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In this note we point out some errors in the paper "On the set of solutions for the Darboux problem for fractional order partial hyperbolic functional differential inclusions" by S. Abbas and M. Benchohra published in Fixed Point Theory, vol. 14, no. 2, 2013, 253-262. [ABSTRACT FROM AUTHOR]

This paper presents the numerical solution of the heat conduction model with a fractional derivative of the Riemann–Liouville type with respect to the spatial variable. The considered mathematical model assumes the dependence on temperature of the material parameters (such as specific heat, density, and thermal conductivity) of the model. In the paper, the boundary conditions of the first and second types are considered. If the heat flux equal to zero is assumed on the left boundary, then the thermal symmetry is obtained, which results in a simplification of the problem and the possibility of considering only half the area. The numerical examples presented in the paper illustrate the effectiveness and convergence of the discussed computational method. [ABSTRACT FROM AUTHOR]

This paper investigates the existence, uniqueness, and continuous dependence of solutions to fractional order nonlocal thermistor two-point boundary value problems on time scales. We employ the Schauder fixed point theorem to establish the existence of solutions, and the contraction principle to prove uniqueness. We also obtain a result on the continuous dependence of solutions. Finally, we present several examples to illustrate our findings. This work is the first to study a fractional model of thermistor on Department of Medical Research,time scales, and it makes a significant contribution to the field of modeling on time scales. The results of this paper can be used to develop new and improved mathematical models for thermistors, which can be used to design more efficient and reliable thermistor-based devices. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating the weak solvability of one model nonlinear viscosity fluid motion with memory along the trajectories of fluid particles determined by the velocity field. We used the topological approximation method for studying hydrodynamic problems, the theory of regular Lagrangian flow, when proving the solvability of the described model. The existence of at least one weak solution of the nonlinear viscosity fluid is proved in the paper. [ABSTRACT FROM AUTHOR]

In this paper, a short review of the calculus of exact finite-differences of integer order is proposed. The finite-difference operators are called the exact finite-differences of integer orders, if these operators satisfy the same characteristic algebraic relations as standard differential operators of the same order on some function space. In this paper, we prove theorem that this property of the exact finite-differences is satisfies for the space of simple entire functions on the real axis (i.e., functions that can be expanded into power series on the real axis). In addition, new results that describe the exact finite-differences beyond the set of entire functions are proposed. A generalized expression of exact finite-differences for non-entire functions is suggested. As an example, the exact finite-differences of the square root function is considered. The use of exact finite-differences for numerical and computer simulations is not discussed in this paper. Exact finite-differences are considered as an algebraic analog of standard derivatives of integer order. [ABSTRACT FROM AUTHOR]

This paper deals with the research of solutions of bounded variation (BV) to evolution inclusion coupled with a time and state dependent maximal monotone operator. Different problems are studied: existence of solutions, unicity of the solution, existence of periodic and bounded variation right continuous (BVRC) solutions. Second-order evolution inclusions and fractional (Caputo and Riemann–Liouville) differential inclusions are also considered. A result of the Skorohod problem driven by a time- and space-dependent operator under rough signal and a Volterra integral perturbation in the BRC setting is given. The paper finishes with some results for fractional differential inclusions under rough signals and Young integrals. Many of the given results are novel. [ABSTRACT FROM AUTHOR]

Grain-oriented silicon steel (GO FeSi) laminations are vital components for efficient energy conversion in electromagnetic devices. While traditionally optimized for power frequencies of 50/60 Hz, the pursuit of higher frequency operation (f ≥ 200 Hz) promises enhanced power density. This paper introduces a model for estimating GO FeSi laminations' magnetic behavior under these elevated operational frequencies. The proposed model combines the Maxwell diffusion equation and a material law derived from a fractional differential equation, capturing the viscoelastic characteristics of the magnetization process. Remarkably, the model's dynamical contribution, characterized by only two parameters, achieves a notable 4.8% Euclidean relative distance error across the frequency spectrum from 50 Hz to 1 kHz. The paper's initial section offers an exhaustive description of the model, featuring comprehensive comparisons between simulated and measured data. Subsequently, a methodology is presented for the localized segregation of magnetic losses into three conventional categories: hysteresis, classical, and excess, delineated across various tested frequencies. Further leveraging the model's predictive capabilities, the study extends to investigating the very high-frequency regime, elucidating the spatial distribution of loss contributions. The application of proportional–iterative learning control facilitates the model's adaptation to standard characterization conditions, employing sinusoidal imposed flux density. The paper deliberates on the implications of GO FeSi behavior under extreme operational conditions, offering insights and reflections essential for understanding and optimizing magnetic core performance in high-frequency applications. [ABSTRACT FROM AUTHOR]

