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This study uses the triple-phase lag model to investigate how higher-order fractional order and memory-dependent derivatives affect reflection at the free surface of an orthotropic piezo-thermoelastic medium. The performance of both kinds of derivatives is studied using the normal mode analysis technique. Four different types of coupled reflected plane waves are identified and explore the impact of various parameters, fractional order parameter, kernel function, the higher-order time differential fractional, and memory-dependent heat conduction parameters on energy distribution with respect to the angle of incidence. The results are presented graphically, providing numerical data for reflected waves, amplitude, and energy ratios. [ABSTRACT FROM AUTHOR]

The long-term stability and safety of underground engineering greatly depend on a thorough understanding of the creep characteristics of fractured rock. This paper discusses the potential application of the damage evolution equation and the fractional derivative operator for describing the creep characteristics of fractured rock. By considering the initial damage of the original fracture and the internal damage induced by external loads, coupled damage evolution equations are proposed to account for different failure mechanisms. The introduction of damage and fractional calculus into basic mechanical elements allows for the deduction of creep constitutive relationships for both viscoelastic and viscoplastic bodies. A range of fractional-order damaged creep models is developed based on the Burgers model and the Nishihara model. Parameter sensitivity analysis shows that the inclusion of the damage evolution equation in the creep model is crucial for accurately capturing the accelerated creep deformation characteristics of rock, with the fractional derivative order primarily affecting the steady creep strain rate. By fitting the creep experimental data of fractured Beishan granite, it is evident that the damaged Nishihara model [H-N(D)/H(D)-N(D)/St. V model], which considers only the damage effect, and the fractional damaged Burgers model [H-N(A)-N(D)/H(D) model], which accounts for both the damage effect and the fractional operator, exhibit significant advantages in characterizing the entire process of creep deformation in fractured granite. Highlights: Coupled damage evolution equations for fractured rock with different inclination angles are proposed to account for different failure mechanisms. Fractional order damaged creep models are developed by incorporating damage equations and fractional calculus into basic mechanical elements. The inclusion of the damage in the creep model is crucial for capturing the accelerated creep deformation characteristics of the rock. The fractional derivative order mainly affects the steady state creep strain rate. The validity and applicability of the model are verified by fitting the experimental data of fractured granite. [ABSTRACT FROM AUTHOR]

The rubber joints mounted on swing arms provide the main part of the yaw stiffness for wheelsets and thus have a significant influence on the dynamic behaviour of railway vehicles. However, their nonlinear dynamic properties concerning the frequency, amplitude, and temperature were not fully considered in previous studies, most of which adopt simple models such as the Kelvin–Voigt model to represent rubber components. This study aims to investigate the radial nonlinear properties of swing arm rubber joints under various conditions and assess their effects on the dynamic performance of high-speed trains. A nonlinear rubber spring model which combines the fractional derivative Zener model with Berg's friction model is established. To achieve a better fit to the measurements of a swing arm rubber joint, an optimisation-based method is employed to identify the model parameters. The viscous and frictional effects of the rubber joint are compared at different frequencies, amplitudes, and temperatures, by determining the predominant element of the nonlinear model. Additionally, the nonlinear rubber spring model is integrated into a high-speed vehicle dynamic model to investigate the effects of the nonlinear properties of swing arm rubber joints on the vehicle dynamic behaviour through the MATLAB/SIMPACK co-simulation method. [ABSTRACT FROM AUTHOR]

In this work, we study the coupled system of fractional integro-differential equations, which includes the fractional derivatives of the Riemann–Liouville type and the fractional q-integral of the Riemann–Liouville type. We focus on the utilization of two significant fixed-point theorems, namely the Schauder fixed theorem and the Banach contraction principle. These mathematical tools play a crucial role in investigating the existence and uniqueness of a solution for a coupled system of fractional q-integro-differential equations. Our analysis specifically incorporates the fractional derivative and integral of the Riemann–Liouville type. To illustrate the implications of our findings, we present two examples that demonstrate the practical applications of our results. These examples serve as tangible scenarios where the aforementioned theorems can effectively address real-world problems and elucidate the underlying mathematical principles. By leveraging the power of the Schauder fixed theorem and the Banach contraction principle, our work contributes to a deeper understanding of the solutions to coupled systems of fractional q-integro-differential equations. Furthermore, it highlights the potential practical significance of these mathematical tools in various fields where such equations arise, offering a valuable framework for addressing complex problems. [ABSTRACT FROM AUTHOR]

In recent years, there has been a growing interest among researchers in the study of shallow water waves, driven by their wide applicability across various scientific disciplines. Within this field, the Camassa–Holm equation has garnered significant attention due to its ability to capture complex wave phenomena. However, researchers have extended their investigations to the Modified Camassa–Holm equation (MCH), which incorporates principles from fractional calculus, thereby offering a more comprehensive framework for understanding intricate dynamics. This article focuses on the analytical solution of the Fractional Modified Camassa–Holm (FMCH) equation, an extension of the MCH, achieved through a novel method known as the Residual Power Series Method (RPSM). By utilizing RPSM and initial conditions, the main aim is to unveil an analytical solution for the FMCH, contributing to an improved understanding and modeling of the complex real-world phenomena observed in waves. [ABSTRACT FROM AUTHOR]

