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This article is aimed at quantifying the health condition of the power transformer insulation system through its frequency-domain dielectric characteristic by using the concept of the resistor and fractional calculus. In this regard, the dielectric model of the fractional Poynting–Thomson (PT) is proposed, and a mathematical model for parameter identification of the proposed model is presented. The shuffled frog leaping algorithm (SFLA) is used to realize the parameter identification and the fractional PT model parameters of the oil-paper samples under different insulation health conditions. The influence of moisture and aging on the fractional PT dielectric model parameters is experimentally studied. Results show that, with the increase in the water content within the insulation system, the parameters of the fractional PT model gradually decrease, and there is a strong correlation between the relaxation time and the water content. This facilitates the employment of the fractional PT model parameters to quantify the aging and moisture content of the power transformer insulation system. [ABSTRACT FROM AUTHOR]

The oil‐paper compound insulation plays a vital role in the insulation structure of power transformer. To obtain the characteristics of the transformer's insulation system, it is of great importance to study the dielectric response of oil‐paper compound insulation. In this study, fractional calculus is applied to model the oil‐paper compound insulation. Both low‐frequency and broadband high‐frequency models are proposed and then they are verified by fitting the measured data of different insulation papers. Finally they are applied to two different cases: (i) the low‐frequency model for fitting polarity reversal property; (ii) the broadband high‐frequency model for studying the influence of the parameters variations on the frequency domain spectroscopy (FDS). The results demonstrate the advantages of the proposed models when compared with traditional ones. [ABSTRACT FROM AUTHOR]

The document is an editorial conclusion for a special issue on "New Theory and Applications of Nonlinear Analysis, Fractional Calculus, and Optimization." It highlights the widespread and significant applications of nonlinear analysis in various fields of mathematics and science. The special issue includes 10 research papers that cover topics such as optimality conditions, fixed point results, determinantal expressions, pricing of credit risk derivatives, pseudo-asymptotically periodic solutions, common fixed points, local unique solvability, finite-time stability, summation and integral representations, and the existence of solutions. The authors express their appreciation to the contributors and reviewers and hope that the papers inspire future research in these areas. [Extracted from the article]

In this work, some properties of the general convolutional operators of general fractional calculus (GFC), which satisfy analogues of the fundamental theorems of calculus, are described. Two types of general fractional (GF) operators on a finite interval exist in GFC that are conventionally called the L-type and T-type operators. The main difference between these operators is that the additivity property holds for T-type operators and is violated for L-type operators. This property is very important for the application of GFC in physics and other sciences. The presence or violation of the additivity property can be associated with qualitative differences in the behavior of physical processes and systems. In this paper, we define L-type line GF integrals and L-type line GF gradients. For these L-type operators, the gradient theorem is proved in this paper. In general, the L-type line GF integral over a simple line is not equal to the sum of the L-type line GF integrals over lines that make up the entire line. In this work, it is shown that there exist two cases when the additivity property holds for the L-type line GF integrals. In the first case, the L-type line GF integral along the line is equal to the sum of the L-type line GF integrals along parts of this line only if the processes, which are described by these lines, are independent. Processes are called independent if the history of changes in the subsequent process does not depend on the history of the previous process. In the second case, we prove the additivity property holds for the L-type line GF integrals, if the conditions of the GF gradient theorems are satisfied. [ABSTRACT FROM AUTHOR]

The application of fractional calculus to obtain dynamic models for supercapacitors represents an alternative approach to obtaining simpler and more accurate models. This paper presents a model for the supercapacitor in the time domain, based on the use of the fractional or non-integer order integral. This fractional model is compared with the conventional simple model, which is typically used in industrial applications. This fractional integral-based model provides satisfactory fits in relation to the number of parameters used in the model. Furthermore, an interpretation of the effect of the application of fractional integration is presented for constant current charging and discharging processes at constant current, using the Riemann–Liouville definition for the non-integer order integral. Supercapacitors are devices that exhibit non-linear behavior, with a distinct charging and discharging operation. There are several methods of dynamic analysis for the characterization of supercapacitors. The information extracted from these methods is essential for understanding the behavior of supercapacitors and, thus, ensuring that processes involving supercapacitors are as efficient as possible. This paper presents a dynamic analysis based on charge and discharge operations with constant currents. The conclusion is that the fractional model provides fairly accurate fits. [ABSTRACT FROM AUTHOR]

