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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))$$ {\mathcal{D}}^{\alpha }y(t)=y(t)\cdotp \left(1+a(t){y}^{\beta }(t)\right) $$ 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]

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]

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]

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. [ABSTRACT FROM AUTHOR]

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

In applied sciences, besides the importance of obtaining the analytical solutions of differential equations with constant coefficients, the qualitative analysis of the solutions of such equations is also very important. Due to this importance, in this study, a qualitative analysis of the solutions of a delayed and constant coefficient fractal differential equation with more than one fractional derivative was performed. In the equation under consideration, the derivatives are the Riemann–Liouville fractional derivatives. In the proof of the obtained results, Laplace transform formulas of the Riemann–Liouville fractional derivative and some inequalities are used. We also provide some examples to check the accuracy of our results. [ABSTRACT FROM AUTHOR]

In this paper, we propose an anisotropic diffusion equation based on fractional‐order tensor diffusion applied on noisy images. We start by investigating the existence and uniqueness results of the proposed model. We also provide a discretization scheme based on the finite difference method. In order to prove the efficacy of our proposed approach, we deliver some simulations, which show robustness and remarkable results against noise while preserving edges and corners, compared to some well‐known models in literature. For a fair comparison, we manage to obtain the optimal parameters of every model thanks to PSNR and SSIM. We demonstrate the robustness efficacy of our model against high level of noise. [ABSTRACT FROM AUTHOR]

We reconstruct the initial functions from the trace of the solution of an initial value problem for the wave equation on the light cone. A method to recover the initial function from the solution of the wave equation on the light cone is already known for odd spatial dimensions. We generalize their work to the fractional wave equation and all dimensions. In other words, we present a method to reconstruct the initial functions from the solution of the fractional wave equation on the light cone. [ABSTRACT FROM AUTHOR]

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

As challenges in gas extraction from coal mines increase, precise measurement of permeability becomes crucial. This study proposes a novel pulse transient method based on a fractional derivative model derived on physical fractal space, incorporating operator algebra and the mechanics–electricity analogy to derive a new control equation that more accurately delineates the permeability evolution in coal. To validate the approach, permeability experiments were conducted on coal samples under mining stress conditions. The results showed that the adoption of a physically meaningful fractional-order relaxation equation provides a more accurate description of non-Darcy flow behaviour in rocks than traditional integer-order control equations. Additionally, the method proved effective across different rock types, verifying its broad applicability. By establishing a new theoretical foundation, this approach illustrates how the microscale fractal structure of rocks is fundamentally linked to their macroscale fractional responses, thereby enhancing the understanding of fractional modelling methods in rock mechanics and related domains. [ABSTRACT FROM AUTHOR]

This paper presents the mathematical and physical analysis, as well as distinct types of exact wave solutions, of an important fluid flow dynamics model called the truncated M-fractional (1+1)-dimensional nonlinear Estevez–Mansfield–Clarkson (EMC) equation. This model is used to explain waves in shallow water, fluid dynamics, and other areas. We obtain kink, bright, singular, and other types of exact wave solutions using the modified extended direct algebraic method and the improved (G ′ / G) -expansion method. Some solutions do not exist. These solutions may be useful in different areas of science and engineering. The results are represented as three-dimensional, contour, and two-dimensional graphs. Stability analysis is also performed to check the stability of the corresponding model. Furthermore, modulation instability analysis is performed to study the stationary solutions of the corresponding model. The results will be helpful for future studies of the corresponding system. The methods used are easy and useful. [ABSTRACT FROM AUTHOR]

A novel morphing activation function is proposed, motivated by the wavelet theory and the use of wavelets as activation functions. Morphing refers to the gradual change of shape to mimic several apparently unrelated activation functions. The shape is controlled by the fractional order derivative, which is a trainable parameter to be optimized in the neural network learning process. Given the morphing activation function, and taking only integer-order derivatives, efficient piecewise polynomial versions of several existing activation functions are obtained. Experiments show that the performance of polynomial versions PolySigmoid, PolySoftplus, PolyGeLU, PolySwish, and PolyMish is similar or better than their counterparts Sigmoid, Softplus, GeLU, Swish, and Mish. Furthermore, it is possible to learn the best shape from the data by optimizing the fractional-order derivative with gradient descent algorithms, leading to the study of a more general formula based on fractional calculus to build and adapt activation functions with properties useful in machine learning. [ABSTRACT FROM AUTHOR]