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

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

In this paper, we consider constructions of first derivative approximations using the generating function. The weights of the approximations contain the powers of a parameter whose modulus is less than one. The values of the initial weights are determined, and the convergence and order of the approximations are proved. The paper discusses applications of approximations of the first derivative for the numerical solution of ordinary and partial differential equations and proposes an algorithm for fast computation of the numerical solution. Proofs of the convergence and accuracy of the numerical solutions are presented and the performance of the numerical methods considered is compared with the Euler method. The main goal of constructing approximations for integer-order derivatives of this type is their application in deriving high-order approximations for fractional derivatives, whose weights have specific properties. The paper proposes the construction of an approximation for the fractional derivative and its application for numerically solving fractional differential equations. The theoretical results for the accuracy and order of the numerical methods are confirmed by the experimental results presented in the paper. [ABSTRACT FROM AUTHOR]

Ebola virus disease (EVD) is a severe and often fatal illness posing significant public health challenges. This study investigates EVD transmission dynamics using a novel fractional mathematical model with five distinct compartments: individuals with low susceptibility ( S 1 ), individuals with high susceptibility ( S 2 ), infected individuals (I), exposed individuals (E), and recovered individuals (R). To capture the complex dynamics of EVD, we employ a Φ -piecewise hybrid fractional derivative approach. We investigate the crossover effect and its impact on disease dynamics by dividing the study interval into two subintervals and utilize the Φ -Caputo derivative in the first interval and the Φ -ABC derivative in the second interval. The study determines the basic reproduction number R 0 , analyzes the stability of the disease-free equilibrium and investigates the sensitivity of the parameters to understand how variations affect the system's behavior and outcomes. Numerical simulations support the model and demonstrate consistent results with the theoretical analysis, highlighting the importance of fractional calculus in modeling infectious diseases. This research provides valuable information for developing effective control strategies to combat EVD. [ABSTRACT FROM AUTHOR]

Diabetes is becoming more and more dangerous, and the effects continue to grow due to the population's ignorance of the seriousness of this phenomenon. The studies that have been carried out have not been able to follow the phenomenon more precisely, which has led to the use of the fractional derivative tool, which has a very great capability to study real problems and phenomena but is somewhat limited on nonlinear models. In this work, we will develop a new fractional derivative model of a diabetic population, the Takagi–Sugeno fractional fuzzy model, which will enable us to study the phenomenon with these nonlinear terms in order to obtain greater precision in the results. We will study the existence and uniqueness of the solution using the Lipschizian theorem and then turn to the new fuzzy model, which leads us to four dynamical systems. The interpretation results show the quality of fuzzy membership in tracking the malleable phenomena of nonlinear terms existing in the system. [ABSTRACT FROM AUTHOR]

Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples. [ABSTRACT FROM AUTHOR]

The fuzzy C-means (FCM) clustering algorithm is a widely used unsupervised learning method known for its ability to identify natural groupings within datasets. While effective in many cases, FCM faces challenges such as sensitivity to initial cluster assignments, slow convergence, and difficulty in handling non-linear and overlapping clusters. Aimed at these limitations, this paper introduces a novel fractional fuzzy C-means (Frac-FCM) algorithm, which incorporates fractional derivatives into the FCM framework. By capturing non-local dependencies and long memory effects, fractional derivatives offer a more flexible and precise representation of data relationships, making the method more suitable for complex datasets. Additionally, a genetic algorithm (GA) is employed to optimize a new least-squares objective function that emphasizes the geometric properties of clusters, particularly focusing on the Fukuyama–Sugeno and Xie–Beni indices, thereby enhancing the balance between cluster compactness and separation. Furthermore, the Frac-FCM algorithm is evaluated on several benchmark datasets, including Iris, Seed, and Statlog, and compared against traditional methods like K-means, SOM, GMM, and FCM. The results indicate that Frac-FCM consistently outperforms these methods in terms of the Silhouette and Dunn indices. For instance, Frac-FCM achieves higher Silhouette scores of most cases, indicating more distinct and well-separated clusters. Dunn's index further shows that Frac-FCM generates clusters that are better separated, surpassing the performance of traditional methods. These findings highlight the robustness and superior clustering performance of Frac-FCM. The Friedman test was employed to enhance and validate the effectiveness of Frac-FCM. [ABSTRACT FROM AUTHOR]