The aim of this paper is to determine the eigenvalue intervals of μk; 1 < k < n for which an iterative systems of a class of fractional-order differential equations with parameterized integral boundary conditions (BCs) has at least one positive solution by means of standard fixed point theorem of cone type. To the best of our knowledge, this will be the first time that we attempt to reach such findings for the topic at hand in the literature. The obtained results in the paper are illustrated with an example for their feasibility. [ABSTRACT FROM AUTHOR]

Introduction: The stress-strain relationship of clays characterized by strain hardening exhibits varying curves under different confining pressures and dry densities. Methods: Considering the viscoelastic properties of clays, a normalized viscoelastic model of strain-hardening clay was established based on fractional derivatives, and normalization factors were proposed. Results: The experimental results showed that the stress-strain relationship of the clay was strain hardening. It shows that Chengdu clay has better normalization conditions. Furthermore, the normalized analysis of this clay through the viscoelastic normalization model revealed that the straight line of normalized data displayed a goodness-of-fit of over 0.98. The obtained values were consistent with experimental results, suggesting the reasonability of the normalized strain-hardening parameters and elastic moduli. Discussion: In addition, the superiority of the developed model was verified by testing the strain-hardening clays in Wuhan, China and Bangkok, Thailand. After analyzing the strain-hardening parameters and normalization factors of our model, it was found that the slope of the normalized line can accurately reflect the strain-hardening ability of the clay. These findings demonstrated that the proposed normalization factor is preferred for a normalized viscoelastic model. It shows that the model proposed in this paper has clearer physical meaning and advancement. [ABSTRACT FROM AUTHOR]

The Helmholtz equation plays a crucial role in the study of wave propagation, underwater acoustics, and the behavior of waves in the ocean environment. The Helmholtz equation is also used to describe propagation through ocean waves, such as sound waves or electromagnetic waves. This paper presents the Elzaki transform residual power series method (E T-RPSM) for the analytical treatment of fractional-order Helmholtz equation. To develop this scheme, we combine Elzaki transform (E T) with residual power series method (RPSM). The fractional derivatives are described in Caputo sense. The E T is capable of handling the fractional order and turning the problem into a recurrence form, which is the novelty of our paper. We implement RPSM in such a way that this recurrence relation generates the results in the form of an iterative series. Two numerical applications are considered to demonstrate the efficiency and authenticity of this scheme. The obtained series are determined very quickly and converge to the exact solution only after a few iterations. Graphical plots and absolute error are shown to observe the authenticity of this suggested approach. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay principles combined with a technique based on vector-valued metrics and convergent to zero matrices. [ABSTRACT FROM AUTHOR]

Loss functions are an essential part in modern data-driven approaches, such as bi-level training scheme and machine learnings. In this paper, we propose a loss function consisting of a r -order (an)-isotropic total variation semi-norms TV r , r ∈ ℝ + , defined via the Riemann–Liouville (RL) fractional derivative. We focus on studying key theoretical properties, such as the lower semi-continuity and compactness with respect to both the function and the order of derivative r , of such loss functions. [ABSTRACT FROM AUTHOR]

This paper explores the boundary stabilization of a degenerate wave equation in the non-divergence form, which includes a drift term and a singular potential term. Additionally, we introduce boundary fractional derivative damping at the endpoint where divergence is absent. Using semi-group theory and the multiplier method, we establish polynomial stability, with a decay rate depending upon the order of the fractional derivative. [ABSTRACT FROM AUTHOR]

