The fractional advection-dispersion equation FADE plays an important role, especially in explaining the process of solute-transport in natural porous media with different degrees of heterogeneity. However, studies of the flow and transport of contaminants in the underground environment are based on the knowledge of many functional parameters of the subsoil. The delay factor caused by nonlinear absorption is one of these parameters inheriting significant ambiguities in its estimation in heterogeneous media. In the study, a fractional advection-dispersion FADE model was developed by the implicit Euler method, based on a Grunwald and Caputo-Liousville approximation to explicitly evaluate and simulate the transport mechanism of contaminants in media with unsaturated porous materials subjected to a nonlinear sorption model. Better solute mass retention was observed for fractional advection-dispersion FADE model. Moreover, the absorption parameter of the retardation coefficient has a better impact on the motion of the solute in a saturated porous medium. In other words, it has a significant effect on the dispersion of pollutants in aquifers. This study highlights the need to characterize the aquifer to predict solute transport in porous media.

In the present paper four passive micromixers designs (G1, G2, G3 and G4) inspired on distillation columns trays were proposed. The devices performance was assessed by numerical simulations. The mixing performance was investigated for a Reynolds number range from 0.01 to 100 and channel height of 200 µm, 500 µm and 1000 µm for oil/ethanol flow. G1 and G4 designs provided high mixing index (> 0.975). The G1 device achieved superior mixing performance with a moderate pressure drop (about 0.5 MPa) due to the induced flow recirculation pattern for a relative high flow rate of 0.21 L/min, highlighting the potential use of such microdevice for scale-up and numbering-up of microdevices in modular chemical plant processing.

Our aim is to implement and test some less known numerical methods that might be used to solve an advection equation. The description is restricted to information necessary to implement and test the presented methods and it is given only for one-dimensional case.

In this work, we proposed a novel design of microfluidic mixer utilizing the amplified Marangoni chaotic advection induced by AC continuous electrowetting of a metal droplet situated in electrolyte solution, due to the linear and quadratic voltage-dependence of flow velocity at small or large voltages, respectively. Unlike previous researchers exploiting the unidirectional surface stress with DC bias at droplet/medium interface for pumping of electrolyte where the resulting flow rate is linearly proportional to the field intensity, dominance of another kind of dipolar flow pattern caused by local Marangoni stress at the drop surface in sufficiently intense AC electric field is demonstrated by both theoretical analysis and experimental observation, which exhibits a quadratic growth trend as a function of the applied voltage. The dipolar shear stress merely appears at larger voltages and greatly enhances the mixing performance by inducing chaotic advection between the neighboring laminar flow. The mixer design developed herein, on the basis of amplified Marangoni chaotic advection around a liquid metal droplet at larger AC voltages, has great potential for chemical reaction and MEMS actuator applications, because of generating high-throughput and excellent mixing performance at the same time.

This paper proposes a definition of fractional line integral, generalising the concept of fractional definite integral. The proposal replicates the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It is based on the concept of fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integrals the Gr\"unwald-Letnikov and Liouville directional derivatives are introduced and their properties described. The integral is defined first for broken line paths and afterwards to any regular curve

In this paper, we analyze the convergence of a finite difference method with the implicit forward time central space (FTCS) scheme for solving the three-dimensional advection-diffusion-reaction equation (ADRE). It is proved that the method is unconditionally convergent. We apply the scheme to obtain numerical solutions for the transport of pollutants in street tunnel problems with various reaction coefficients and various rates of change of concentrations of sources or sinks of pollution.

The weakness of the dispersion-managed link combined with optical phase conjugation to compensate for optical signal distortion caused by chromatic dispersion and nonlinear Kerr effect of standard single mode fiber is its limited structural flexibility. We propose dispersion map that can simultaneously compensate for the distorted wavelength division multiplexed signal while increasing the configurational flexibility. Each residual dispersion per span (RDPS) in the former half of the proposed link is randomly determined, and in the latter half, the arrangement order of RDPS is the same as or inverted in the former half. We confirm that the dispersion maps in which the RDPS distribution pattern in the latter half is opposite to the arrangement order in the former half are more effective in compensation, and rather, the compensation effect is better than in the dispersion map of the conventional scheme. The notable result of this paper is that the increase of flexibility can be achieved through random arrangement of RDPS in the former half, and the compensation improvement can be achieved by through inverse arrangement in the latter half which make the distribution profile of each half link roughly symmetric with respect to the midway optical phase conjugator.

In this paper we give approximations to the Mittag-Leffler functions in terms of elementary functions using different methods. This allowed us to establish a practical method we called integerization principle. This principle states that many fractional nonlinear oscillators may be solved by means of the solution to some integer-order Duffing oscillator equation. The accuracy of the obtained results is illsutrated in concrete examples. Formulas for estimating the errors in the approximations are also provided.

Autonomous differential equations of fractional order and non-singular kernel are solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we seek an implicit analytical solution.

This experimental research presents an easier model for fitting experimental results that can be simulate the dynamical behaviour of a single degree freedom system, SDFS, subjected in a forced vibration in a shaking table. The fractional damping using a Kelvin-Voigt fractional derivative rheological model, KVFRdm, to represent a fractional oscillator and compare with the experimental model.

Power series expansions are useful in approximation theory and mathematical physics. The manuscript presents several types of fractional Taylor expansions of sufficiently smooth functions. This is achieved by employing an incremental regularization procedure to the computation of the derivative. The series are constructed in an algorithmic way, which allows for its implementation in computer algebra systems.

In this present paper, we deal with the generalized (k, s)-fractional integral and differential operators recently defined by Nisar et al. and obtain some generalized (k, s)-fractional integral and differential formulas involving the class of a function as its kernels. Also, we investigate a certain number of their consequences containing the said function in their kernels.

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm's accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of problem. Under Caputo meaning the obtained results are illustrated numerically, and graphically. Geometrically, the behavior of the solution declares that the changing of the fractional derivative parameter values' in their domain alters the style of the attained solution in a symmetrical meaning and fully coinciding to the ordinary derivative value'. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for Integral, and integro-differental equations of fractional order.

