution equation based on fractional derivative to address the IIH and noise issues in image segmentation, which integrates the information from the input and guidance images. The proposed model is a linear diffusion equation of level set function with nonlinear edge-stopping and guidance sources, in which the edge-stopping source is derived from the input image and a local statistical model (Bayesian formulation) in the level set framework, whereas the guidance source is derived from the guidance image and the fitting term of the popular Chan-Vese model. The guidance source ensures swift contour movement toward the boundary of the objects and enhances the robustness to noise and contour initialization, while the edge-stopping source is used to stop the evolving contour on the boundary of the objects later in the evolution. To tackle the challenge of obtaining accurate guidance images for many applications, an adaptive variable-order of fractional derivative robust to IIH and noise is introduced to generate the desired guidance image from the input image. The proposed diffusion equation with sources is solved numerically by the series splitting method and finite difference scheme. Experimental results on synthetic and real images show that the proposed model has higher performance in terms of accuracy and efficiency of segmentation, compared to several state-of-the-art level set models. • Propose guided active contours via level set evolution for noisy images with intensity inhomogeneity (IIH). • Design an adaptive order of fractional derivative that is basically not affected by IIH and noise. • Present a fractional derivative based method for generating guidance image from the input image. • Our model is experimentally very effective and efficient in segmenting images with IIH and noise. [ABSTRACT FROM AUTHOR]

This paper proposes a fractional structure and texture aware Retinex (FSTAR) model for image decomposition with application to low-light enhancement. First, a novel structure aware measure called Maximum Fractional Difference (MFD) is introduced, which is the maximum of fractional differences of the input image in eight symmetric directions. Then fractional structure and texture aware maps are built based on MFD and Huber function from robust statistics, which are used as weighted matrices in the regularization terms of reflectance and illumination components. To ensure the intrinsic physical constraints of Retinex on two components, two extra penalty terms are incorporated into the objective function of FSTAR to penalize deviations of the estimated components from the corresponding constraints. The FSTAR is solved via an alternating iterative algorithm; i.e., the objective function is minimized with respect to one variable with another variable fixed, and this process is done alternately until termination condition is met. Qualitative and quantitative evaluations of FSTAR on three low-light image datasets, compared to ten state-of-the-art methods, show its superior performance in Retinex decomposition and low-light image enhancement. • Present texture aware measure based on fractional directional difference for texture extraction. • Propose fractional structure-texture aware Retinex model with constraints on two components. • Model is faithful to intrinsic physical meaning of Retinex in theory and easy to solve numerically. • Model has outstanding performance in terms of image decomposition and low-light enhancement. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive investigation into the construction of electromagnetic timelike particles within the frame of Minkowski space, employing the Bishop model. Our study involves deriving the fractional derivatives of key Lorentz forces namely Ω (t) , Ω (v 1) , and Ω (v 2) . Furthermore, we delve into the computation of essential normalizing and recursion operators tailored to magnetic vector fields, drawing from the principles outlined in the Bishop model. Additionally, we extend our analysis to determine the Fermi–Walker (FW) fractional derivatives, which play a crucial role in understanding the behavior and dynamics of the normalizing and recursion operators. Through this thorough exploration, we aim to provide valuable insights into the electromagnetic properties of timelike particles within the context of Minkowski space. [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]

The primary focus of this paper is the investigation of the truncated M fractional Kuralay equation, which finds applicability in various domains such as engineering, nonlinear optics, ferromagnetic materials, signal processing, and optical fibers. As a result of its capacity to elucidate a vast array of complex physical phenomena and unveil more dynamic structures of localized wave solutions, the Kuralay equation has received considerable interest in the scientific community. To extract the solutions, the recently developed integration method, referred to as the modified generalized Riccati equation mapping (mGREM) approach, is utilized as the solving tool. Multiple types of optical solitons, including mixed, dark, singular, bright-dark, bright, complex, and combined solitons, are extracted. Furthermore, solutions that are periodic, hyperbolic, and exponential are produced. To acquire a valuable understanding of the solution dynamics, the research employs numerical simulations to examine and investigate the exact soliton solutions. Graphs in both two and three dimensions are presented. The graphical representations offer significant insights into the patterns of voltage propagation within the system. The aforementioned results make a valuable addition to the current body of knowledge and lay the groundwork for future inquiries in the domain of nonlinear sciences. The efficacy of the modified GREM method in generating a wide range of traveling wave solutions for the coupled Kuralay equation is illustrated in this study. [ABSTRACT FROM AUTHOR]