In this paper, a fractional food chain system consisting of a Holling type II functional response was studied in view of a fractional derivative operator. The considered fractional derivative operator provided nonsingular as well as a nonlocal kernel which was significantly better than other derivative operators. Fractional order modeling of a model was also useful to model the behavior of real systems and in the investigation of dynamical systems. This model depicted the relationship among four types of species: prey, susceptible intermediate predators (IP), infected intermediate predators, and apex predators. One of the significant aspects of this model was the inclusion of Michaelis-Menten type or Holling type II functional response to represent the predator-prey link. A functional response depicted the rate at which the normal predator consumed the prey. The qualitative property and assumptions of the model were discussed in detail. The present work discussed the dynamics and analytical behavior of the food chain model in the context of fractional modeling. This study also examined the existence and uniqueness related analysis of solutions to the food chain system. In addition, the Ulam-Hyers stability approach was also discussed for the model. Moreover, the present work examined the numerical approach for the solution and simulation for the model with the help of graphical presentations. [ABSTRACT FROM AUTHOR]

This article is devoted to the investigation of the weak solvability to the initial boundary value problem, which describes the viscoelastic fluid motion with memory. The memory of the fluid is considered not at a constant position of the fluid particle (as in most papers on this topic), but along the trajectory of the fluid particle (which is more physical). This leads to the appearance of an unknown function z, which is the trajectory of fluid particles and is determined by the velocity v of a fluid particle. However, in this case, the velocity v belongs to L 2 (0 , T ; V 1) , which does not allow the use of the classical Cauchy Problem solution. Therefore, we use the theory of regular Lagrangian flows to correctly determine the trajectory of the particle. This paper establishes the existence of weak solutions to the considered problem. For this purpose, the topological approximation approach to the study of mathematical hydrodynamics problems, constructed by Prof. V. G. Zvyagin, is used. [ABSTRACT FROM AUTHOR]

This paper proposes some modifications to the method of studying the fractional mass-spring-damper system in order to meet the high reliability requirements of the mechanical system. These changes mainly depend on two things, the discovered drawback of the fractional methods, and the general behavior of the mechanical damping system. The defects discovered in fractional calculus are summarized in the results of differences between the variance of fractional definitions and the constant coefficients. Therefore, it became necessary to introduce some modifications and suggestions for compatibility with the system to make the proposed fractional definition more intuitive and usable. In this paper, the numerical and theoretical results were compared using our proposed definition and Caputo's. We found that there is a clear difference in the results between them. We believe that our proposed method distinguished by its compatibility with the basics of damped mechanical system behavior. In first method the vibration takes place around the reference axis, while in Caputo's method vibration occurs around different axes when changing the fractional differential orders. The method proposed is based on a modification in conformable fractional derivative. [ABSTRACT FROM AUTHOR]

This paper explores the truncated M -fractional three-component coupled nonlinear Schrödinger (tc-CNLS) equation, that regulates the behavior of optical pulses in optical fibers. These equations are utilized in various scientific and engineering fields, including nonlinear fiber optics, electromagnetic field waves, and signal processing through optical fibers. The study of multi-component NLS equations has gained significant attention due to their ability to elucidate various complex physical phenomena and exhibit more dynamic structures of localized wave solutions. The freshly invented integration tools, known as the fractional modified Sardar subequation method (MSSEM) and fractional enhanced modified extended tanh-expansion method (eMETEM), are employed to ensure the solutions. The study focuses on extracting various types of optical solitons, including bright, dark, singular, bright-dark, complex, and combined solitons. Optical soliton propagation in optical fibers is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. Furthermore, hyperbolic, periodic and exponential solutions are generated. The utilized methodology is effective in explaining fractional nonlinear partial differential equations (FNLPDEs) as it offers pre-existing solutions and additionally derives novel exact solutions by mixing outcomes from various procedures. Furthermore, we plot the visualizations of solutions by plotting 3D, 2D, and contour graphs with the corresponding parameter values. The findings of this paper can improve the understanding of the nonlinear dynamical behavior of a specific system and demonstrate the efficacy of the methodology used. We anticipate that our study will provide substantial benefits to a considerable group of engineering model experts. The findings demonstrate the efficacy, efficiency, and applicability of the computational method employed, particularly in dealing with intricate systems. [ABSTRACT FROM AUTHOR]