The space time fractional Korteweg-de Vries equation, modified Equal Width equation and Benney-Luke Equations are solved by using simple hyperbolic tangent Ansatz method. A simple compatible wave transformation in one dimension is employed to reduce the governing equations to integer ordered ODEs. Then, the ansatz approximation is used to derive exact solutions. Some illustrative examples are presented for some particular choices of parameters and derivative orders.

Spraying various liquids (primarily aqueous solutions of various substances) is widely used in various technological processes. For most of them, high dispersibility and narrow droplet size distribution are essential. This facilitates ultrasonic spray technique. However, the higher the dispersity of droplets produced by ultrasonic spray, the lower the spray productivity. To solve this problem, we proposed a method of multistage spraying, consisting in generation of a large number of particles of large initial size (to ensure high productivity) and their subsequent destruction by propagation in a periodic ultrasonic field (to ensure small droplet sizes at high productivity). To experimentally determine the capabilities of this technique, a tubular emitter in the shape of a cylinder of stepped-variable cross-section was designed. Configuration of the ultrasonic field inside the emitter (22.2 kHz; 182 dB) implements three-stage spraying (the number of sputtering stages corresponds to the number of anti-nodes along the emitter axis). The effectiveness of proposed and developed technique suitable for generation of droplets sized less than 40 µm with performance exceeding the known ultrasonic spray techniques at least 10 times has been confirmed in the course of conducted research.

In the land use planning framework in the neighbourhood of industrial facilities, the current approach to predicting the consequences of massive toxic gas releases is generally based on Gaussian or integral models. For many years, CFD models have been more and more used in this context, in accordance with the development of High-Performance Computing (HPC). The present paper focuses on harmonizing input data for Atmospheric Transport and Dispersion (AT&D) modelling between the widely used approaches. First, a synthesis of the practice’s harmonization for atmospheric dispersion modelling within the framework of risk assessment is presented. Then, these practices are applied to a large-scale INERIS ammonia experimental release. For illustration purposes, the impact of the proposed harmonization will be evaluated using different approaches: the SLAB model, the FDS model and the Code_Saturne model. The two main focuses of this paper are the adaptation of the source term dealing with a massive release and the wind flow modelling performance using an experimental signal for CFD models inflow. Finally, comparisons between the modelling and experimental results enable checking the consistency of these approaches and reinforce the importance of the input data harmonization for each AT&D modelling approach.

The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and showed that it cannot be considered as a derivative.

In this paper, we propose to explore a nonlinear fractional mathematical model of HFMD- Hand Foot Mouth Disease with vaccinated compartment. Firstly, we prove the non negativity and boundedness for the fractional order dynamical model. The system’s existence and uniqueness are studied using the formulation of the Caputo derivative operator. The fixed-point approach was used to get results indicating the presence of at least one solution. In numerical simulation, we obtained the approximate solution by using fractional Euler method.

This paper deals with the analysis of the temperatures in a group of 36 African countries. By looking at the maximum, minimum and the range (the difference between the maximum and the minimum) and using a long memory model based on fractional integration and cointegration, we first show that all series display a long memory pattern, with a significant positive time trend in 29 countries for the maximum temperatures and in 33 for the minimum ones. Looking at the range, the estimated value for the order of integration is smaller than the one based on maximum or minimum temperatures in 17 countries. Performing fractional cointegration tests between the maximum and minimum temperatures, our results indicate that the two series cointegrate in the classical sense (i.e., with a short memory equilibrium relationship) in a group of 11 countries, and there is another group of eight countries displaying cointegration in a fractional sense. The remaining 17 countries with no evidence of cointegration are therefore at the very high risk of climate change due to the absence of long-term comovement in their maximum and minimum temperatures. Findings in this paper are of tremendous interpretations and relevance for the analysis and climate projections in Africa.

In this paper, we establish the generalized Qi-type inequality involving conformable fractional integrals. The results presented here would provide extensions of those given in earlier works.

Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.

In this paper, the 1st level general fractional derivatives of arbitrary order are defined and investigated for the first time. We start with a generalization of the Sonin condition for the kernels of the general fractional integrals and derivatives and then specify a set of the kernels that satisfy this condition and posses an integrable singularity of power law type at the origin. The 1st level general fractional derivatives of arbitrary order are integro-differential operators of convolution type with the kernels from this set. They contain both the general fractional derivatives of arbitrary order of the Riemann-Liouville type and the regularized general fractional derivatives of arbitrary order considered in the literature so far. For the 1st level general fractional derivatives of arbitrary order, some important properties including the 1st and the 2nd fundamental theorems of Fractional Calculus are formulated and proved.

Recently, Gohar et al. introduced a novel, local, and well-behaved fractional calculus. It possesses all the classical properties, and Its locality imposes simplicity and accuracy in modeling fractional order systems. In this article, we further develop the definitions and extend the classical properties of Gohar fractional calculus to address some of the open problems in Calculus. The fractional Gronwall's integral inequality, Taylor power series expansion, and Laplace transform are defined and applied to overcome some of the limitations in the classical integer-order calculus. The fractional Laplace transform is applied to solve Bernoulli-type logistic and Bertalanffy nonlinear fractional differential equations, and the criteria under which it can be applied to solve linear differential equations are investigated.

Under suitable continuity and uniqueness conditions, solutions of an α order Caputo fractional boundary value problem are differentiated with respect to boundary values and boundary points. This extends well-known results for nth order boundary value problems. The approach used is a standard technique and makes heavy use of recent results for differentiation of solutions of Caputo fractional intial value problems with respect to initial conditions and continuous dependence for Caputo fractional boundary value problems.