The 1st-level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann–Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the fractional calculus literature. In this paper, we first construct an operational calculus of the Mikusiński type for the 1st-level GFDs. In particular, it includes the operational calculi for the GFDs of the Riemann–Liouville type and for the regularized GFDs as its particular cases. In the second part of the paper, this calculus is applied for the derivation of the closed-form solution formulas to the initial-value problems for the linear fractional differential equations with the 1st-level GFDs. [ABSTRACT FROM AUTHOR]

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems. [ABSTRACT FROM AUTHOR]

Mathematical models play a crucial role in evaluating real-life processes qualitatively and quantitatively. They have been extensively employed to study the spread of diseases such as hepatitis B, COVID-19, influenza, and other epidemics. Many researchers have discussed various types of epidemiological models, including deterministic, stochastic, and fractional order models, for this purpose. This article presents a comprehensive review and comparative study of the transmission dynamics of fractional order in epidemiological modeling. A significant portion of the paper is dedicated to the graphical simulation of these models, providing a visual representation of their behavior and characteristics. The article further embarks on a comparative analysis of fractional-order models with their integer-order counterparts. This comparison sheds light on the nuances and subtleties that differentiate these models, thereby offering valuable insights into their respective strengths and limitations. The paper also explores time delay models, non-linear incidence rate models, and stochastic models, explaining their use and significance in epidemiology. It includes studies and models that focus on the transmission dynamics of diseases using fractional order models, as well as comparisons with integer-order models. The findings from this study contribute to the broader understanding of epidemiological modeling, paving the way for more accurate and effective strategies in disease control and prevention. [ABSTRACT FROM AUTHOR]

This document titled "Fractional Calculus and Hypergeometric Functions in Complex Analysis" explores the impact of fractional calculus on various scientific and engineering disciplines. It emphasizes the significance of fractional operators in the study of fractional calculus and their applications in complex analysis research, specifically in the theory of univalent functions. The document also introduces hypergeometric functions and their connection to the theory of univalent functions. It compiles 12 research papers that cover topics such as geometric properties of fractional differential operators, logarithmic-related problems of univalent functions, and the study of generalized bi-subordinate functions. This document serves as a valuable resource for researchers interested in these subjects and their applications in complex analysis. Additionally, it provides a summary of three articles published in the Special Issue on "Fractional Calculus and Hypergeometric Functions in Complex Analysis." The first article explores the use of the Sălăgean q-differential operator for meromorphic multivalent functions, introducing new subclasses of functions. The second article presents three general double-series identities using Whipple transformations for terminating generalized hypergeometric functions, which can be used to derive additional identities. The third article defines a new generalized domain based on the quotient of two analytic functions and investigates the upper bounds of certain coefficients and determinants. The authors anticipate that these findings will inspire further research in the field. [Extracted from the article]

Many techniques have been recently employed by researchers to address the challenges posed by fractional differential equations. In this paper, we investigate the concept of Ulam-Hyers stability for a class of neutral fractional stochastic differential equations by using the Banach fixed point theorem and the stochastic analysis techniques. An example is presented at the end of the paper to show the interest and the applicability of the results. [ABSTRACT FROM AUTHOR]

Calculus of Variations In the paper I On a Non-Newtonian Calculus of Variations i , Delfim F. M. Torres presents, for the first time in the literature, a non-Newtonian calculus of variations that involves the minimization of a function defined by a non-Newtonian integral with a Lagrangian depending on the non-Newtonian derivative. This publication is an editorial for the Special Issue of Axioms "Calculus of Variations, Optimal Control and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th birthday". This Special Issue is dedicated to Professor Delfim F. M. Torres on the occasion of his 50th birthday, as recognition of his significant contributions to Mathematics, in particular in the calculus of variations, optimal control, and mathematical biology. [Extracted from the article]

