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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 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]

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]

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]

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]

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]

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]

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]

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 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]

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]

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, 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]

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]

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]

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]

This paper was concerned with the existence and uniqueness results for a coupled system of nonlinear generalized fractional differential equations supplemented with a new class of nonlocal coupled multipoint boundary conditions containing Riemann-Stieltjes and generalized fractional integrals. The nonlinearities in the given system depend on the unknown functions as well as their lower order generalized fractional derivatives. We made use of the Leray-Schauder alternative and Banach contraction mapping principle to obtain the desired results. An illustrative example was also discussed. The paper concluded with some interesting observations. [ABSTRACT FROM AUTHOR]

Fractional calculus is defined by expanded integer order integration and differentiation. In this paper, multiple mathematical models of a nonlinear dual-tank device are precisely formulated by fractional calculus. Using the accurate model, a multivariable fractional-order controller is designed for nonlinear devices. The merits of the fractional-order design include: (1) control of multivariable nonlinearities, (2) compensation of uncertainties, and (3) elimination of coupling effects. Simulations and experiments are conducted to verify the precision of the fractional order models and the effectiveness of the multivariable fractional-order control system design. [ABSTRACT FROM AUTHOR]

In recent years, fractional calculus (FC) has filled in a hole in traditional calculus in terms of the effect of memory, which lets us know things about the past and present and guess what will happen in the future. It is very important to have this function, especially when studying biological models and integral equations. This paper introduces developed mathematical strategies for understanding a direct arrangement of fractional integro-differential equations (FIDEs). We have presented the least squares procedure and the Legendre strategy for discussing FIDEs. We have given the form of the Caputo concept fractional order operator and the properties. We have presented the properties of the shifted Legendre polynomials. We have shown the steps of the technique to display the solution. Some test examples are given to exhibit the precision and relevance of the introduced strategies. Mathematical outcomes show that this methodology is a comparison between the exact solution and the methods suggested. To show the theoretical results gained, the simulation of suggested strategies is given in eye-catching figures and tables. Program Mathematica 12 was used to get all of the results from the techniques that were shown. [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 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]

In this paper, we propose a novel approach to the fractional power of the Laplace-Bessel operator Δv, defined as... The fractional power of this operator is introduced as a pseudo-differential operator through the multidimensional Bessel transform. Our primary contributions encompass a normalized singular integral representation, Bochner subordination, and intertwining relations. [ABSTRACT FROM AUTHOR]

In designing control systems, it is known that fractional-order proportional integral derivative (FOPID) controllers often provide greater flexibility than conventional proportional integral derivative (PID) controllers. This higher level of flexibility has proven to be extremely valuable for various applications such as vibration suppression in structural engineering. In this paper, we study the optimization of FOPID controllers using twelve well-established algorithms to minimize structural responses under seismic excitations. The algorithms include crystal structure algorithm (CryStAl), stochastic paint optimizer, particle swarm optimization, krill herd, harmony search, ant colony optimization, genetic algorithm, grey wolf optimizer, Harris hawks optimization, sparrow search algorithm, hippopotamus optimization algorithm, and duck swarm algorithm. In addition to highlighting the benefits of fractional calculus in structural control, this study provides a detailed analysis of FOPID controllers as well as a brief description of the algorithms used to optimize them. To evaluate the efficiency of the proposed techniques, two building models with different numbers of stories are examined. FOPID controllers are designed based on oustaloup's approximation and the El Centro earthquake data. Using five well-known metrics, the performances of the developed methods are evaluated against five earthquake scenarios, including the recent earthquake in Turkey. A non-parametric (Friedman) test is also employed to compare the algorithms based on their corresponding vibration reduction. The findings of this analysis show that CryStAl consistently performs better than the other algorithms for both building models, thus resulting in superior vibration suppression. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study to fractional calculus for class of Ψ function. The present study is designed to study generalized fractional derivatives and find their generalized transforms called Ψ-Laplace transform and Ψ-Sumudu transform. Moreover, find the analytical solutions of some applications in physics the form of generalized fractional derivatives by transform technique. [ABSTRACT FROM AUTHOR]

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the C 0 algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map's states in commensurate and incommensurate cases. [ABSTRACT FROM AUTHOR]

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie's modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots. [ABSTRACT FROM AUTHOR]