In this study, we consider delta fractional Sturm--Liouville (DFSL) initial value problems in the sense of delta Caputo and delta Riemann-Liouville (R--L) operators. One of the properties of delta fractional difference operators which makes it different from nabla counterpart is to shift its domain. This feature makes it more complex than the nabla fractional operator. We obtain sum representation of solutions for DFSL initial value problems with the help of $\mathcal{Z}-$ transformation. Moreover, we get analytical solutions of homogeneous DFSL problem within Riemann-Liouville (R--L) and Caputo sense, discrete Sturm--Liouville (DSL) problem, continuous fractional Sturm--Liouville (FSL) problem in the sense of R--L and Caputo operators, and continuous Sturm--Liouville (SL) differential problem. From this point of view, we compare all the solutions with each other. Consequently, we show that all results for these four eigenvalue problems are compatible with each other and approach to each other while the orders tends to one, i.e. $\Delta^{\mu }\left( \Delta x\left( t-\mu \right) \right)\cong D_{0^{+}}^{\mu }\left( x^{\prime }\left( t\right) \right)\cong \Delta^2x(n-1) \cong x^{\prime \prime }\left( t\right) =\lambda x\left( t\right),\ \mu\rightarrow1 $ . We support our results comparatively by tables and simulations in detail.

Here are given tight probabilistic inequalities that provide nearly best estimates for the Csiszar's f-divergence. These use the right and left psi -Hilfer fractional derivatives of the directing function f. Csiszar's f- divergence or the so called Csiszar's discrimination is used as a measure of dependence between two random variables which is a very essential aspect of stochastics, we apply our results there. The Csiszar's discrimination is the most important and general measure for the comparison between two probability measures. We give also other applications.

In this paper, we pursue and investigate the solutions for fractional kinetic equations, involving Bessel-Struve function by means of their Sumudu transforms. In the process, one Important special case is then revealed, and analyzed. The results obtained in terms of Bessel-Struve function are rather general in nature and can easily construct various known and new fractional kinetic equations.

Recently, representation formulae and monotonicity properties of generalized k-Bessel functions, Wk v,c., were established and studied by SR Mondal [24]. In this paper, we pursue and investigate some of their image formulae. We then extract solutions for fractional kinetic equations, involving Wk v,c, by means of their Sumudu transforms. In the process, Important special cases are then revealed, and analyzed.

Polyaniline or PANI is a conductive polymer (CPs) with flexibility in turning into conducting properties. The drawback is mainly in terms of limited mechanical properties improvement which attracted the interest in fabrication of PANI with other polymeric materials. PANI-PVL composites film was synthesized and fabricated in this work with chronological steps according to the objectives aligned in two different phases. The first phase determines the optimum parameters to synthesize nanosized PANI. The ideal initial molar ratio, r was 1, stirring speed of 600 rpm, purified by filtration, synthesis temperature of 25 oC, washed by dopant acid, acetone and distilled water. The usage of non-ionic surfactant, Triton X-100 (TX100) with 0.1 % concentration favours the formation of smaller particle size of PANI and good dispersibility compared to the usage of anionic sodium dodecyl sulphate (SDS). The synthesis of PANI in nanosized dimension has the potential to be applied for antimicrobial applications such as gloves and antimicrobial coating of frequent contact points in the near future.

Background: Previous studies have associated slowed ventricular conduction in the arrhythmogenesis mediated by the gap junction and sodium channel inhibitor heptanol in mouse hearts but did not study the propagation patterns that might contribute to the arrhythmic substrate. This study used a multi-electrode array mapping technique, to further investigate different conduction abnormalities in Langendorff-perfused mouse hearts exposed to 0.1 or 2 mM heptanol. Methods: Multi-electrode array recordings were made from the left ventricular epicardium in spontaneously beating hearts, during right ventricular regular 8 Hz pacing or S1S2 pacing. Results: In spontaneously beating hearts, heptanol at 0.1 and 2 mM significantly reduced the heart rate from 314±25 to 189±24 and 157±7 bpm, respectively (ANOVA, P<0.05 and P<0.001). During regular 8 Hz pacing, Mean LATs were increased by 0.1 and 2 mM heptanol from 7.1±2.2 ms to 19.9±5.0 ms (P<0.05) and 18.4±5.7 ms (P<0.05). The standard deviation of mean LATs was increased from 2.5±0.8 ms to 10.3±4.0 ms and 8.0±2.5 ms (P< 0.05), and the median of phase differences was significantly increased from 1.7±1.1 ms to 13.9±7.8 ms and 12.1±5.0 ms by 0.1 and 2 mM heptanol (P<0.05). P5 took a value of 0.2±0.1 ms and was not significantly altered by heptanol at 0.1 or 2 mM (1.1±0.9 ms and 0.9±0.5 ms, P>0.05). P50 was increased from 7.3±2.7 ms to 24.0±12.0 ms by 0.1 mM heptanol and then to 22.5±7.5 ms by 2 mM heptanol (P< 0.05). P95 was increased from 1.7±1.1 ms to 13.9±7.8 ms by 0.1 mM heptanol and to 12.1±5.0 ms by 2 mM heptanol (P<0.05). These changes led to increases in the absolute inhomogeneity in conduction (P5-95) from 7.1±2.6 ms to 31.4±11.3 ms, 2 mM: 21.6±7.2 ms, respectively (P<0.05). The inhomogeneity index (P5-95/P50) was significantly reduced from 3.7±1.2 to 3.1±0.8 by 0.1 mM and then to 3.3±0.9 by 2 mM heptanol (P<0.05). Conclusion: Increased activation latencies, reduced CVs and increased dispersion index of conduction were associated with both spontaneous and induced ventricular arrhythmias.

In this paper, two types of the general fractional derivatives of distributed order as well as a corresponding fractional integral of distributed type are defined and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of the one-parametric Sonin kernels with a power law singularities at the origin. The conventional fractional derivatives of distributed order based on the Riemann-Liouville and the Caputo fractional derivatives are particular cases of the general fractional derivatives of distributed order introduced in this paper.

Our aim in this paper is to establish some new Hermite-Hadamard- Mercer type integral inequalities by utilizing the fractional proportional-integral operators.For this purpose, Hermite-Hadamard-Mercer inequalities for di¤er- antiable mappings whose derivatives in absolute value are convex.