Most engineering structures serve in complex and harsh dynamic environments and suffer from nonlinear vibrations generated by random excitations such as strong winds and waves, which need to be mitigated. To this end, a novel viscoelastic nonlinear energy sink (VNES) device is proposed for the vibration control of structures, whereby the random vibration of a nonlinear oscillator with a VNES device is analyzed in this paper. Specifically, the mathematical model of the structure-VNES is formulated. An approximate technique for the treatment of viscoelastic damping element modeled by the fractional derivative is developed, whereby an equivalent nonlinear system without fractional derivative is derived. The averaged Itô equations associated with the amplitude envelopes are then obtained by resorting to the stochastic averaging method. The closed-form solution of the stationary response is solved from the Fokker–Planck–Kolmogorov (FPK) equation and verified by the Monte Carlo simulation (MCS) data simultaneously. The effects of system parameters and noise intensity on the stationary response are sequentially examined. By comparing the VNES system to the case without attachment and one equipped with the commonly used tuned mass damper (TMD) device, the performance and robustness of the VNES are highlighted. The conclusions of this work may contribute to the efficacy of the application of the VNES in practice. [ABSTRACT FROM AUTHOR]

In this paper, we have considered the problem of reconstructing the time dependent potential term for the third order time fractional pseudoparabolic equation from an additional data at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of engineering, mechanics and physics. The existence of unique solution to the problem has been discussed by means of the contraction principle on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output potential), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the potential term. The proposed problem is discretized using the cubic B-spline (CB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine $lsqnonlin$ tool has been used to expedite the numerical computations. Both perturbed data and analytical are inverted and the numerical outcomes for two benchmark test examples are reported and discussed. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the existence of positive solutions to the fractional boundary value problem at resonance - (D a + α , ρ x) (t) = f (t , x (t) , D a + α - 1 , ρ x (t)) , t ∈ (a , b) , x (a) = 0 , x (b) = ∫ a b x (t) d A (t) , where 1 < α ≤ 2 , and D a + α , ρ is a Katugampola fractional derivative, which generalizes the Riemann–Liouville and Hadamard fractional derivatives, and ∫ a b x (t) d A (t) denotes a Riemann–Stieltjes integral of x with respect to A, where A is a function of bounded variation. Coincidence degree theory is applied to obtain existence results. This appears to be the first work in the literature to deal with a resonant fractional differential equation with a Katugampola fractional derivative. Examples are given to illustrate the application of their results. [ABSTRACT FROM AUTHOR]

In this paper, we present the design of a fractional-order sliding mode observer (FO-SMO) for actuator fault estimation in a quadrotor unmanned aerial vehicle (QUAV) system. Actuator faults can significantly compromise the stability and performance of QUAV systems; therefore, early detection and compensation are crucial. Sliding mode observers (SMOs) have recently demonstrated their accuracy in estimating faults in QUAV systems under matched uncertainties. However, existing SMOs encounter difficulties associated with chattering and sensitivity to initial conditions and noise. These challenges significantly impact the precision of fault estimation and may even render fault estimation impossible depending on the magnitude of the fault. To address these challenges, we propose a new fractional-order SMO structure based on the Caputo derivative definition. To demonstrate the effectiveness of the proposed FO-SMO in overcoming the limitations associated with classical SMOs, we assess the robustness of the FO-SMO under three distinct scenarios. First, we examined its performance in estimating actuator faults under varying initial conditions. Second, we evaluated its ability to handle significant chattering phenomena during fault estimation. Finally, we analyzed its performance in fault estimation under noisy conditions. For comparison purposes, we assess the performance of both observers using the Normalized Root-Mean-Square Error (NRMSE) criterion. The results demonstrate that our approach enables more accurate actuator fault estimation, particularly in scenarios involving chattering phenomena and noise. In contrast, the performance of classical (non-fractional) SMO suffers significantly under these conditions. We concluded that our FO-SMO is more robust to initial conditions, chattering phenomena, and noise than the classical SMO. [ABSTRACT FROM AUTHOR]