This study focuses on the mathematical and physical analysis of a truncated M-fractional (2+1)-dimensional nonlinear Kadomtsev–Petviashvili-modified equal-width model. The distinct types of the exact wave solitons of an important real-world equation called the truncated M-fractional (2+1)-dimensional nonlinear Kadomtsev–Petviashvili-modified equal-width (KP-mEW) model are achieved. This model is used to explain ocean waves, matter-wave pulses, waves in ferromagnetic media, and long-wavelength water waves. The diverse patterns of waves on the oceans are yielded by the Kadomtsev–Petviashvili-modified equal-width (KP-mEW) equation. We obtain kink-, bright-, and periodic-type soliton solutions by using the exp a function and modified extended tanh function methods. The solutions are more valuable than the existing results due to the use of a truncated M-fractional derivative. These solutions may be useful in different areas of science and engineering. The methods applied are simple and useful. [ABSTRACT FROM AUTHOR]

Using the position as an independent variable, and time as the dependent variable, we derive the function P (±) = ± 2 m (H - V (q)) , which generates the space evolution under the potential V (q) and Hamiltonian H . No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian ( - H ). While the classical dynamics do not change, the corresponding Quantum operator P ^ (±) naturally leads to a 1/2-fractional time evolution, consistent with a recent proposed space–time symmetric formalism of the Quantum Mechanics. Using Dirac’s procedure, separation of variables is possible, and while the two-coupled position-independent Dirac equations depend on the 1/2-fractional derivative, the two-coupled time-independent Dirac equations lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of P ^ (±) and the kinetic energy K 0 (separation constant) is the coupling strength. Thus, we obtain a pair of coupled states for systems with finite forces, not necessarily stationary states. The potential shifts for the harmonic oscillator (HO) are ± ħ ω / 2 , and the corresponding pair of states are coupled for K 0 ≠ 0 . No time evolution is present for K 0 = 0 , and the ground state with energy ħ ω / 2 is stable. For K 0 > 0 , the ground state becomes coupled to the state with energy - ħ ω / 2 , and this coupling allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of K 0 = k ħ ω ( k = 1 , 2 , … ). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case K 0 = 0 leads to plane-waves-like solutions at the threshold. Above the threshold ( K 0 > 0 ), we obtain a plane-wave-like solution, and for the bounded states ( K 0 < 0 ), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

An adaptive finite difference scheme for variable-order fractional-time subdiffusion equations in the Caputo form is studied. The fractional-time derivative is discretized by the L1 procedure but using nonhomogeneous timesteps. The size of these timesteps is chosen by an adaptive algorithm to keep the local error bounded around a preset value, a value that can be chosen at will. For some types of problems, this adaptive method is much faster than the corresponding usual method with fixed timesteps while keeping the local error of the numerical solution around the preset values. These findings turn out to be similar to those found for constant-order fractional diffusion equations. [ABSTRACT FROM AUTHOR]

In this article, new results are investigated in the context of the recently introduced Abu-Shady–Kaabar fractional derivative. First, we solve the generalized Legendre fractional differential equation. As in the classical case, the generalized Legendre polynomials constitute notable solutions to the aforementioned fractional differential equation. In the sense of the fractional derivative of Abu-Shady–Kaabar, we establish important properties of the generalized Legendre polynomials such as Rodrigues formula and recurrence relations. Special attention is also devoted to another very important property of Legendre polynomials and their orthogonal character. Finally, the representation of a function f ∈ L α 2 ([ − 1 , 1 ]) in a series of generalized Legendre polynomials is addressed. [ABSTRACT FROM AUTHOR]