Fractional-time Schrödinger equation (FTSE) describes the evolution of quantum processes endowed with memory effects. FTSE manifestly breaks all consistency requirements of quantum field theory (unitarity, locality and compliance with the clustering theorem), unless the order of fractional differentiation and integration ( ) falls close to the standard index . Working in the context of the minimal fractal manifold (where , ), we confirm here that FTSE approximates the attributes of gravitational metric and provides an unforeseen generation mechanism for massive fields.

The gradient descent algorithm has become the standard algorithm for computing the extreme values of functions, but for multivariate functions this algorithm is mostly ineffective. This is because the convergence rate of each element is inconsistent in most cases. However, in this paper, we found that the gradient sign is a self-optimizing operator, ensuring that the convergence rate is consistent across all elements. This also explains, from an optimization perspective, the success of the Fast Gradient Sign Method (FGSM) in generating adversarial samples that are indistinguishable from the normal input, but can easily fool neural networks. We also found that the fractional order gradient is also self-optimizing, and that the convergence speed of this algorithm can be controlled by adjusting the order of the gradient. Experiments suggest that this algorithm not only generates adversarial samples faster than other algorithms, but that a single source image can generate many such samples. This algorithm is also more effective than others at generating adversarial samples from simple images.

A stochastic dominance (SD) relation can be defined by two different perspectives: One from the view of distributions, and the other one from the view of expected utilities. In early days, Fishburn investigated SD from the view of distributions and we refer this perspective as Fishburn’s SD. One of his many results was the development of fractional-order SD for continuous distributions. However, discrete fractional-order SD may not be generalized directly since some properties of fractional calculus do not have a discrete counterpart. In this paper, we develop a discrete analogue of fractional-order SD from the view of distributions. We generalize the order of SD by Lizama’s fractional delta operator, show the preservation of SD hierarchy, and formulate the utility classes that are congruent with our SD relations. This work brings a message that some results of discrete SD cannot be generalized directly from continuous SD. We characterize the difference between discrete and continuous fractional-order SD, as well as the way to handle them.

Fractional calculus gained a lot of attention in the last couple of years. Researchers discovered that processes in various fields follow rather fractional dynamics than ordinary integer-ordered dynamics, meaning the corresponding differential equations feature non-integer valued derivatives. There are several arguments for why this is the case, one of them being that fractional derivatives’ inherit spatiotemporal memory and/or the ability to express complex naturally occurring phenomena. Another popular topic nowadays is machine learning, i.e., learning behavior and patterns from historical data. In our ever-changing world with ever-increasing amounts of data, machine learning is a powerful tool for data analysis, problem-solving, modeling, and prediction. It further provides many insights and discoveries in various scientific disciplines. As these two modern-day topics provide a lot of potential for combined approaches to describe complex dynamics, this article reviews combined approaches of fractional derivatives and machine learning from the past, puts them into context, and thus provides a list of possible combined approaches and the corresponding techniques. Note, however, that this article does not deal with neural networks, as there already is profound literature on neural networks and fractional calculus. We sorted past combined approaches from the literature into three categories, i.e., preprocessing, machine learning &amp; fractional dynamics, and optimization. The contributions of fractional derivatives to machine learning are manifold as they provide powerful preprocessing and feature augmentation techniques, can improve physically informed machine learning, and are capable of improving hyperparameter optimization. Thus, this article serves to motivate researchers dealing with data-based problems, to be specific machine learning practitioners, to adopt new tools and enhance their existing approaches.

The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are constructed and a Cauchy-type problem is studied. It should be noted that the considered method is based on the construction of normalized systems of functions with respect to a differential operator of fractional order.

Anyon is collective excitation of two dimensional electron gas subjected to strong magnetic field, carrying fractional charges and exotic statistical character beyond fermion and boson. So far, anyons with serial fractional charges only exist in fractional quantum Hall effect. It is still a challenge to find new serial of fractional charges in other physical system and develop an unified mathematical physics theory based on the same root. Here a topological path fusion theory of propagating electrons in magnetic flux lattice is proposed to explore the physical origin of fractional charges based on a generalization of Feynman's path integral theory and Thurston's train track theory. This mathematical physics theory generated the existed serial of fractional charges in fractional quantum Hall effect and predicted new serial of fractional charges. A serial of irrational charges are predicted in one dimensional lattice of magnetic fluxes. Fractionally charged anyons are also generated in two dimensional and three dimensional lattice of train tracks of electric currents, revealing an exact correspondence between knot lattice model and train track model. A new explanation for the modular symmetry of complex Hall conductance and composite fermion in fractional quantum Hall effect is also derived from this topological path fusion theory. Experimental observation of anyon in three dimensions can be realized by constructing three dimensional interlocking magnetic fluxes or mapping magnetic fluxes into forbidden zones in multi-connected space filled by solid state material. A photonic crystal with porous nano-structures is a promising system for detecting fractional charges and paves a new way for topological quantum computation.

Here we present Hilfer-Polya, psi-Hilfer Ostrowski and psi-Hilfer-Hilbert- Pachpatte types fractional inequalities. They are univariate inequalities involving left and right Hilfer and -Hilfer fractional derivatives. All estimates are with respect to norms ||.||p, 1 <_ p <_infinite. At the end we provide applications.

This paper presents Chebyshev Type inequalities for the Riemann-Liouville variable-order fractional integral operator using two synchronous functions on the set of real numbers. It is the first result of its kind in the current literature using variable-order Riemann-Liouville fractional integral operator. Some special cases for the result obtained in the paper are discussed.

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time fractional Mittag-Leffler process. The approach of construction of good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

Keywords: liquid alloys, microstratification, dispersed particles, spontaneous dispersion, criteria for true (entropic) spontaneous dispersion, mechanisms of quasi-spontaneous dispersion, methods and results for assessing size of dispersed particles.