This paper explores the Apéry-like series and demonstrates the derivation of closed-form expressions using fractional calculus. We consider a variety of Apéry-like functions, which were categorized by their functional forms and coefficients by applying the Riemann–Liouville fractional integral and derivative to examine their properties across various domains. The study focuses on establishing rigorous mathematical frameworks that unveil new insights into the behaviors of these series, contributing to a deeper understanding of number theory and mathematical analysis. Key results include proofs of convergence and divergence within specified intervals and the derivation of closed-form solutions through fractional integration and differentiation. This paper also introduces a method aimed at conjecturing mathematical constants through continued fractions as an application of our results. Finally, we provide the proof of validation for three unproven conjectures of continued fractions obtained from the Ramanujan Machine. [ABSTRACT FROM AUTHOR]

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

In this paper, we study the existence result of solutions for fuzzy nonlinear fractional differential equations involving Caputo differentiability of an arbitrary order 0 < q < 1. As application, an example is included to show the applicability of our result. [ABSTRACT FROM AUTHOR]

The continuous Hopfield network (CHN) is a common recurrent neural network. The CHN tool can be used to solve a number of ranking and optimization problems, where the equilibrium states of the ordinary differential equation (ODE) related to the CHN give the solution to any given problem. Because of the non-local characteristic of the "infinite memory" effect, fractional-order (FO) systems have been proved to describe more accurately the behavior of real dynamical systems, compared to the model's ODE. In this paper, a fractional-order variant of a Hopfield neural network is introduced to solve a Quadratic Knap Sac Problem (QKSP), namely the fractional CHN (FRAC-CHN). Firstly, the system is integrated with the quadratic method for fractional-order equations whose trajectories have shown erratic paths and jumps to other basin attractions. To avoid these drawbacks, a new algorithm for obtaining an equilibrium point for a CHN is introduced in this paper, namely the optimal fractional CHN (OPT-FRAC-CHN). This is a variable time-step method that converges to a good local minima in just a few iterations. Compared with the non-variable time-stepping CHN method, the optimal time-stepping CHN method (OPT-CHN) and the FRAC-CHN method, the OPT-FRAC-CHN method, produce the best local minima for random CHN instances and for the optimal feeding problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider and study in detail the generalized Fox–Wright function Ψ ˜ q p introduced in our recent work as an extension of the Fox–Wright function Ψ q p . This special function can be seen as an important case of the so-called I-functions of Rathie and H ¯ -functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, Ψ ˜ q p is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of Ψ ˜ q p are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys "Going Next after 'A Guide to Special Functions in Fractional Calculus': A Discussion Survey", and "A Guide to Special Functions of Fractional Calculus", published recently in this journal. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain R + n with positive coordinates. Due to the generality of the Fox H-function, many special integral transforms have the form studied in this paper, including operators with such kernels as generalized hypergeometric functions, classical hypergeometric functions, Bessel and modified Bessel functions and so on. Moreover, most important fractional integral operators, such as the Riemann–Liouville type, are covered by the class under consideration. The mapping properties in Lebesgue-weighted spaces, such as the boundedness, the range and the representations of the considered transformation, are established. In special cases, it is applied to the specific integral transforms mentioned above. We use a modern technique based on the extensive use of the Mellin transform and its properties. Moreover, we generalize our own previous results from the one-dimensional case to the multi-dimensional one. The multi-dimensional case is more complex and needs more delicate techniques. [ABSTRACT FROM AUTHOR]

This survey paper complements to our previous review papers on Lyapunov-type inequalities and contains some of the most recent results on these inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. In precise terms, we have included the results related to Riemann-Liouville, Caputo, mixed Riemann-Liouville and Caputo, Riesz-Caputo, ψ -Caputo, Hadamard, Katugampola, Hilfer, ψ - Hilfer, proportional, variable order Hadamard, partial, systems of Riemann-Liouville, bi-ordinal Hilfer-Katugampola, and ψ -Hilfer fractional derivative operators. The Lyapunov-type inequalities for discrete fractional boundary value problems are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider a time-fractional wave equation for positive operators, including the classical Laplacian with the Dirichlet boundary condition. Determinations of initial velocity and perturbation are investigated. It is also shown that these inverse problems of determining the initial data are ill-posed. Moreover, under some conditions of well-posedness properties of the inverse problems are proved. As an appendix, we also provide some proof of the direct problems. Here, we develop the theoretical part of the inverse problems of finding the initial data for the time-fractional wave equations. [ABSTRACT FROM AUTHOR]