In this paper, the time-resolved model of the photoacoustic signal for samples with a complex inner structure is derived including local non-equilibrium of structural elements with multiple degrees of freedom, i.e., structural entropy of the system. The local non-equilibrium is taken into account through the fractional operator. By analyzing the model for two types of time-dependent excitation, a very short pulse and a very long pulse, it is shown that the rates of non-equilibrium relaxations in complex samples can be measured by applying the derived model and time-domain measurements. Limitations of the model and further directions of its development are discussed. [ABSTRACT FROM AUTHOR]

This paper aims to examine an approach that studies many Euler–Maclaurin-type inequalities for various function classes applying Riemann–Liouville fractional integrals. Afterwards, our results are provided by using special cases of obtained theorems and examples. Moreover, several Euler–Maclaurin-type inequalities are presented for bounded functions by fractional integrals. Some fractional Euler–Maclaurin-type inequalities are established for Lipschitzian functions. Finally, several Euler–Maclaurin-type inequalities are constructed by fractional integrals of bounded variation. [ABSTRACT FROM AUTHOR]

Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the limitations of traditional tools. Fractional calculus is a valuable tool for these issues, with applications ranging from membrane diffusion to electrical response of complex fluids, particularly electrolytic cells like liquid crystal cells. This paper presents the main fractional tools to formulate a diffusive model regarding time-fractional derivatives and modify the continuity equations stating the conservation laws. We explore two possible ways to introduce time-fractional derivatives to extend the continuity equations to the field of arbitrary-order derivatives. This investigation is essential, because while the mathematical description of neutral particle diffusion has been extensively covered by various authors, a comprehensive treatment of the problem for electrically charged particles remains in its early stages. For this reason, after presenting the appropriate mathematical tools based on fractional calculus, we demonstrate that generalizing the diffusion equation leads to a generalized definition of the displacement current. This modification has strong implications in defining the electrical impedance of electrolytic cells but, more importantly, in the formulation of the Maxwell equations in material systems. [ABSTRACT FROM AUTHOR]

This document is a preface to a special issue of the journal Mathematics, focusing on abstract fractional integro-differential equations and fixed point theory with applications. The preface explains that fractional calculus is used to model complex systems in various disciplines, and abstract fractional integro-differential equations arise from different scientific research areas. The special issue includes 12 research papers selected from 36 submissions, covering topics such as almost periodic solutions, stability analysis, and convex analysis. The authors express their appreciation to the contributors and reviewers, and hope that the papers will inspire future research in these areas. [Extracted from the article]

Smoking is globally a challenging issue that causes many fatal health problems. In this paper, a nonlinear fractional smoking mathematical model is proposed in the context of a modified form of the Caputo fractional-order derivative. The analytical and approximate-analytical solutions are obtained for the proposed mathematical model via the fractional differential transform method (FDTM) and Laplace Adomian decomposition method (LADM). The obtained solution is provided as a rapidly convergent series. Simulation results are provided in this paper to compare the obtained solutions by FDTM, LADM, Runge Kutta (RK) method, and reduced differential transforms method (RDTM) with the exact solution of the proposed problem. By comparing both FDTM and LADM solutions, the FDTM solution is closer to the exact solution than the LADM solution. All obtained solutions have been analyzed and compared graphically to validate the effiency and applicability of all results. [ABSTRACT FROM AUTHOR]

In [[10]], the authors investigate certain image formulas related to the product of Srivastava polynomials and extended Wright functions using fractional integral and differential operators, including Marichev-Saigo-Maeda, Lavoie-Trottier, and Oberhettinger. The paper [[13]] investigates the Dirichlet and modified Dirichlet average of the I R i function, i.e., the Mittag-Leffler-type function. Among the numerous applications of the theory of fractional calculus in almost all applied sciences, applications in numerical analysis and various fields of physics and engineering stand out. [Extracted from the article]

In this paper, we introduce a ground-breaking approach to defining fractional calculus for a selected class of analytic functions. Our new definitions, based on a novel and intuitive understanding of fractional derivatives and integrals, offer improved mathematical tractability for a variety of applications, including physics, engineering and finance. Our approach significantly simplifies the complexity of mathematical functions compared to the traditional Riemann-Liouville approach, by using simple functions rather than special functions, while preserving the intrinsic sense of fractional calculus. This article not only presents our proposed definitions but also provides a thorough analysis of their properties and advantages. The conclusion of this paper discusses the potential for future research in the field of fractional calculus. [ABSTRACT FROM AUTHOR]