Objective image quality assessment involves the use of mathematical models to quantitatively describe image quality. FR-IQA (Full-Reference Image Quality Assessment) methods using reference images are also often used to evaluate image processing and computer vision algorithms. Quality indices often use gradient operators to express relevant visual information, such as edges. Fractional calculus has been applied in the last two decades in various fields such as signal processing, image processing, and pattern recognition. Fractional derivatives are generalizations of integer-order derivatives and can be computed using various operators such as the Riemann-Liouville, Caputo-Fabrizio, and Grünwald-Letnikov operators. In this paper, we propose a modification of the FSIMc image quality index by including fractional derivatives to extract and enhance edges. A study of the usefulness of fractional derivative in the FSIMc model was conducted by assessing Pearson, Spearman and Kendall correlations with MOS scores for images from the TID2013 and KADID-10k databases. Comparison of FD_FSIMc with the classic FSIMc shows an increase of several percent in the correlation coefficients for the modified index. The results obtained are superior to those other known approaches to FR-IQA that use fractional derivatives. The results encourage the use of fractional calculus. [ABSTRACT FROM AUTHOR]

The challenge of predicting time series with limited data has evolved over time due to nonlinearity, complexity, and limited information. It can be perceived as a mapping of dynamical systems in one-dimensional space. This article proposes a neural grey system to tackle this challenge. The system enhances its ability to fit nonlinearity by employing polynomials, captures complexity through a fractional-order cumulant operator, and resolves information-poor uncertainty by utilizing grey system modeling techniques. The model effectively integrates research findings from neural computing, uncertainty theory, and complexity theory at a theoretical level. It accurately describes dynamic processes of complex systems. Additionally, we have reduced the complexity of calculations in the algorithm design. We selected a dataset of total retail sales of consumer goods to test the model's validity and applicability. Our experiments demonstrate that the newly proposed grey forecasting model can effectively forecast time series with small samples, offering good forecasting outcomes and generalization ability. [ABSTRACT FROM AUTHOR]

On the unit ball of , some Bergman type operators are defined with the use of fractional derivatives and reproducing kernels of Poisson–Bergman type, which act boundedly in the Lipschitz spaces. In a special case, these operators continuously map the Lipschitz space of sufficiently smooth functions onto its harmonic subspace. [ABSTRACT FROM AUTHOR]

Vector-borne infections are a class of human diseases resulting from the transmission of pathogenic microorganisms, including bacteria, viruses, and parasites, through various vectors. Yellow fever, prevalent in both the American and African continents, stands as a prominent example of vector-borne infections. In this paper, we structure the transmission dynamics of yellow fever with vaccine and treatment through non-integer derivative. The fixed point theorem introduced by Banach and Schaefer is utilized to examine the existence and uniqueness of solutions for the proposed yellow fever system. The sufficient conditions of the Ulam-Hyers stability has been established for our system. The solution routes are highlighted using the Laplace Adomian decomposition method to show the influence of input factors on yellow fever. In order to visualise the impacts of fractional-order, vaccine, biting rate, and treatment on the infection level, numerical simulations are performed. We proposed the most attractive parameters of the system for the prevention and control of the infection. Furthermore, it is proposed that the biting rate of mosquitoes is dangerous and can increase the possibility of infection in the community. We suggest that the index of memory, treatment and vaccination are attractive parameters which can reduce the level of infection. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to suggest a novel technique based on the operational matrix approach for solving a fractional weakly singular two-dimensional partial Volterra integral equation (FWS2DPVIE) using numerical methods. In this technique, Boubaker polynomials are used to create operational matrices. The technique consists of two major phases. In the first step, Boubaker polynomials are employed to generate operational matrices, which help in transforming the problems into systems of algebraic equations. In the second step, the algebraic equations are numerically solved.The suggested technique is also compared with existing approaches. The results show that the suggested technique outperforms its counterparts, demonstrating its superiority. [ABSTRACT FROM AUTHOR]

Calcium is a decisive messenger for neuronal vivid functions. The calcium intracellular sequestering major unit is the Endoplasmic Reticulum (ER). Brownian motion of calcium could be bound to different buffers like S100B, calmodulin, etc, and different organelles. Plasma membrane channels like voltage-gated calcium channels (VGCC) and Plasma Membrane Calcium ATPase (PMCA), Orai channel could perturb the calcium concentration. To investigate the calcium interplay for intracellular signaling we have developed the two-dimensional time fractional reaction–diffusion equation. To solve this model analytically, we have used the Laplace and Fourier cosine integral transform method. By using Green's function we obtained the compact solution in closed form with Mainardi's function and Wright's function. Uniqueness and existence proved the more fundamental approach to the fractional reaction–diffusion problem. The fractional Caputo approach gives better insight into this real-life problem by its nonlocal nature. Significant effects of different parameters on free calcium ions were obtained and the results are interpreted with normal and Alzheimeric cells. Non-local property and dynamical aspects are graphically presented which might provide insight into the Stromal interaction molecule (STIM) and S100B parameters. [ABSTRACT FROM AUTHOR]