This paper considers a nonlinear fractional-order boundary value problem   H D a , g α 1 , β , μ x (t) + f (t , x (t) , H D a , g α 2 , β , μ x (t)) = 0 , for t ∈ [ a , b ] , α 1 ∈ (1 , 2 ] , α 2 ∈ (0 , 1 ] , β ∈ [ 0 , 1 ] with appropriate integral boundary conditions on the Hölder spaces. Here, f is a real-valued function that satisfies the Hölder condition, and   H D a , g α , β , μ represents the tempered-Hilfer fractional derivative of order α > 0 with parameter μ ∈ R + and type β ∈ [ 0 , 1 ] . The corresponding integral problem is introduced in the study of this issue. This paper addresses a fundamental issue in the field, namely the circumstances under which differential and integral problems are equivalent. This approach enables the study of differential problems using integral operators. In order to achieve this, tempered fractional calculus and the equivalence problem of the studied problems are introduced and studied. The selection of an appropriate function space is of fundamental importance. This paper investigates the applicability of these operators on Hölder spaces and provides a comprehensive rationale for this choice. [ABSTRACT FROM AUTHOR]

The article discusses the advances in fractional-order neural networks and their applications in various fields such as biology, biochemistry, engineering, economics, and physics. Volume II of the Special Issue focuses on the synchronization of fractional-order neural networks, with four papers addressing this topic. Other papers explore variable-order fractionals, non-periodicity of complex maps, impulsive delayed reaction-diffusion gene regulatory networks, split-plot designs, consensus in multi-agent systems, and the mathematical theory of nabla tempered fractional calculus. The editors express their gratitude to the authors and reviewers for their contributions, and believe that the selected papers will enrich readers' knowledge in this field. [Extracted from the article]

In this paper, we prove an identity for differentiable convex functions related to conformable fractional integrals. Moreover, some parameterized inequalities are established by using conformable fractional integrals. To be more precise, parameterized inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Furthermore, previous and new results are presented by using special cases of the obtained theorems. [ABSTRACT FROM AUTHOR]

The variational theory is an inextricable part of both continuum mechanics and physics, and plays an important role in mathematics and nonlinear science, however it is difficult to find a variational formulation for a nonlinear system, and it is more difficult for a fractional differential system. This paper is to search for a variational formulation for the Schrödinger-KdV system with M-fractional derivatives. The fractional complex transformation is used to convert the system into a traditional differential system, and the semi-inverse method is further applied to establish a needed variational principle. [ABSTRACT FROM AUTHOR]

In this paper, the research discussed involves fractional calculus applied to a q -operator. Fractional integrals applied to the q -analogue of the multiplier transformation gives a new operator, and the research is conducted applying the differential subordination and superordination theories. The best dominant and the best subordinant are obtained by the theorems and corollaries discussed. Combining the results from the both theories, sandwich-type results are presented as a conclusion of this research. In this paper, the research discussed involves fractional calculus applied to a -operator. Fractional integrals applied to the -analogue of the multiplier transformation gives a new operator, and the research is conducted applying the differential subordination and superordination theories. The best dominant and the best subordinant are obtained by the theorems and corollaries discussed. Combining the results from the both theories, sandwich-type results are presented as a conclusion of this research. [ABSTRACT FROM AUTHOR]

This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order λ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular. [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]