In this work, we introduce a scalable and efficient GPU-accelerated methodology for volumetric particle advection and finite-time Lyapunov exponent (FTLE) calculation, focusing on the analysis of Lagrangian Coherent Structures (LCS) in large-scale Direct Numerical Simulation (DNS) datasets across incompressible, supersonic, and hypersonic flow regimes. LCS play a significant role in turbulent boundary layer analysis, and our proposed methodology offers valuable insights into their behavior in various flow conditions. Our novel owning-cell locator method enables efficient, constant-time cell search, and the algorithm draws inspiration from classical search algorithms and modern multi-level approaches in numerical linear algebra. The proposed method is implemented for both multi-core CPUs and Nvidia GPUs, demonstrating strong scaling up to 32,768 CPU cores and up to 62 Nvidia V100 GPUs. By decoupling particle advection from other problems, we achieve modularity and extensibility, resulting in consistent parallel efficiency across different architectures. Our methodology was applied to calculate and visualize the FTLE on four turbulent boundary layers at different Reynolds and Mach numbers, revealing that coherent structures grow more isotropic proportional to the Mach number, and their inclination angle varies along the streamwise direction. We also observed increased anisotropy and FTLE organization at lower Reynolds numbers, with structures retaining coherency along both spanwise and streamwise directions. Additionally, we demonstrated the impact of lower temporal frequency sampling by upscaling with an efficient linear upsampler, preserving general trends with only 10% of the required storage. In summary, we present a particle search scheme for particle advection workloads in the context of visualizing LCS via FTLE that exhibits strong scaling performance and efficiency at scale. Our proposed algorithm is applicable across various domains requiring efficient search algorithms in large structured domains. While this manuscript focuses on the methodology and its application to LCS, an in-depth study of the physics and compressibility effects in LCS candidates will be explored in a future publication.

This paper uses the $\exp(-\Phi(\xi))$-Expansion method to investigate solitons to the M-fractional nonlinear Schrödingers equation with cubic nonlinearity. The results obtained are dark solitons, trigonometric function solutions, hyperbolic solutions and rational solutions. Thus, the constraint relations between the model coefficients and the traveling wave frequency coefficient for the existence of solitons solutions are also derived.

The theory of fractional calculus extends the order of classical integer calculus from integer to non-integer. As a new engineering application tool, it has made many important research achievements in many fields, including image processing. This paper mainly studies the application of fractional calculus theory in image enhancement and denoising, including the basic theory of fractional calculus and its amplitude frequency characteristics, the application of fractional differential operator in image enhancement, and the application of fractional integral operator in image denoising. The experimental results show that the fractional calculus theory has more special advantages in image enhancement and denoising. Compared with the existing integer order image enhancement operators, the fractional differential operator can more effectively enhance the "weak edge" and "strong texture" details of the image. The fractional order integral image denoising operator can not only improve the signal-to-noise ratio of the image compared to traditional denoising methods, but also better preserve detailed information such as edges and textures of the image.

In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations as approximations of some type of fractional nonlinear birth--death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While FDEs appear more flexible in fitting empirical data, our ODEs offered better fits to two out of three data sets. Important differences in transient dynamics between these modeling approaches are discussed.

The exact solutions in the wave form are derived for the time fractional KdV and the time fractional Burgers' equations in conformable fractional derivative sense. The fractional variable change using the fundamental properties of the conformable derivative reduces both equations to some nonlinear ODEs. The predicted solution is assumed to be a finite series form of a function satisfying a particular first-order ODE whose solution contains an exponential function in the denominator. The solutions are represented in the explicit forms and illustrated by some choices of the parameters for various fractional orders of the equations.

In this research article we solve the problem of synchronization and anti-synchronization of chaotic systems described by discrete and time-delayed variable fractional order differential equations. To guarantee the synchronization and anti-synchronization of these systems, we use the well-known PID control theory and the Lyapunov-Krasovskii stability theory for discrete systems of variable fractional order.We illustrate the results obtained through simulation with examples, in which it can be seen that our results are satisfactory, thus achieving synchronization and anti-synchronization of chaotic systems of variable fractional order with discrete time delay.

Plasmonics is the study of resonant oscillations of free electrons in metals caused by incident electromagnetic radiation. Surface plasmons can focus and steer light on the subwavelength scale. Apart from metals, plasmonic phenomena can be observed in soft matter systems such as electrolytes where resonant charge oscillations can be induced for ions in solution. Due to their larger mass, they are plasmon-active in a lower frequency regime and on a larger wavelength scale. Spatial confinement allows increasingly strong charge interactions and gives rise to nonlocality or spatial dispersion effects. These effects manifest as additional longitudinal propagation modes and are known to cause plasmonic broadening and resonance shifts in metal nanostructures. We derive and discuss the nonlocal optical response of ionic plasmons using a hydrodynamic, two-fluid model in a planar homogeneous three-layer system with electrolyte-dielectric interfaces. Studying such systems enables us to identify and understand plasmonic phenomena in biological and chemical systems.

This manuscript investigates the dispersion of cetyltrimethylammonium bromide (CTAB) treated cellulose nanocrystals (CNCs) in polypropylene (PP) by melt mixing. FTIR spectra of CTAB-modified CNCs and unmodified CNCs in a PP matrix were used to evaluate the surface treatments. The surface energies and morphologies of CTAB surface modified CNCs in polypropylene (PP) were used to interpret the effects of nanoparticle dispersion on the crystallization and rheological and mechanical properties of the PP matrix.

Acoustic metasurface are artificial structures which could manipulate the wavefront in sub-wavelength dimensions and the previous proposed acoustic metasurface were mostly realized with single materials. An acoustic metasurface with composite structure is proposed for underwater acoustic stealth considering both wavefront manipulation and sound absorption. The unit cells of the metasurface are composed by metallic supporting lattice, interconnecting polymer materials and mass balancing columns. With the gradual modulations of equivalent physical properties along the horizontal direction of metasurface, the incident acoustic wave is reflected to other directions. Meanwhile, the polymer material inside the unit cells would dissipate the acoustic wave energy due to inherent damping properties. With the simultaneous modulations of reflected wave direction and scattering acoustic amplitude, significant improvement of underwater stealth effect is achieved. Compared with single-phase metasurface, the Far-Field Sound Pressure Level (FFSPL) of the multiphase metasurface decreased by 4.82 dB within the frequency range of 3kHz~30kHz. Under a hydrostatic pressure of 5MPa, the linearized mean stress for multiphase metasurface is only 1/3 of that of single-phase metasurface due to much thicker struts and much more uniform stress distribution. The proposed composite structure possesses potential applications due to acceptable thickness (80mm) and low equivalent density (1100 kg/m3).