In this paper, many important formulas of the subtrigonometric, subhyperbolic, pretrigonometric, prehyperbolic, supertrigonometric, and superhyperbolic functions sin Wiman class are developed for the first time. The subsine, subcosine, subhyperbolic sine, and subhyperbolic cosine associated with Kohlrausch–Williams–Watts (KWW) function and their scaling-law ODEs are proposed. The supersine, supercosine, superhyperbolic sine, and superhyperbolic cosine functions associated with Mittag-Leffler (ML) function and their fractional ODEs are obtained. The conjectures for the supercosine functions containing ML function are presented in detail. [ABSTRACT FROM AUTHOR]

In this paper, a new fractional derivative model for the non-Darcian seepage within the exponential decay kernel is addressed for the first time. The new fractional derivative model is for high-speed non-Darcian and low-speed non-Darcian seepage, in which the applied zone is enlarged. [ABSTRACT FROM AUTHOR]

Creep deformation control is the key point and difficulty in the construction of loess high fill, the creep characteristics of loess with different compactness and confining pressure conditions has been studied. To achieve the objective, the influence of different conditions on the creep deformation characteristics of loess were analyzed by the triaxial creep tests of loess under different compaction and confining pressures. The results show that the instantaneous deformation of the soil samples gradually decrease with increasing compactness, and the total creep deformation also decreases. With the increase of confining pressure, the deformation increase gradually with the compactness decreased. Finally, in order to describe the creep process of loess, the creep model suitable for different compactness is proposed, which is based on fractional calculus and the continuous damage theory. Through fitting verification, the applicability of the theoretical model to the description of loess creep characteristics is proved, and then the whole stages of loess creep can be better described. [ABSTRACT FROM AUTHOR]

In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this article was to establish the Hölder regularities of mild solutions, classical solutions, and strict solutions. We introduced an interpolation space in terms of an analytic resolvent to lower the spatial regularity of initial value data. By virtue of the properties of analytic resolvent and the interpolation space, the Hölder regularities were obtained. As applications, the main conclusions were applied to the regularities of fractional Rayleigh–Stokes problems. [ABSTRACT FROM AUTHOR]

The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey's fear of the predator and its carry-over effects. This study examines the state evolution of fractional-order systems and compares their dynamic behavior with integer-order systems. By utilizing the Routh–Hurwitz condition and the stability theory of fractional differential equations, this paper establishes the local stability conditions of the model through the application of the Jacobi matrix and eigenvalue method. Furthermore, the conditions for the Hopf bifurcation generation are determined. Subsequently, chaos control techniques based on the Lyapunov stability theory are employed to stabilize the unstable trajectory at the equilibrium point. The theoretical findings are validated through numerical simulations. These results enhance our understanding of the stability properties and chaos control mechanisms in fractional-order three-population food chain models. [ABSTRACT FROM AUTHOR]

Machine learning is a branch of artificial intelligence that dates back more than 50 years. It is currently experiencing a boom in research and technological development. With the rise of machine learning, the need to propose improved optimizers has become more acute, leading to the search for new gradient-based optimizers. In this paper, the ancient concept of fractional derivatives has been applied to some optimizers available in PyTorch. A comparative study is presented to show how the fractional versions of gradient optimizers could improve their performance on generative adversarial networks (GAN) and natural language applications with Bidirectional Encoder Representations from Transformers (BERT). The results are encouraging for both state-of-the art algorithms, GAN and BERT, and open up the possibility of exploring further applications of fractional calculus in machine learning. [ABSTRACT FROM AUTHOR]

In this paper we consider a boundary value problem for a loaded fractional diffusion equation. The loaded term has the form of the Riemann-Liouville fractional derivative or integral. The BVP is considered in the open right upper quadrant. The problem is reduced to an integral equation that, in some cases, belongs to the pseudo-Volterra type, and its solvability depends on the order of differentiation in the loaded term and the behavior of the support line of the load in a neighborhood of the origin. All these cases are considered. In particular, we establish sufficient conditions for the unique solvability of the problem. Moreover, we give an example showing that violation of these conditions can lead to nonuniqueness of the solution. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish the results of a double Dirichlet average of the generalized Bessel-Maitland function by using fractional derivative. We obtain the solution in the compact form of a double Dirichlet average of the generalized Bessel-Maitland function. Further, several special cases involving a number of well-known functions such as the Bessel-Maitland function, Mittag Leffler functions, Wright-hypergeometric functions and H-functions etc. have been established. Numerous expansions of the generalised Bessel-Maitland function, which reduces to the Mittag-Leffler function, have been studied and applied to solve a wide range of problems in physics, biology, chemistry and engineering. In the context of fractional calculus, Bessel-Maitland function, Mittag-Leffler functions, hypergeometric function and H-functions etc. have been widely studied. Since Carlson first introduced the concept of the Dirichlet average and its various forms. The Dirichlet average of elementary function like power function, exponential function etc is given by many notable mathematicians. These averages have been investigated and utilized in a number of different fields. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence, uniqueness, and other properties of solution of differential equation of fractional order involving the Caputo fractional derivative. The tool employed in the analysis is based on the application of a new three steps iteration process introduced by V. Karakaya, Y. Atalan, K. Dogan, and NH. Bouzara [26]. Furthermore, the study of various properties such as dependence on initial data, the closeness of solutions, and dependence on parameters and functions involved therein. The results obtained are illustrated through an example. [ABSTRACT FROM AUTHOR]