In this paper, a technique is developed for determining the nonstationary response statistics of linear oscillators endowed with fractional derivative elements. Notably, fractional operators are particularly effective in modeling solid mechanics problems as they offer the option of influencing both the elasticity and the energy dissipation capacity of the system. In this paper, particular attention is devoted to the case of fractional derivatives of rational order that approximates reasonably well any real order model. The oscillators are subjected to stationary stochastic excitations, and the pertinent nonstationary response statistical moments are determined by first introducing a finite number of oscillator response related states; this is afforded by the rational number order of the fractional operator. Next, the technique involves proceeding to treating the problem in the Laplace transform domain. This leads to multiple convolution integrals determined by representing the transfer function of the oscillator in a partial fraction form by a pole-residue formulation. In this manner, the response evolutionary power spectral density of the fractional oscillator is derived in a closed form, while nonstationary second-order statistics can be obtained by mundane numerical integration in the frequency domain. Applications to oscillators comprising one or two fractional derivative elements are presented, considering the case of a white noise excitation and of a random process possessing the classical Kanai-Tajimi spectrum. Reliability of the developed technique is assessed by juxtaposing its analytical results with pertinent Monte Carlo simulation data. [ABSTRACT FROM AUTHOR]

This paper attempts to investigate the existence of at least one positive solution and the existence of a unique positive solution to an iterative and conformable fractional system associated with the Sturm–Liouville type conditions. The approach employed in this work relies on the well-known Guo–Krasnosel'skii fixed point theorem and Wardowski fixed point theorem. To ensure the validity of our key findings, examples are included. [ABSTRACT FROM AUTHOR]

This paper deals with a one-dimensional system in the linear isothermal theory of swelling porous elastic soils subject to fractional derivative-type boundary damping. We apply the semigroup theory. We prove well-posedness by the Lumer–Phillips theorem. We show the lack of exponential stability and strong stability is proved by using general criteria due to Arendt–Batty. Polynomial stability result is obtained by applying the Borichev–Tomilov theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper to explain the role and importance of fractional derivatives for mass and heat transfer in Casson nanofluids including clay nanoparticles. These particles can be found in water, kerosene, and engine oil. The physical flow phenomena are illustrated using PDE and thermophysical nanoparticle properties, and this paper addresses the Casson fractional fluid along with chemical reaction and heat generation. The heat and mass fluxes are generalized using the constant proportional Caputo fractional derivative. The present flow model are solved semi-analytically using the Laplace transform. We generated several graphs to understand how various flow factors affect velocity. The acquired results reveal that fractional parameters have dual behavior in velocity profiles. Velocity and temperature are also compared to previous studies. Compared to the other fractional derivatives results, field variables and proposed hybrid fractional derivatives showed a more decaying trend. Furthermore, significant results of clay nanoparticles with various base fluids have been obtained. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical solution of fractional differential equations with interval parameters in terms of derivatives describing anomalous diffusion processes. Computational algorithms for solving initial–boundary value problems as well as the corresponding inverse problems for equations containing interval fractional derivatives with respect to time and space are presented. The algorithms are based on the previously developed and theoretically substantiated adaptive interpolation algorithm tested on a number of applied problems for modeling dynamical systems with interval parameters; this makes it possible to explicitly obtain parametric sets of states of dynamical systems. The efficiency and workability of the proposed algorithms are demonstrated in several problems. [ABSTRACT FROM AUTHOR]

This paper concerns a new semi analytical method for approximate solution of the first type generalized Abel integral equations. An especial numerical fractional derivative which is L1 method will be used for solving this kind of equations. Also, existence, uniqueness, convergence and stability of the given scheme will be studied through some theorems and lemmas. Moreover, some numerical examples are presented and the results are compared with their exact solutions and some other numerical methods to illustrate the capability of this algorithm for solving these generalized Abel integral equations. [ABSTRACT FROM AUTHOR]

In this paper we obtain sufficient condition for univalence of analytic functions defined by Hurwitz-Lerach Zeta function. [ABSTRACT FROM AUTHOR]