Ultrashort pulses are severely distorted even by low dispersive media. While the mathematical analysis of dispersion is well known, the technical literature focuses on pulses, which keep their shape: Gaussian and Airy pulses. However, the cases where the shape of the pulse is unaffected by dispersion is the exception rather than the norm. It is the object of this chapter to present a variety of pulses profiles, which have analytical expressions but can simulate real-physical pulses with greater accuracy. In particular, the dynamics of smooth rectangular pulses, physical Nyquist-Sinc pulses, and slowly rising but sharply decaying ones (and vice-versa) are presented. Besides the usage of this chapter as a handbook of analytical expressions for pulses' propagations in a dispersive medium, there are several new findings. The main ones are: Analytical expressions for the propagation of chirped rectangular pulses, which converge to extremely short pulses; analytical approximation for the propagation of Super-Gaussian pulses; the propagation of Nyquist Sinc Pulse with smooth spectral boundaries and an analytical expression for a physical realization of an attenuation compensating Airy pulse.

Alumina dispersion-strengthened copper, Glidcop, is used widely in high-heat-load ultra-high-vacuum components for synchrotron light sources (absorbers), accelerator components (beam intercepting devices) and in nuclear power plants. Glidcop has similar thermal and electrical properties to OFE (oxygen free electrical) copper, but has superior mechanical properties, thus making it a feasible structural material; its yield and ultimate strength are equivalent to those of mild-carbon steel. The purpose of this work has been to develop a brazing technique to join Glidcop to Mo, using a commercial Cu-based alloy. The effects of the excessive diffusion of the braze along the grain boundaries on the interfacial chemistry and joint microstructure, as well as on the mechanical performance of the brazed joints, has been investigated. In order to prevent the diffusion of the braze into the Glidcop alloy, a copper barrier layer has been deposited on Glidcop by means of RF-sputtering.

It is familiar that the Casimir self-energy of a homogeneous dielectric ball is divergent, although a finite self-energy can be extracted through second order in the deviation of the permittivity from the vacuum value. The exception occurs when the speed of light inside the spherical boundary is the same as that outside, so the self-energy of a perfectly conducting spherical shell is finite, as is the energy of a dielectric-diamagnetic sphere with $\varepsilon\mu=1$, a so-called isorefractive or diaphanous ball. Here we re-examine that example, and attempt to extend it to an electromagnetic $\delta$-function sphere, where the electric and magnetic couplings are equal and opposite. Unfortunately, although the energy expression is superficially ultraviolet finite, additional divergences appear that render it difficult to extract a meaningful result in general, but some limited results are presented.

We consider fractional integral operators $(I-T)^d, d \in (-1,1)$ acting on functions $g: \mathbb{Z}^{\nu} \to \mathbb{R}, \nu \ge 1 $, where $T $ is the transition operator of a random walk on $\mathbb{Z}^{\nu}$. We obtain sufficient and necessary conditions for the existence, invertibility and square summability of kernels $\tau (\mathbf{s}; d), \mathmb{s} \in \mathbb{Z}^{\nu}$ of $(I-T)^d $. Asymptotic behavior of $\tau (\mathbf{s}; d)$ as $|\mathbf{s}| \to \infty$ is identified following local limit theorem for random walk. A class of fractionally integrated random fields $X$ on $\mathbb{Z}^{\nu}$ solving the difference equation $(I-T)^d X = \varepsilon$ with white noise on the right-hand side is discussed, and their scaling limits. Several examples including fractional lattice Laplace and heat operators are studied in detail.

The Sine-Gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in Regularized Long Wave(RLW)-class. Compatible wave transform reduces the governing equation to classical ordinary differential equation. The homogeneous balance procedure gives the order of the predicted polynomial-type solution that is inspired from well-known Sine-Gordon equation. The substitution of this solution follows the previous step. Equating the coefficients of the powers of predicted solution leads a system of algebraic equations. The solution of resultant system for coefficients gives the necessary relations among the parameters and the coefficients to be able construct the solutions. Some solutions are simulated for some particular choices of parameters.

Exact solutions to conformable time fractional (3+1)- dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE. The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo-Miwa, the Zakharov-Kuznetsov and the modified Zakharov-Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.

The fractional reduced differential transform method is a finite iterative method based on infinite fractional expansions. The result obtained is the approximation of the real value. There are few reports on the approximate error and the applicable condition. In this paper, according to the fractional expansions, we study the factors related to the approximate errors. Our research shows that the approximate errors relate not only to fractional order but also to time $t$ and increase rapidly with time $t$. This method can only be applied within a certain time range and the time range is relevant to fractional order and fractional expansions. Then, many obtained achievements may be incorrect if the applicable conditions are not satisfied. Some examples presented in this paper verify our analysis.

In the particular configuration of the scalar field K-essence in the Wheeler-DeWitt quantum equation, for some age in the Bianchi type I anisotropic cosmological model, a fractional differential equation for the scalar field arises naturally. The order of the fractional differential equation is $\beta=\frac{2\alpha}{2\alpha - 1}$. This fractional equation belongs to different intervals, depending on the value of the barotropic parameter; when $\omega_{X} \in [0,1]$, the order belongs to the interval $1\leq \beta \leq 2$, and when $\omega_{X}\in[-1,0)$, the order belongs to the interval $0<\beta \leq 1$. In the quantum scheme, we introduce the factor ordering problem in the variables $(\Omega,\phi)$ and its corresponding momenta $(\Pi_\Omega, \Pi_\phi)$, obtaining a linear fractional differential equation with variable coefficients in the scalar field equation, then the solution is found using a fractional power series expansion. The corresponding quantum solutions are also given. We found the classical solution in the usual gauge N obtained in the Hamiltonian formalism and without a gauge. In the last case, the general solution is presented in a transformed time $T(\tau)$, however in the dust era we found a closed solution in the gauge time $\tau$.