We establish new characterizations of the Bloch space B which include descriptions in terms of classical fractional derivatives. Being precise, for an analytic function f (z) = ∑ n = 0 ∞ f ^ (n) z n in the unit disc D , we define the fractional derivative D μ (f) (z) = ∑ n = 0 ∞ f ^ (n) μ 2 n + 1 z n induced by a radial weight μ , where μ 2 n + 1 = ∫ 0 1 r 2 n + 1 μ (r) d r are the odd moments of μ . Then, we consider the space B μ of analytic functions f in D such that ‖ f ‖ B μ = sup z ∈ D μ ^ (z) | D μ (f) (z) | < ∞ , where μ ^ (z) = ∫ | z | 1 μ (s) d s . We prove that B μ is continously embedded in B for any radial weight μ , and B = B μ if and only if μ ∈ D = D ^ ∩ D ˇ . A radial weight μ ∈ D ^ if sup 0 ≤ r < 1 μ ^ (r) μ ^ 1 + r 2 < ∞ and a radial weight μ ∈ D ˇ if there exist K = K (μ) > 1 such that inf 0 ≤ r < 1 μ ^ (r) μ ^ 1 - 1 - r K > 1. [ABSTRACT FROM AUTHOR]

This paper employs the fractional derivative to investigate the effect of curves in Lorentz–Minkowski space, which is of crucial significance in geometry and physics. In the method of examining this effect, the conformable fractional derivative is chosen because it best fits the algebraic structure of differential geometry. Therefore, with the aid of conformable fractional derivatives, numerous special curves and the Frenet frame that were previously derived using classical derivatives have been reinterpreted in Lorentz–Minkowski three-space. [ABSTRACT FROM AUTHOR]

In this work, we study the modulation instability (MI) and closed‐form soliton solution of the modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV‐ZK) equation with a truncated M‐fractional derivative. The mKdV‐ZK equation can be used to describe the behavior of ion‐acoustic waves in plasma and the propagation of surface waves in deep water with nonlinear and dispersive effects in fluid dynamics. To execute a closed soliton solution, we implement two dominant techniques, namely, the improved F‐expansion scheme and unified solver techniques for the mKdV‐ZK equation. Under the condition of parameters, the obtained solutions exhibit hyperbolic, trigonometric, and rational functions with free parameters. Using the Maple software, we present three‐dimensional (3D) plots with density plots and two‐dimensional (2D) graphical representations for appropriate values of the free parameters. Under the conditions of the numerical values of the free parameters, the obtained closed‐form solutions provided some novel phenomena such as antikink shape wave, dark bell shape, collision of kink and periodic lump wave, periodic wave, collision of antikink and periodic lump wave, collision of linked lump wave with kink shape, periodic lump wave by using improved F‐expansion method and kink shape, diverse type of periodic wave, singular soliton, and bright bell and dark bell‐shape wave phenomena by using unified solver method. The comparative effects of the fractional derivative are illustrated in 2D plots. We also provided a comparison between the results obtained through the suggested scheme and those obtained by other approaches, showing some similar solutions and some that are different. Besides, to check of stability and instability of the solution, the MI analysis of the given system is investigated based on the standard linear stability analysis and the MI gain spectrum analysis. With the use of symbolic calculations, the applied approach is clear, simple, and elementary, as demonstrated by the more broad and novel results that are obtained. It may also be applied to more complex phenomena. [ABSTRACT FROM AUTHOR]

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

The dynamic electromechanical coupling behavior of composite materials is highly dependent on external excitation frequency. While degradable biomolecular materials typically exhibit lower piezoelectric coefficients compared to ceramics, neglecting their frequency‐dependent performance in the design of piezoelectric devices further leads to less efficient utilization of their piezoelectric properties. This oversight greatly hinders the practical application of these materials. To address this, a novel fractional derivation (FD) theory‐assisted model is introduced to reversely design the glycine‐polyvinyl alcohol (PVA) thin films for versatile enhanced bio‐applications. An electromechanical coupling model incorporating FD theory is developed to learn the relationships between FD parameters, film dimensions, and dynamic electromechanical properties. This model accurately predicts the electromechanical performance of the films across a wide frequency range, validated by both finite element simulations and experimental results. This therefore allows to establish key design principles for piezoelectric thin film in bioenergy harvesting and sensing, by tailoring thin film parameters to enhance the piezoelectric performance at specific stimuli frequencies. Demonstrations of glycine‐PVA film devices guided by this model reveal excellent performance in ultrasonic energy harvesting and carotid artery bio‐signal sensing. This study provides a robust theoretical framework for designing and optimizing biodegradable piezoelectric materials for various practical applications. [ABSTRACT FROM AUTHOR]