One of the modern, recently developed mathematical approaches for modeling various complex chaotic processes (the bacteria migration is apparently one of them), is the application of fractional differential equations. Introduction of fractional derivatives is also a very effective approach for investigation of the reactive processes (growth of bacteria in our case). Our recent advances in application of fractional differential equations for modeling the anomalous transport of reactive and non-reactive contaminants (see our recent publications in the References) allow us to expect that the anomalous transport of growing bacteria can also be effectively described by the models with fractional derivatives. Based on these modern approaches, utilizing fractional differential equations, in this paper we developed a reliable mathematical model that could be properly calibrated and, consequently, provide an adequate description of the growing bacteria transport. This model accounts for the memory effects in the bacteria transport due to the random character of bacteria trapping and release by the porous matrix. Two types of bacteria in the saturated porous medium are considered: mobile and immobile bacteria. Bacteria in the mobile phase are migrating in the fluid and have the velocity of the bulk flow, whereas bacteria in the immobile phase are the bacteria that are captured by the porous matrix. These bacteria have zero velocity and can cause clogging of some pores (therefore, porosity is possibly not constant). Examining different conventional models and comparing computations based on these models, we show that this extremely complex character of bacteria transport cannot be described by the traditional approach based on classical partial differential equations. In this paper we suggest fractional differential equations as a simple but very effective tool that can be used for constructing the proper model capable of simulating all the above-mentioned effects associated with migration of alive bacteria. Using this approach, a reliable model of the growing bacteria transport in the porous medium is developed and validated by comparison with experimental laboratory results. We proved that this novel model can be properly linearized and calibrated, so that an excellent agreement with available experimental results can be achieved. This simple model can be used in many applications, for example, as a part of more general mathematical models for predicting the outcomes of the bioremediation of contaminated soils. [ABSTRACT FROM AUTHOR]

Asymmetry plays a significant role in the transmission dynamics in novel discrete fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed discrete models that incorporate different symmetry developmental stages. This paper introduces a Taylor monomial falling function and presents some properties of this function in a delta fractional model with Green's function kernel. In the deterministic case, Green's function will be non-negative, and this shows that the function has an upper bound for its maximum point. More precisely, in this paper, based on the properties of the Taylor monomial falling function, we investigate Lyapunov-type inequalities for a delta fractional boundary value problem of Riemann–Liouville type. [ABSTRACT FROM AUTHOR]

Genetic oscillators have been widely used in modeling key processes of biological systems, especially cell cycles and circadian rhythms. In particular, repressilatory genetic oscillators have been employed in modeling the dynamics of mRNA and protein interactions with transcriptional and translational feedback loops at the molecular level. In addition, synchronization of these oscillators is crucial for understanding the underlying mechanisms of the associated biological processes. In this paper, models of fractional-order genetic oscillators and their coupling are established, where the aspects of time delay, coupling strength, noise, and stability are all taken into consideration. Communication in the proposed coupling model is based on quorum sensing. The synchronization of the fractional-order repressilator model has been examined through simulations which show three main findings. Firstly, the synchronization of the fractional-order repressilator model can be optimized through coupling weight selection. Secondly, the synchronization can be enhanced by increasing the fractional order and decreasing the time delay and the noise intensity. Finally, transitions between the states of the fractional-order repressilatory oscillator can be achieved through varying the fractional order. The simulation results verify the biological relevance of the genetic oscillator models, and their potential for explaining the underlying mechanisms of the associated biological processes. [ABSTRACT FROM AUTHOR]

A previously introduced operator defined by applying the Riemann-Liouville fractional integral to the convex combination of well-known Ruscheweyh and Salagean differential operators is used for defining a new fuzzy subclass. The authors suggest that the operator introduced here can be utilized to define other classes of analytic functions or to generalize other types of differential operators. The new operator defined in this paper can be used to introduce other specific subclasses of analytic functions, and quantum calculus can be also investigated in future studies. The fractional differential operator and the Mittag-Leffler functions are combined to formulate and arrange a new operator of fractional calculus. [Extracted from the article]

In this paper, we have explored the existence and uniqueness of solutions for a pair of nonlinear fractional integro-differential equations comprising of the Ψ-Caputo fractional derivative and the Ψ-Riemann–Liouville fractional integral. These equations are subject to nonlocal boundary conditions and a variable coefficient. Our findings are drawn upon the Mittage–Leffler function, Babenko's attitude, and topological degree theory for condensing maps and the Banach contraction principle. To further elucidate our principal outcomes, we have presented two illustrative examples. [ABSTRACT FROM AUTHOR]