TThis paper delves into the extension and characterization of radial positive definite functions into non-integer dimensions. We provide a thorough investigation by employing the Riemann-Liouville fractional integral and fractional Caputo derivatives, enabling a comprehensive understanding of these functions. Additionally, we introduce a secondary characterization based on the Bernstein characterization of completely monotone functions. The practical significance of our study is showcased through an examination of the positivity of the fundamental solution of the space-fractional Bessel diffusion equation, highlighting the real-world applicability of the developed concepts. Through this work, we contribute to the broader understanding of radial positive definite functions and their utility in diverse mathematical and applied contexts.

In this paper the D 3 dihedral logistic map of fractional order is introduced. The map 1 presents a dihedral symmetry D 3 . It is numerically shown that the construction and interpretation 2 of the bifurcation diagram versus the fractional order require special attention. The system stability 3 is determined and the problem of hidden attractors is analyzed. Also, analytical and numerical 4 results show that the chaotic attractor of integer order, with D 3 symmetries, looses its symmetry 5 in the fractional-order variant.

In view of the problem of cracking easily and partial distortion of images after encryption or decryption, a novel image encryption and decryption algorithm based on Arnod Transform and fractional chaotic is proposed. To begin with, the Arnold transform is used to encrypt. So that the spatial confidence of the original image has been comprehensively disturbed. Secondly, the XOR involving the fractional order chaotic sequence is used to encrypt. The key sequence is dynamically generated to ensure the randomness and difference of key generation. When decryption is required, the first decryption is performed using the key and XOR. Then the second decryption is carried out by using the inverse Arnold transform, and finally the decrypted image is obtained. Experimental results show that the improved algorithm has achieved better performance in encryption and decryption.

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approxi- mation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

The exact solutions of some conformable time fractional PDEs are presented explicitly. The modified Kudryashov method is applied to construct the solutions to the conformable time fractional Regularized Long Wave-Burgers (RLW-Burgers, potential Korteweg-de Vries (KdV) and clannish random walker's parabolic (CRWP) equations. Initially, the predicted solution in the finite series of a rational form of an exponential function is substituted to the ODE generated from the conformable time fractional PDE by using wave transformation. The coefficients used in the finite series are determined by solving the algebraic system derived from the coefficients of the powers of the predicted solution.

In the paper, we develop some numerical schemes including finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulted numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by a integral operator with zero boundary condition. Second, because of the solution of fractional diffusion equation is usually singular near the boundary, we apply finite difference scheme to discrete fractional Laplacian operator and fractional diffusion equation by fractional interpolation function. Moreover, it is find that the differential matrix of the above scheme is symmetric matrix, and strictly row-wise diagonally dominant in special fractional interpolation function. Third, we show finite volume scheme to discrete fractional diffusion equation by fractional interpolation function, and analyze the properties of differential matrix. Finally, the numerical experiments are given, and verify the correctness of theoretical results and the efficiency of the schemes.

Fractional differential equations have become a fundamental modelling approach for understanding and simulating the many aspects of nonlocality and spatial heterogeneity of complex materials and systems. Yet, while real-order fractional operators are nowadays widely adopted, little progress has been made in extending such operators to complex-order counterparts. In this work, we introduce new definitions for the complex-order fractional Laplacian, fully consistent with the theory of averaging of smooth functions over fractal sets, and present tailored spectral methods for their numerical treatment. The proposed complex-order operators exhibit a dual particle-wave behaviour, with solutions manifesting wave-like features depending on the magnitude of the imaginary part of the fractional order. Reaction-diffusion systems driven by the complex-order fractional Laplacian exhibit unique spatio-temporal dynamics, such as equilibrium of diffusion in random materials by interference of scattered waves, conduction block and highly fractionated propagation, or the generation of completely novel Turing patterns. Taken together, our results support that the proposed complex-order operators hold unparalleled capabilities to advance the description of multiscale transport phenomena in physical and biological processes highly influenced by the heterogeneity of complex media.

The differential transformation, an approach based on Taylor's theorem, is proposed as convenient for finding exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem results in a reliable and expected outcome.

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension for the sake of simplicity. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with high solution gradients or any other special features. No interpolation procedure is employed, thus an unnecessary solution smearing is avoided. Thus, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.

This study explores the near-surface dispersion mechanisms of contaminants in coastal waters, leveraging a comprehensive method that includes using dye and drifters as tracers, coupled with diverse observational platforms like drones, satellites, in-situ sampling, and HF radar. The aim is to deepen our understanding of surface currents' impact on contaminant dispersion, thereby improving predictive models for environmental incident management, including pollutant releases. Rhodamine WT dye, chosen for its significant fluorescent properties and detectability, along with drifter data, allowed us to investigate the dynamics of near-surface physical phenomena such as the Ekman current, Stokes drift, and wind-driven currents. Our research emphasizes the importance of integrating scalar tracers and Lagrangian markers in experimental designs, revealing differential dispersion behaviors due to near-surface vertical shear caused by the Ekman current and Stokes drift. The elongation direction of the dye patch aligns with the Ekman current direction during slow current conditions. Analytical calculations of vertical shear, based on the Ekman current and Stokes drift, closely matched those derived from tracer observations. Over a 7-hour experiment, the vertical diffusivity of near-surface water was observed at 2×10-4 m2/s, and the horizontal eddy diffusivity of the dye patch and drifters reached the order of 1 m2/s and at a 1000m length scale. Particle tracking models demonstrate that while HF radar currents can effectively predict the trajectories of tracers near the surface, incorporating near-surface currents, including Ekman current, Stokes drift, and windage, is essential for a more accurate prediction of the fate of surface floats.