Richards in [35] proposed a modification of the logistic model to model growth of biological populations. In this paper, we give a new representation (or characterization) of the solution to the Richards‐type fractional differential equation Dαy(t)=y(t)·(1+a(t)yβ(t))$$ {\mathcal{D}}&amp;#x0005E;{\alpha }y(t)&amp;#x0003D;y(t)\cdotp \left(1&amp;#x0002B;a(t){y}&amp;#x0005E;{\beta }(t)\right) $$ for t≥0$$ t\ge 0 $$, where a:[0,∞)→ℝ$$ a:\left[0,\infty \right)\to \mathrm{\mathbb{R}} $$ is a continuously differentiable function on [0,∞),α∈(0,1)$$ \left[0,\infty \right),\alpha \in \left(0,1\right) $$ and β$$ \beta $$ is a positive real constant. The obtained representation of the solution can be used effectively for computational and analytic purposes. This study improves and generalizes the results obtained on fractional logistic ordinary differential equation. [ABSTRACT FROM AUTHOR]

This study employs the Lie symmetry technique to explore the symmetry features of the time fractional Kupershmidt equation. Specifically, we use the Lie symmetry technique to derive the symmetry generators for this equation, which incorporates a conformal fractional derivative. We use the symmetry generators to transform the fractional partial differential equation into a fractional ordinary differential equation, thereby simplifying the analysis. The obtained reduced equation is of fourth order nonlinear ordinary differential equation. To find the wave solutions, F/G‐expansion process has been used to obatin different types of solutions of the time‐fractional Kuperschmidt equation. The obtained wave solutions are hyperbolic and trigonometric in nature. We then use Maple software to visually depict these wave solutions for specific parameter values, providing insights into the behaviour of the system under investigation. Peak and kink wave solutions are achieved for the given problem. [ABSTRACT FROM AUTHOR]

Mass transfer limitations may considerably affect the rate of a heterogeneous catalytic process. The catalyst effectiveness factor is a quantitative measure of the impact of the diffusion process inside a catalyst particle. The effectiveness factor is derived from the solution of the steady-state reaction-diffusion problem. Herein, we simulate the steady-state reaction-diffusion equation with space-fractional derivative and linear reaction kinetics. The solution to the problem is obtained numerically using the Haar wavelet collocation method. The effect of the anomalous diffusion exponent on the catalyst effectiveness factor and process parameters, e.g. reactor volume and catalyst mass, is demonstrated. We anticipate that the process efficiency will be notably improved by changing the diffusion regime from standard to superdiffusive. [ABSTRACT FROM AUTHOR]

In the current contribution, we handle the inverse problem of identifying and reconstructing a space-depend source term in a degenerate time-fractional parabolic equation from a final observed data. By the use the optimal control framework, the inverse problem is first reformulated as an optimization problem. After discussing the well-posedness of the state problem, we prove the existence of a minimizer and establish the first-order necessary condition. This last one is used to deduce our main results concerning the uniqueness and stability. In the numerical aspect, we use the decent gradient method to solve the associated optimization problem. Some numerical examples are included to illustrate the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

The article focuses on the presentation and comparison of selected heuristic algorithms for solving the inverse problem for the anomalous diffusion model. Considered mathematical model consists of time-space fractional diffusion equation with initial boundary conditions. Those kind of models are used in modelling the phenomena of heat flow in porous materials. In the model, Caputo's and Riemann-Liouville's fractional derivatives were used. The inverse problem was based on identifying orders of the derivatives and recreating fractional boundary condition. Taking into consideration the fact that inverse problems of this kind are ill-conditioned, the problem should be considered as hard to solve. Therefore,to solve it, metaheuristic optimization algorithms popular in scientific literature were used and their performance were compared: Group Teaching Optimization Algorithm (GTOA), Equilibrium Optimizer (EO), Grey Wolf Optimizer (GWO), War Strategy Optimizer (WSO), Tuna Swarm Optimization (TSO), Ant Colony Optimization (ACO), Jellyfish Search (JS) and Artificial Bee Colony (ABC). This paper presents computational examples showing effectiveness of considered metaheuristic optimization algorithms in solving inverse problem for anomalous diffusion model. [ABSTRACT FROM AUTHOR]

This study presents the application of the ϕ 6 model expansion technique to find exact solutions for the (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation under Jumarie's modified Riemann–Liouville derivative (JMRLD). The suggested method captures dark, periodic, traveling, and singular soliton solutions, providing deep insights into wave behavior. Clear graphics demonstrate that the solutions are greatly affected by changes in the fractional order, deepening our understanding and revealing the hidden dynamics of wave propagation. The considered equation has several applications in fluid dynamics, plasma physics, and nonlinear optics. [ABSTRACT FROM AUTHOR]