The aim of this article is to find a geometric and physical interpretation of fractional order derivatives for a general class of functions defined over a bounded or unbounded domain. We show theoretically and geometrically that the absolute value of the fractional derivative value of a function is inversely proportional to the area of the triangle. Further, we prove geometrically that the fractional derivatives are inversely proportional to the classical integration in some sense. The established results are verified numerically for non‐monotonic, trigonometric, and power functions. Further, this article establishes a significant connection between the area of the projected fence and the area of triangles. As the area of triangles decreases, the area of the projected fence increases, and vice versa. We calculate the turning points of the fractional derivative values of different functions with respect to order α$$ \alpha $$, including non‐monotonic, trigonometric, and power functions. In particular, we demonstrate that for the power function xβ$$ {x}^{\beta } $$, with β$$ \beta $$ being a positive real number, the value α=0.5$$ \alpha =0.5 $$ is a turning point when x=β$$ x=\beta $$. However, for x>β$$ x>\beta $$, the turning point shifts to the left of point (0.5,0)$$ \left(0.5,0\right) $$ and shifts to the right of point (0.5,0)$$ \left(0.5,0\right) $$ for x<β.$$ x<\beta. $$ We discuss the physical interpretation of fractional order derivatives in terms of fractional divergence. We present some applications of fractional tangent lines in the field of numerical analysis. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

The study consists of the distinct types of the exact soliton solutions to an important model called the beta-time fractional (1 + 1)-dimensional non-linear Van der Waals equation. This model is used to explain the motion of molecules and materials. The Van der Waals equation explains the phase separation phenomenon. Noncovalent Van der Waals or dispersion forces usually have an effect on the structure, dynamics, stability, and function of molecules and materials in different branches of science, including biology, chemistry, materials science, and physics. Solutions are obtained, including dark, dark-singular, periodic wave, singular wave, and many more exact wave solutions by using the modified extended tanh function method. Using the fractional derivatives makes different solutions different from the existing solutions. The gained results will be of high importance in the interaction of quantum-mechanical fluctuations, granular matters, and other applications of the Van der Waals equation. The solutions may be useful in distinct fields of science and civil engineering, as well as some basic physical ones like those studied in geophysics. The results are verified and represented by two-dimensional, three-dimensional, and contour graphs by using Mathematica software. The obtained results are newer than the existing results. Stability analysis is also performed to check the stability of the concerned model. Furthermore, modulation instability is studied to study the stationary solutions of the concerned model. The results will be helpful in future studies of the concerned system. In the end, we can say that the method used is straightforward and dynamic, and it will be a useful tool for debating tough issues in a wide range of fields. [ABSTRACT FROM AUTHOR]

To model several engineering and physical models, the approach of the fractional derivative is highly anticipated. As compared to the ordinary derivatives, the fractional derivatives with more flexibility can estimate the data due to the involvement of the fractional-order derivatives. Due to these advantages of the fractional approach, this study communicates with the determination of the fractional-based exact outcomes of an oscillatory rectangular duct problem of a generalized second-grade fluid. The approach of the fractional operator is involved in the relationship of the constitutive equations. For cosine oscillation of the rectangular duct, exact results of the magnetized unsteady flow problem are evaluated through the technique of Laplace transform with double finite Fourier sine transform. This study concludes that the velocity field exhibits escalating behavior relative to the improved fractional parameter. Moreover, the magnetic parameter with increasing values declines the flow field while the accelerating values of the fluid parameter enhance the velocity field. [ABSTRACT FROM AUTHOR]

In this article, we investigate the fractional Volterra-Fredholm integro-differential equations. These equations appear in several applications such as control theory, biology, and particle dynamics in physics. We derive a numerical method based on the operational matrix method to solve this class of integro-differential equations. We prove the existence and uniqueness of the exact solution. Additionally, we demonstrate the uniform convergence of the numerical solutions to the exact solution. We present several numerical examples to show the numerical efficiency of the proposed method. In the first example, we choose a linear problem and find that the approximate solution converges to the exact solution when the number of block pulse functions is very large. In the next two examples, we consider the nonlinear case and compute the L2-local truncation error since exact solutions are not available. The error was of order 10−12. Furthermore, we sketch the graph of the approximate solutions for different values of the fractional derivative to observe the influence of the fractional derivative on the profile of the solutions. Theoretical and numerical results show that the proposed method is accurate and can be applied to other nonlinear problems in science. [ABSTRACT FROM AUTHOR]