This paper aims to demonstrate a numerical strategy via finite difference formulations for time fractional reaction–diffusion models which are ubiquitous in chemical and biological phenomena. The time-fractional derivative is considered in the Caputo sense for both linear and nonlinear problems. First, the Caputo derivative is replaced with a quadrature formula, then an implicit method is used for the remaining part. In the linear case, the proposed strategy reduces the time fractional models into linear simultaneous equations. In nonlinear cases, Quasilinearization is utilized to tackle the nonlinear parts. With this strategy, solutions of the fractional system transform into linear algebraic systems which are easy to solve. Next, the Von Neumann method is implemented to examine the stability of the scheme which discloses that the scheme is unconditionally stable. Further, the applicability of the presented scheme is tested with different linear and nonlinear models which include the one dimensional Schnakenberg and Gray–Scott models, and one and two dimensional Brusselator models. To analyze the accuracy of the present technique two norms namely, L ∞ and L 2 , and relative error are addressed. Moreover, the obtained outcomes are shown tabulated and graphically which identifies that the scheme properly works for the time fractional reaction–diffusion systems. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of signal modeling using fractional-order linear prediction. In this research, we obtain the closed-form expression of the optimum sampling frequency of the One-Parameter Fractional-order Linear Prediction (OPFLP) and examine the performance when the fractional order (α) is in 0≤α≤2. Our investigation focuses on determining optimum a within the individual ranges of 0≤ α ≤1 and 1≤ α ≤2 while considering various influential parameters, such as sampling frequency and environmental interferences. We initiate our study by examining the impact of the sampling frequency, a critical parameter that demands meticulous selection, on the optimal value of a. Simulation results demonstrate that if the sampling rate falls within five to six times the maximum frequency of the signal under scrutiny, the optimal range for a resides within 1≤ α ≤2.Conversely, when the sampling frequency exceeds six times the maximum signal frequency, the optimal a shifts to 0≤ α ≤1. This observation underscores the crucial relationship between sampling frequency and the appropriate selection of the fractional order a for effective OPFLP performance. In the next step, we assess the robustness of OPFLP in handling challenging signal processing tasks, particularly in hands-free speech acquisition applications. We evaluate the model's performance and robustness against environmental interferences in three scenarios: noisy environments, reverberant environments, and noisy-reverberant settings. Simulation outcomes highlight OPFLP's superior robustness compared to second-order LP in handling environmental interferences. Furthermore. our investigations elucidate that noise exerts a more detrimental impact on OPFLP performance than reverberation, emphasizing the nuanced effects of these interferences on the model's efficacy. [ABSTRACT FROM AUTHOR]

This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order 1 < q < 2 . Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects of finding mild solutions for the Hilfer fractional stochastic differential equation. Subsequently, we determined that the specified system is approximately controllable. Finally, an example displays the theoretical application of the results. [ABSTRACT FROM AUTHOR]

This paper presented the formulation and solution of the time fractional q-deformed tanhGordon equation, a new extension to the traditional tanh-Gordon equation using fractional calculus, and a q-deformation parameter. This extension aimed to better model physical systems with violated symmetries. The approach taken involved the controlled Picard method combined with the Laplace transform technique and the Caputo fractional derivative to find solutions to this equation. Our results indicated that the method was effective and highlighted our approach in addressing this equation. We explored both the existence and the uniqueness of the solution, and included various 2D and 3D graphs to illustrate how different parameters affect the solution’s behavior. This work aimed to contribute to the theoretical framework of mathematical physics and has potential applications across multiple interdisciplinary fields. [ABSTRACT FROM AUTHOR]

This paper focuses on introducing and investigating the class of generalized n-fractional polynomial s-type convex functions within the framework of fractional calculus. Relationships between the novel class of functions and other kinds of convex functions are given. New integral inequalities of Hermite-Hadamard and Ostrowski-type are established for our novel generalized class of convex functions. Using some identities and fractional operators, new refinements of Ostrowskitype inequalities are presented for generalized n-fractional polynomial s-type convex functions. Some special cases of the newly obtained results are discussed. It has been presented that, under some certain conditions, the class of generalized n-fractional polynomial s-type convex functions reduces to a novel class of convex functions. It is interesting that, our results for particular cases recaptures the RiemannLiouville fractional integral inequalities and quadrature rules. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes, and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields. [ABSTRACT FROM AUTHOR]