In this work, we investigate the exact solutions of (2+1)-dimensional coupled resonant Davey-Stewartson equation (DSE) with the properties of truncated M-fractional derivative. It is a significant equation system that models wave packets in different fields. DSE and its coupling with other system have interesting properties and many applications in the fields of nonlinear sciences. The concept of resonant is quite important in optics, plasma physics, magneto-acoustic waves and fluid dynamics. In order to use newly designed integration method known as modified Sardar subequation method (MSSEM), we first convert the (2+1)-dimensional fractional coupled resonant DSE into a set of nonlinear ordinary diferential equations. To acquire the exact solutions, the ordinary differential equation is solved by applying the homogeneous balance method between the highest power terms and the highest derivative of the ordinary differential equation. The optical soliton solutions of the resultant system are investigated using different cases and physical constant values. The aforementioned technique is applied to the considered model, yielding several kinds of soliton solutions, such as mixed, dark, singular, bright-dark, bright, complex and combined solitons. In addition, exponential, periodic, and hyperbolic solutions are also obtained. Also, we plot the 2D, and 3D graphs with the associated parameter values to visualize the solutions. The findings of this work will help to identify and clarify some novel soliton solutions and it is expected that the solutions obtained will play a vital role in the fields of physics and engineering. [ABSTRACT FROM AUTHOR]

The article is about the study of reflected waves through the surface of an elastic solid. The medium considered for the propagation of waves is homogeneous isotropic with semiconductor properties. The thermoelastic properties of the medium are a function of temperature. The governing equations are formulated by using non-local elastic theory. The conduction process of heat is studied by using the concept of three-phase lag along with fractional order time derivative. After reflecting through the surface one transverse and three longitudinal waves travel back into the media. The reflection coefficients and their energy ratios are computed analytically. Numerically simulated results are obtained for the intrinsic semiconductor material Silicon to depict the effect of temperature dependency parameters, nonlocal parameters, and time derivative fractional order on the different reflection coefficients. Some published results are also discussed as special cases. [ABSTRACT FROM AUTHOR]

The aim of this research is to explore the influence of fractional derivatives on solutions of various physical forms within a single mathematical model. By examining the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equation with the inclusion of temporal Atangana-conformable derivatives, and utilizing two effective methods, we observe distinct variations in the impact of the fractional derivative on altering the inherent physical properties of the proposed model. This research highlights an important function of the fractional derivative, indicating its role as a memory transmitter. This role illustrates how the physical characteristics inherent in the proposed application evolve as the value of the fractional derivative changes within the range of (0, 1) and nears that of the integer derivative. Finally, we provide illustrative 2D-plots to reinforce the findings of this study. [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]

The article deals with fractional viscoelastic models, including conformable derivatives. The Maxwell model and Zener model involving conformable derivative are studied for relaxation modulus as well as for creep compliance. We obtain some mechanical properties from both models, which is very useful for studying material viscoelasticity. Interesting results are extracted and compared to experimental data. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with introducing a new fractional derivative operator with a generalized exponential kernel. The benefit of the new definition over existing exponential kernel operators is that it is possible to extend the new operator to a fractional derivative operator with a singular kernel. We introduced the corresponding fractional integral operator and the extended derivative operator which includes integrable singular kernel. Furthermore, we expressed the new fractional derivative and integral operators as convergent series in terms of the Riemann–Liouville integral operator. Some relationships, characteristics and comparisons with other operators were studied. Finally, we discussed the dynamics of a nonlinear fractional order model incorporating the proposed extended fractional derivative. [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]

In recent years, there has been a growing interest among researchers in the study of shallow water waves, driven by their wide applicability across various scientific disciplines. Within this field, the Camassa–Holm equation has garnered significant attention due to its ability to capture complex wave phenomena. However, researchers have extended their investigations to the Modified Camassa–Holm equation (MCH), which incorporates principles from fractional calculus, thereby offering a more comprehensive framework for understanding intricate dynamics. This article focuses on the analytical solution of the Fractional Modified Camassa–Holm (FMCH) equation, an extension of the MCH, achieved through a novel method known as the Residual Power Series Method (RPSM). By utilizing RPSM and initial conditions, the main aim is to unveil an analytical solution for the FMCH, contributing to an improved understanding and modeling of the complex real-world phenomena observed in waves.

Abstract A fractional model for the kinetics of hepatitis B transmission was developed. The hepatitis B virus significantly affects the world’s economic and health systems. Acute and chronic carrier phases play a crucial part in the spread of the HBV infection. The Hepatitis B infection can be spread by chronic carriers even though they show no symptoms. In this article, we looked into the Hepatitis B virus’s various stages of infection-related transmission and built a nonlinear epidemic. Then, a fractional hepatitis B virus model using a Caputo derivative and vaccine effects is created. First, we determined the proposed model’s essential reproductive value and equilibria. With the aid of Fixed Point Theory, a qualitative analysis of the problem’s approximative root has been produced. The Adams-Bashforth predictor-corrector scheme is used to aid in the iterative approximate technique’s evaluation of the fractional system under consideration that has the Caputo derivative. In the final section, a graphical representation compares various noninteger orders and displays the discovered scheme findings. In this study, we’ve utilized Artificial Neural Network (ANN) techniques to partition the dataset into three categories: training, testing, and validation. Our analysis delves deep into each category, comprehensively examining the dataset’s characteristics and behaviors within these divisions. The study comprehensively analyzes the fractional HBV transmission model, incorporating both mathematical and computational approaches. The findings contribute to a better understanding of the dynamics of HBV infection and can inform the development of effective public health interventions.