Closed-form expressions for the impulse and step responses of commensurable linear time-invariant systems are deduced and exemplified. The algorithm is based on obtaining the solution in terms of the α-exponential monomials that generate a vector space containing the solutions of such systems. Several examples are considered, together with the numerical aspects involved in calculating the related series, which show the accuracy and effectiveness of the approach. [ABSTRACT FROM AUTHOR]

We further extend the results of other researchers on existence theory to homogeneous fractional Cauchy–Euler equations ∑ i = 1 m d i x α i D α i u (x) + μ u (x) = 0 , α i > 0 , with the derivatives in Caputo or Riemann–Liouville sense. Unlike the existing works, we consider multi-term equations without any restrictions on the order of fractional derivatives. The results are based on the characteristic equations which generate the solutions. Depending on the roots of the characteristic equations (real, multiple, or complex), we construct the corresponding solutions and prove their linear independence. [ABSTRACT FROM AUTHOR]

The characterisation of edges in medical images is critical for disease diagnosis. However, existing systems are still deficient in this task. Traditionally, integer-based derivative operators are employed due to their efficiency in time complexity but lack the ability to track nonlocal and non-singular edge maps. This study proposes a new mask based on Atangana-Beleanu fractional operator. This operator has the same complexity as the state-of-the-art integer-order derivative mask known as the Canny edge detector but has the added advantage to characterise more efficiently nonlocal and nonsingular edge maps. Performance evaluation of the proposed mask reveals an enhanced performance in the context of robustness to noise and quality edge extraction, a significant contribution to literature. The metric for the study is the signal-to-noise ratio, and the structural similarity index and appropriate mask observed is a mask of dimension greater than five. [ABSTRACT FROM AUTHOR]

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

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

In this study, we investigate the effect of Clarke's subdifferential type on the optimal control results for fractional differential systems of order 1

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

This article presents a parameter uniform convergence numerical scheme for solving time fractional order singularly perturbed parabolic convectiondiffusion differential equations with a delay. We give a priori bounds on the exact solution and its derivatives obtained through the problem's asymptotic 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 parameter, 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 parameter, 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 FROM AUTHOR]

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

Summary: Time history analysis is sometimes used in an estimation of the wind‐induced response of a tall building. However, time history analysis for the wind‐induced behavior by the ensemble‐averaging wind force costs much computation time. This paper provides a reliable prediction method for the wind‐induced response of the viscoelastic (VE)‐damped system considering its frequency dependency coupling with frame damping effect with frequency spectral method. VE damper used in high‐rise buildings can dissipate energy from excessive vibration induced by seismic or wind excitation. Fractional derivative (FD) model of VE dampers can express VE frequency dependency clearly, but it is computationally complicated. The herein proposed prediction method is based on frequency spectral method and evaluated wind‐induced responses of the single degree of freedom (SDOF) VE‐damped system with a FD VE damper subjected to the respective 1st modal along‐ and across‐wind force. The maximum error of wind‐induced responses of the VE‐damped system, such as the root mean square value of responses, the total input energy, and the total energy dissipation, is within 5%. In summary, the proposed method had high accuracy in the prediction of wind‐induced responses of the VE‐damped system considering its frequency dependency with the coupling effect of frame damping. [ABSTRACT FROM AUTHOR]

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.

AbstractIn this article, truncated M-fractional coupled nonlinear Schrodinger equation (NLSE) with quadratic–cubic nonlinearity is under observation. The studied model is composed of chromatic dispersion, magneto-optic parameter and inter-modal dispersion. The NLSE is the most significant physical model to explain the fluctuations of optical soliton proliferation. The NLSEs have become more popular because of the clarity with which they explain a wide range of complex physical phenomena and the depth with which they display dynamical patterns via localized wave solutions. Optical soliton propagation in magneto-optic is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. The optical solitons are secured in the forms of bright, dark, singular and combo solitons. In addition, hyperbolic, periodic and exponential function solutions have been recovered. The modified Sardar subequation and enhanced modified extended tanh-expansion approaches recently developed integration tools are adopted in this study for securing the solutions. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. The effect of parameters have been observed by allotting suitable values and sketching the different shapes of the graphs.

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.