Motion information from image sequences (videos) plays a key role in solving a number of real-world problems such as surveillance, traffic management, fire identification, weather prediction, and COVID-19 detection, etc. Generally, this motion is estimated in terms of optical flow between two consecutive image frames. Optical flow is a 2D vector field that illustrates object motion behavior. This paper presents a nonlinear fractional order variational model for estimating optical flow. The objective of this work is to provide dense and discontinuity preserving optical flow for different spectra and make the model robust against outliers. In particular, the presented model generalizes the integer order variational models for fractional order (0, 1). For this purpose, the proposed model is formulated with the help of Charbonnier norm and Caputo fractional derivative. Charbonnier norm is nonlinear in nature, which makes the model robust against noise and outliers, whereas Caputo fractional derivatives are well-capable to deal with discontinuous functions such as images, and therefore, preserve motion discontinuities. The Caputo derivative also manifests long-term memory effect and allows to choose the optimal value of the fractional order that corresponds to a stable solution. The numerical implementation of the formulated variational functional is performed by discretizing the fractional derivative using the Grünwald–Letnikov derivative. The resulting system of equations is solved using multi-variable fixed point iteration scheme. The entire framework is embedded in coarse-to-fine warping strategy, which helps in finding the global extremal. The experimental results are carried out on 20 different application oriented datasets such as fire and smoke, fluid, CXR, etc. The performance of the model is tested using different error measures and demonstrated against several outliers. Error concentration is shown through 3D histograms. The edge preserving nature is also realized through intensity distribution in RGB color channels. A detailed convergence analysis is also provided for the presented model. The validity of the proposed model is also verified through comparisons with other existing models. [ABSTRACT FROM AUTHOR]

Accurate estimation and prediction of battery state of health (SOH) is the focus of battery reliability research. Traditional algorithms ignore the coupling of linear and nonlinear parameters in the battery SOH model, leading to additional errors. To estimate the battery SOH more quickly and accurately, a variable projection algorithm based on truncated variable order fractional gradient descent (VP-TVF) is proposed in this paper. By choosing appropriate variable order constants, the VP-TVF algorithm can accurately estimate model parameters while converging quickly. The efficacy of our proposed algorithm is further substantiated through two comprehensive simulation examples. [ABSTRACT FROM AUTHOR]

The paper examines the solvability and stability of a particular set of fractional Langevin equations under anti-periodic boundary conditions. Utilizing the Krasnoselskii fixed point theorem, the Banach contraction mapping theorem, and properties of the Mittag-Leffler function, we establish less stringent criteria for the existence and uniqueness of solutions compared to previous findings in the literature. Furthermore, we present illustrative examples with specific parameters that highlight the reduced conditions necessary for ensuring the existence of a unique solution. [ABSTRACT FROM AUTHOR]

This paper aims to incorporate the fractional derivative viscoelastic model into a finite element analysis. Firstly, based on the constitutive equation of the fractional derivative three-parameter solid model (FTS), the constitutive equation is discretized by using the Grünwald–Letnikov definition of the fractional derivative, and the stress increment and strain increment relationship and Jacobian matrix are obtained by using the difference method. Subsequently, we degrade the model to establish stress increment and strain increment relationships and Jacobian matrices for the fractional derivative Kelvin model (FK) and fractional derivative Maxwell model (FM). Finally, we further degrade the fractional derivative viscoelastic model to derive stress increment and strain increment relationships and Jacobian matrices for a three-component solid model and Kelvin and Maxwell models. Based on these developments, a UMAT subroutine is implemented in ABAQUS 6.14 finite element software. Three different loading modes, including static load, dynamic load, and mobile load, are analyzed and calculated. The calculations primarily involve a convergence analysis, verification of numerical solutions, and comparative analysis of responses among different viscoelastic models. [ABSTRACT FROM AUTHOR]

The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs. [ABSTRACT FROM AUTHOR]