A hybrid nanofluid is a kind of nanofluid that is made by combining a base fluid with two distinct types of nanomaterials. Compared to nanofluids, they have been discovered to have better thermal properties and stability, which makes them viable options for thermal applications such as heat sinks, solar thermal systems, automotive cooling systems, and thermal energy storage. Moreover, the research of nanofluids is typically limited to models with partial differential equations of integer order, which neglect the heredity characteristics and memory effect. To overcome these shortcomings, this study seeks to enhance our understanding of heat transfer in hybrid nanofluids by considering fractional Maxwell models. In time-fractional problems, one of the most significant and useful tools is the Caputo fractional derivative. Therefore, the fractional-order derivatives are approximated using the Caputo derivative. However, the integer-order derivatives are discretized using an implicit finite difference method, namely, the Crank–Nicolson method. It is an unconditionally stable and a second-order method in time. The impact of pertinent flow parameters on fluid motion and heat transfer characteristics is examined and displayed in numerous graphs. The results indicate that the volume concentration of hybrid nanoparticles boosts temperature and Nusselt number. Moreover, increasing the magnetic parameter increases Lorentz’s resistive forces, which reduces the velocity and raises the temperature of the fluid, and these effects are more dominant at t=5t=5.

This article presents a parameter uniform convergence numerical scheme for solving time fractional order singularly perturbed parabolic convection-diffusion differential equations with a delay. We give a priori bounds on the exact solution and its derivatives obtained through the problem’s asymp-totic analysis. The Euler’s method on a uniform mesh in the time direction and the extended cubic B-spline method with a fitted operator on a uniform mesh in the spatial direction is used to discretize the problem. The fitting factor is introduced for the term containing the singular perturbation pa-rameter, and it is obtained from the zeroth-order asymptotic expansion of the exact solution. The ordinary B-splines are extended into the extended B-splines. Utilizing the optimization technique, the value of μ (free param-eter, when the free parameter μ tends to zero the extended cubic B-spline reduced to convectional cubic B-spline functions) is determined. It is also demonstrated that this method is better than some existing methods in the literature.

Abstract We took the silt soil in the Yellow River flood area of Zhengzhou City as the research object and carried out triaxial shear and triaxial creep tests on silt soil with different moisture contents (8%, 10%, 12%, 14%) to analyze the effect of moisture content on silt soil. In addition, the influence of moisture contents on soil creep characteristics and long-term strength was analyzed. Based on the fractional derivative theory, we established a fractional derivative model that can effectively describe the creep characteristics of silt soil in all stages, and used the Levenberg–Marquardt algorithm to inversely identify the relevant parameters of the fractional derivative creep model. The results show that the shear strengths of silt soil samples with moisture contents of 8%, 10%, 12% and 14% are 294 kPa, 236 kPa, 179 kPa and 161 kPa, respectively. The shear strength of silt soil decreases with increasing moisture content. When the moisture content increases, the cohesion of the silt soil decreases. Under the same deviatoric stress, the higher the moisture content of the silt soil, the greater the deformation will be. The long-term strength of silt soil decreases exponentially with the increase of moisture content. If the moisture content is 12%, the long-term strength loss rate of silt soil is the smallest, with a value of 32.96%. The calculated values of our creep model based on fractional derivatives have a high goodness of fit with the experimental results. This indicates that our model can better simulate the creep characteristics of silt soil. This study can provide a theoretical basis for engineering construction and geological disaster prevention in silt soil areas in the Yellow River flood area.

Abstract This study explores the fractional form of modified Korteweg-de Vries-Kadomtsev-Petviashvili equation. This equation offers the physical description of how waves propagate and explains how nonlinearity and dispersion may lead to complex and fascinating wave phenomena arising in the diversity of fields like optical fibers, fluid dynamics, plasma waves, and shallow water waves. A variety of solutions in different shapes like bright, dark, singular, and combo solitary wave solutions have been extracted. Two recently developed integration tools known as generalized Arnous method and enhanced modified extended tanh-expansion method have been applied to secure the wave structures. Moreover, the physical significance of obtained solutions is meticulously analyzed by presenting a variety of graphs that illustrate the behaviour of the solutions for specific parameter values and a comprehensive investigation into the influence of the nonlinear parameter on the propagation of the solitary wave have been observed. Further, the governing equation is discussed for the qualitative analysis by the assistance of the Galilean transformation. Chaotic behavior is investigated by introducing a perturbed term in the dynamical system and presenting various analyses, including Poincare maps, time series, 2-dimensional 3-dimensional phase portraits. Moreover, chaotic attractor and sensitivity analysis are also observed. Our findings affirm the reliability of the applied techniques and suggest its potential application in future endeavours to uncover diverse and novel soliton solutions for other nonlinear evolution equations encountered in the realms of mathematical physics and engineering.