As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a "new" appropriate fractional integral operator, which is the right inverse of the Caputo derivative on some Hölder spaces of critical orders less than 1. A series of illustrative examples and counter-examples substantiate the necessity of our research. As an application, we use our method to discuss the BVP for the Langevin fractional differential equation d ψ β , μ d t β d ψ α , μ d t α + λ x (t) = f (t , x (t)) , t ∈ [ a , b ] , λ ∈ R , for f ∈ C [ a , b ] × R and some critical orders β , α ∈ (0 , 1) , combined with appropriate initial or boundary conditions, and with general classes of ψ -tempered Hilfer problems with ψ -tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied. [ABSTRACT FROM AUTHOR]

Traditional computer vision techniques aim to extract meaningful information from images but often depend on manual feature engineering, making it difficult to handle complex real-world scenarios. Fractional calculus (FC), which extends derivatives to non-integer orders, provides a flexible way to model systems with memory effects and long-term dependencies, making it a powerful tool for capturing fractional rates of variation. Recently, neural networks (NNs) have demonstrated remarkable capabilities in learning complex patterns directly from raw data, automating computer vision tasks and enhancing performance. Therefore, the use of fractional calculus in neural network-based computer vision is a powerful method to address existing challenges by effectively capturing complex spatial and temporal relationships in images and videos. This paper presents a survey of fractional calculus neural network-based (FC NN-based) computer vision techniques for denoising, enhancement, object detection, segmentation, restoration, and NN compression. This survey compiles existing FFC NN-based approaches, elucidates underlying concepts, and identifies open questions and research directions. By leveraging FC's properties, FC NN-based approaches offer a novel way to improve the robustness and efficiency of computer vision systems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and thoroughly examine new generalized  ψ -conformable fractional integral and derivative operators associated with the auxiliary function  ψ (t) . We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit  ψ -conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications. [ABSTRACT FROM AUTHOR]

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results. [ABSTRACT FROM AUTHOR]

This study explores the extension of Milne-type inequalities to the realm of Katugampola fractional integrals, aiming to broaden the analytical tools available in fractional calculus. By introducing a novel integral identity, we establish a series of Milne-type inequalities for functions possessing extended s-convex first-order derivatives. Subsequently, we present an illustrative example complete with graphical representations to validate our theoretical findings. The paper concludes with practical applications of these inequalities, demonstrating their potential impact across various fields of mathematical and applied sciences. [ABSTRACT FROM AUTHOR]

Measles is the most infectious disease with a high basic reproduction number ( R 0 ). For measles, it is reported that R 0 lies between 12 and 18 in an endemic situation. In this paper, a fractional order mathematical model for measles disease is proposed to identify the dynamics of disease transmission following a declining memory process. In the proposed model, a fractional order differential operator is used to justify the effect and success rate of vaccination. The total population of the model is subdivided into five sub-compartments: susceptible (S), exposed (E), infected (I), vaccinated (V), and recovered (R). Here, we consider the first dose of measles vaccination and convert the model to a controlled system. Finally, we transform the control-induced model to an optimal control model using control theory. Both models are analyzed to find the stability of the system, the basic reproduction number, the optimal control input, and the adjoint equations with the boundary conditions. Also, the numerical simulation of the model is presented along with using the analytical findings. We also verify the effective role of the fractional order parameter alpha on the model dynamics and changes in the dynamical behavior of the model with R 0 = 1 . [ABSTRACT FROM AUTHOR]

Multiplicative calculus, or geometric calculus, is an alternative to classical calculus that relies on division and multiplication as opposed to addition and subtraction, which are the basic operations of classical calculus. It offers a geometric interpretation that is especially helpful for simulating systems that degrade or expand exponentially. Multiplicative calculus may be extended to fractional orders, much as classical calculus, which enables the analysis of systems having fractional scaling properties. So, in this paper, the well-known Sturm-Liouville problem in fractional calculus is reformulated in multiplicative fractional calculus. The considered problem consists of the Sturm-Liouville operator using multiplicative conformable derivatives on the equation and on boundary conditions. This research aimed to explore some of the problem's spectral aspects, like being selfadjointness of the operator, orthogonality of different eigenfunctions, and reality of all eigenvalues. In this specific situation, Green's function is also recreated. [ABSTRACT FROM AUTHOR]