The visual quality of images captured under sub-optimal lighting conditions, such as over and underexposure may benefit from improvement using fusion-based techniques. This paper presents the Caputo Differential Operator-based image fusion technique for image enhancement. To effect this enhancement, the proposed algorithm first decomposes the overexposed and underexposed images into horizontal and vertical sub-bands using Discrete Wavelet Transform (DWT). The horizontal and vertical sub-bands are then enhanced using Caputo Differential Operator (CDO) and fused by taking the average of the transformed horizontal and vertical fractional derivatives. This work introduces a fractional derivative-based edge and feature enhancement to be used in conjuction with DWT and inverse DWT (IDWT) operations. The proposed algorithm combines the salient features of overexposed and underexposed images and enhances the fused image effectively. We use the fractional derivative-based method because it restores the edge and texture information more efficiently than existing method. In addition, we have introduced a resolution enhancement operator to correct and balance the overexposed and underexposed images, together with the Caputo enhanced fused image we obtain an image with significantly deepened resolution. Finally, we introduce a novel texture enhancing and smoothing operation to yield the final image. We apply subjective and objective evaluations of the proposed algorithm in direct comparison with other existing image fusion methods. Our approach results in aesthetically subjective image enhancement, and objectively measured improvement metrics. [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]

In this paper, we primarily investigate the methodology for the hybrid complex projective synchronization (HCPS) scheme in non-identical complex fractional order chaotic systems via an active complex synchronization technique (ACST). Appropriate controllers of a nonlinear type are designed in view of master–slave composition and Lyapunov's stability criterion (LSC). The HCPS is an extended version of the previously designed projective synchronization scheme. In the HCPS scheme, by using a complex scale matrix, the system taken as slave system is asymptotically synchronized with another system taken as the master system. By utilizing a complex scale matrix, the unpredictability and security of communication are increased along with image encryption. An efficient computational method has been employed to validate and visualize the HCPS method's efficacy by performing numerical simulation outcomes in MATLAB (version 2021). [ABSTRACT FROM AUTHOR]

The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of eigenfunctions and associated functions of the corresponding spectral problems are studied. Theorems on the existence and uniqueness of solutions to the studied problems are proved. [ABSTRACT FROM AUTHOR]

This paper investigates a two-dimensional Riemann–Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliable methods need to be explored for solving the RLDO-SFDE numerically. We develop an alternating direction implicit scheme and prove that the numerical method is unconditionally stable and convergent with an accuracy of O (σ 2 + ρ 2 + τ + h x + h y) . After employing an extrapolated technique, the convergence order is improved to second order in time and space. Furthermore, a fast algorithm is constructed to reduce computational costs. Two numerical examples are presented to verify the effectiveness of the numerical methods. This study may provide more possibilities for simulating diffusion complexities by fractional calculus. [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 paper, we propose a nonuniform numerical formula for the Caputo fractional derivative at the half-grid based on the piecewise linear interpolation and construct a difference scheme for the time-fractional nonlinear fourth-order reaction–diffusion equation. By virtue of two discrete tools: the discrete orthogonal convolution kernels and the discrete complementary convolution kernels, we obtain the positive definiteness of the discrete time-fractional derivative. Then, a discrete variational energy dissipation law of the proposed difference scheme is established for the time-fractional nonlinear fourth-order reaction–diffusion equation, which is asymptotically compatible with the associated energy dissipation law for the classical equation as the value of fractional order approaches to one. Numerical experiments demonstrate the effectiveness and the energy dissipation of the proposed difference scheme with an adaptive time-stepping strategy. [ABSTRACT FROM AUTHOR]