This article discussed a new fractional model that included governing equations describing mass and thermal diffusion in elastic materials. We formulated the thermal and mass diffusion equations using the Atangana-Baleanu-Caputo (ABC) fractional derivative and the Moore-Gibson-Thomson (MGT) equation. In addition to the fractional operators, this improvement included incorporating temperature and diffusion relaxation periods into the Green and Naghdi model (GN-Ⅲ). To verify the proposed model and analyze the effects of the interaction between temperature and mass diffusion, an infinite thermoelastic medium with a cylindrical hole was considered. We analyzed the problem under boundary conditions where the concentration remained constant, the temperature fluctuated and decreased, and the surrounding cavity was free from any external forces. We applied Laplace transform techniques and Mathematica software to generate calculations and numerical results for various field variables. We then compared the obtained results with those from previous relevant models. We have graphically depicted the results and extensively examined and evaluated them to understand the effects of the relationship between temperature and mass diffusion in the system.

Abstract As delays are common, persistent, and ingrained in daily life, it is imperative to take them into account. In this work, we explore the averaging principle for impulsive Atangana–Baleanu fractional stochastic delay differential equations driven by Lévy noise. The link between the averaged equation solutions and the equivalent solutions of the original equations is shown in the sense of mean square. To achieve the intended outcomes, fractional calculus, semigroup properties, and stochastic analysis theory are used. We also provide an example to demonstrate the practicality and relevance of our research.

Abstract The challenge of predicting time series with limited data has evolved over time due to nonlinearity, complexity, and limited information. It can be perceived as a mapping of dynamical systems in one-dimensional space. This article proposes a neural grey system to tackle this challenge. The system enhances its ability to fit nonlinearity by employing polynomials, captures complexity through a fractional-order cumulant operator, and resolves information-poor uncertainty by utilizing grey system modeling techniques. The model effectively integrates research findings from neural computing, uncertainty theory, and complexity theory at a theoretical level. It accurately describes dynamic processes of complex systems. Additionally, we have reduced the complexity of calculations in the algorithm design. We selected a dataset of total retail sales of consumer goods to test the model’s validity and applicability. Our experiments demonstrate that the newly proposed grey forecasting model can effectively forecast time series with small samples, offering good forecasting outcomes and generalization ability.

Artificial frozen soil can be regarded as the blending of ideal solid and ideal fluid in a certain proportion. Its mechanical properties neither comply with the Hooke’s law nor the Newton’s viscosity law, but obey certain relationship between them. Fractional derivative can well describe this blending phenomenon. Uniaxial compression tests were performed on the expansive soils of Hefei under different freezing temperatures, and the influence law of freezing temperature on stress and strain were obtained. The fractional derivative was introduced into the exponential model, and the improved exponential model was the fractional exponential model of stress-strain under uniaxial compression of artificial frozen soil. By taking natural logarithms on both sides of the improved model, the stress-strain linear equations at different temperatures were obtained, and the fractional derivative model parameters were determined by solving the established equations. To further verify the applicability of the established model, a set of triaxial shear tests of frozen silty clay in Nanjing were quoted and the influence of confining pressure was taken into account in the fractional order coefficient. The stress-strain fractional order exponential model was improved to take the influence of confining pressure into account. Comparing the calculated results of the improved stress strain exponential equation of artificial frozen soil with the experimental results, the results show that the calculated results are in good agreement with the experimental results and can accurately predict the changing trend of the shear stress strain curves under uniaxial compression and triaxial compression. The improved fractional derivative model has few parameters and definite physical meaning, which is convenient for engineering application. The current model is only applicable to the strain hardening type. In order to further describe the mechanical properties of the strain softening type, the further study is to establish a fractional exponential model of the strain softening type by considering the damage in the model. At the same time, how to reflect the influence of the structure and anisotropy of frozen soil on the stress-strain in the model will also be investigated.

In this paper a new definitionn of fractional derivative and fractional integral in the sense ofconformable derivative type is presented. This form of definition shows that it is more compatible withclassical natural definition of derivative and is more convinient fractional derivative one. We will definethis for 0 ≤ α < 1 and n − 1 ≤ α < n and further, if α = 1 the definition coincides with the classicaldefinition of derivative of first order.

This work presents a significant contribution to the field of mathematical analysis by introducing θ-Fℵ-contractions of Darbo-type. These newly introduced contractions serve as pivotal tools within our study. By employing these contractions, we systematically establish several pivotal fixed point theorems that allow us to guarantee the existence of fixed points in the Banach spaces. We construct particular examples in order to support the validity and effectiveness of our theoretical results. Moreover, we extend the utility of our established theorems by applying them to solve real-world problems. Specifically, we direct our attention to exploring the existence of solutions to the fractional order Lymphatic filariasis infection model.