In this paper, a discrete-time fractional flocking control algorithm of multi-agent systems is put forward to address the slow convergence issue of multi-agent systems. Firstly, by introducing Grünwald-Letnikov (G-L) fractional derivatives, the algorithm allows agents to utilize historical information when updating their states. Secondly, based on the Lyapunov stability theory, the convergence of the algorithm is proven. Finally, simulations are conducted to verify the effectiveness of the proposed algorithm. Comparisons are made between the proposed algorithm and other methods. The results show that the proposed algorithm can effectively improve the convergence speed of multi-agent systems. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the initial boundary value direct problem and zeroorder determination inverse problem for the time-fractional wave equation. First, we study the direct problem (using Robin-type boundary condition), and then using its properties, we investigate the inverse problem. We prove the local solvability of the inverse problem’s solution and will show its stability. [ABSTRACT FROM AUTHOR]

In this paper, utilizing the Dunkl transform and a generalized translation operator, we derive an analogue of the Riesz-Feller fractional derivative for a class of functions satisfying Lipschitz conditions. [ABSTRACT FROM AUTHOR]

This paper explores the application of fuzzy theory to solve fractional heat equations using a novel approach, the Optimal Homotopy Asymptotic Method (OHAM). We introduce a semi-analytical method to address fuzzy fractional-order heat equations, aiming to overcome the limitations of existing approaches. Our methodology leverages the generalized Caputo-Katugampola (CK) definition with two parameters α and ρ, to define fractional derivatives. Through this research, we present a comprehensive framework for tackling this challenging problem. To illustrate the effectiveness and feasibility of our method, we provide several practical examples. The results are presented in tables and figures, and our approach is compared to the exact solutions. This study not only contributes to the field but also offers a powerful and efficient way to address fuzzy fractional heat equations with increased accuracy and reduced computational effort. [ABSTRACT FROM AUTHOR]

Fractional derivatives are useful tools for many applications in different branch of science such as optics and engineering. In this paper, the ∧-fractional frame that is obtained along a space curve by using the ∧-fractional derivative is being examined in Euclidean E 3 space. In addition, the Darboux vector of the ∧-fractional Frenet frame is constructed. Then the curvatures of the standard Frenet frame, the ∧-fractional Frenet frame and the ∧-fractional Darboux vector are compared geometrically. [ABSTRACT FROM AUTHOR]

In this paper, we study a Cauchy type problem in Banach spaces for various classes of functional inclusions with causal multioperators. Based on the topological degree theory for condensing multimaps, we prove a global theorem on the existence of trajectories for systems governed by functional inclusions. As an application, we obtain generalizations of existence theorems for a Cauchy type problem for semilinear second order differential inclusions of this type and semilinear differential inclusions of the fractional order 1 < q < 2. [ABSTRACT FROM AUTHOR]

Stabilization and observation for nonlinear fractional derivative systems remain open problems in automatic due to the fractional nature and nonlinearity of these systems. The present paper studies global stability by the return output for fractional systems. First, we give some definitions of fractional calculus and the quasi-polynomial (QP) and Lotka-volterra (LV) systems. Then, we analyze their stabilities as well as linear (LMI) and bilinear (BMI) matrix inequalities. In order to solve the controller design problem. The goal of this paper is to investigate the global and local stability of a dynamic fractional order system using the quasi-polynomial and LV representation. Then, we use the LMI to study the stabilization of this fractional system. [ABSTRACT FROM AUTHOR]

The focus of the research presented in this paper is on a new generalized family of degenerate three-variable Hermite–Appell polynomials defined here using a fractional derivative. The research was motivated by the investigations on the degenerate three-variable Hermite-based Appell polynomials introduced by R. Alyosuf. We show in the paper that, for certain values, the well-known degenerate Hermite–Appell polynomials, three-variable Hermite–Appell polynomials and Appell polynomials are seen as particular cases for this new family. As new results of the investigation, the operational rule for this new generalized family is introduced and the explicit summation formula is established. Furthermore, using the determinant formulation of the Appell polynomials, the determinant form for the new generalized family is obtained and the recurrence relations are also determined considering the generating expression of the polynomials contained in the new generalized family. Certain applications of the generalized three-variable Hermite–Appell polynomials are also presented showing the connection with the equivalent results for the degenerate Hermite–Bernoulli and Hermite–Euler polynomials with three variables. [ABSTRACT FROM AUTHOR]