Gold nanohole arrays are periodic metastructures gathering huge interest for biosensing applica-tions. The grating-like structure defines their plasmonic mode features and their angular disper-sion can be used to highlight efficiently their interaction with the fluorescence emission of a properly tuned fluorophore. Resorting to a properly tuned/engineered/optimised plasmonic na-nohole array and a commercial organic dye, we performed an accurate resolved angle-resolved optical characterization. The plasmonic fingerprints was consequently identified as a modification of the dye photoluminescence signal in terms of both spectral redistribution and enhancement. Interestingly, by analyzing the results, an advantageous measurement configuration can be iden-tified to better engineer a suitable platform for efficiency maximization in plasmon-enhanced flu-orescence-based applications. Therefore, the coupling mechanism with the plasmonic modes, and nanostructure properties can be carefully co-optimized.

A new method is proposed for performance assessment of lubricating compositions based on industrial lubricants that contain nano-additives of different chemical composition for higher wear resistance of heavy-loaded steel friction pairs. The method is based on the joint analysis of the experimental data on the wear kinetics, change of the relative hardness, and damage in the surface layers of the metal of friction pairs. The level of structural damage to the materials in the contact zone is determined by using the statistical parameters of dispersion of the hardness values. The paper investigates the known experimental results of assessment of the level of damage to structural materials of various grades under the conditions of long-term, cyclic, static loading, and lubricated friction. It is demonstrated that the structural changes occurring in the materials during loading correlate to the changes in the statistical dispersion characteristics of hardness measurement results. Experimental substantiation of the method is carried out for steel friction pairs using lubricating compositions based on industrial oil Greaseline Lithium BIO Rail 000 by AIMOL and nano-additives of copper of M2 grade, magnesium alloy of MA2 grade, aluminum alloy of Al-Mg system, graphite of medium carbon steel grades, i.e. rail and industrial. Under system of indicators presented in the paper, the highest efficiency in terms of increase in the wear resistance of steel friction pairs is demonstrated by the lubricating composition based on industrial lubricant with the additive of nano-powder made of rail steel, as it provided the optimum value of friction coefficient, minimum damage and wear of friction surface.

Derelict asbestos mine sites in South Africa pose a considerable risk to human, environmental and socio-economic health. Comprehensive mineralogical and geochemical datasets for the existing hazardous geological materials still exposed in Southern African derelict asbestos mines remain largely non-existent, as very little published and up-to-date literature is available. In this study three representative types of asbestos mineral fibres from derelict asbestos mines in Southern Africa, namely chrysotile from Havelock mine, amosite from Penge mine and crocidolite from Prieska mine are characterized mineralogically and geochemically to critically evaluate real-life hazards in rural and asbestos-fibre contaminated regions. The samples were examined using polarising light microscopy, X-ray fluorescence (major and trace elemental analysis), X-ray diffraction (including Rietveld refinement), specific surface area analyses and bio-durability tests. Data are discussed in view of their potential toxicities on both human health and the environment in the context of developing countries. Finally, information on the mineralogical and geochemical status of asbestos mine waste and its importance as baseline data for rehabilitation considerations is also evaluated.

The statistical properties of a good estimator include robustness, unbiasedness, efficiency, and consistency. However, the commonly used estimators of dispersion have lack or are weak in one or more of these properties. In this paper, I proposed statistical mirroring as a good alternative estimator of dispersion around defined location estimates or points. In the main part of the paper, attention is restricted to Gaussian distribution and only estimators of dispersion around the mean that functionalize with all the observations of a dataset were considered at this time. The different estimators were compared with the proposed estimators in terms of alternativeness, scale and sample size robustness, outlier biasedness, and efficiency. Monte Carlo simulation was used to generate artificial datasets for application. The proposed estimators (of statistical meanic mirroring) turn out to be suitable alternative estimators of dispersion that is less biased (more resistant) to contaminations, robust to scale and sample size, and more efficient to a random distribution of variable than the standard deviation, variance, and coefficient of variation. However, statistical meanic mirroring is not suitable with a mean (of a normal distribution) close to zero, and on a scale below ratio level.

Integral atmospheric dispersion models are used widely for flammable hazard and its risk analysis. There is a widespread belief that flammable distances from these models are conservative when flammable ranges are calculated using the 0.5 lower flammability limits (LFL) concentration threshold. This is erroneous. This paper traces through the development of these models and the research that led to the Birch Guidance. It shows that the 0.5 LFL is a necessary factor to transform the results of dispersion models designed for environmental assessment to applications to flammable hazard assessment in quiescent conditions. Current applications do not take account of turbulence due to wind, large and small obstructions, etc. A set of simple guidance is given in the paper to manage flammable hazards based on results from atmospheric dispersion models, including topics for future research.

We show that an experimentally plausible system consisting of a modulated wire medium hosted in a metal cavity can preserve the longitudinal field profile shaping predicted by Boyd et al. (2018) on the basis of a perfectly periodic wire-only structure. These new frequency domain numerical results are a significant step towards justifying the construction of an experimental apparatus to test the field profile shaping in practise.

We present an exact expression for the upscaled dynamic dispersion coefficient (D) for one-dimensional transport by Hagen-Poiseuille flow which is the basis for modeling transport in porous media idealized as capillary tubes. The theoretical model is validated by comparing the breakthrough curves (BTCs) from a 1D advection-dispersion model with dynamic D to that from direct numerical solutions utilizing a 2D advection-diffusion model. Both Taylor dispersion theory and our new theory are good predictors of D at lower Peclet Number (Pe) regime, but gradually fail to capture most parts of BTCs as Pe increases. However, our model generally predicts the mixing and spreading of solutes better than Taylor’s theory since it covers all transport regimes from molecular diffusion, through anomalous transport, and to Taylor dispersion. The model accurately predicts D based on the early part of BTCs even at relatively high Pe regime (~62) where the Taylor’s theory fails. Furthermore, the model allows for calculation of the time scale that separates Fickian from non-Fickian transport. Therefore, our model can readily be used to calculate dispersion through short tubes of arbitrary radii such as the pore throats in a pore network model.

No doubt pollution is a global problem which must be holistically tackled. In doing this, adequate knowledge of the sources of pollution is important, therefore the aim of this paper is to review source apportionment with reference to top-down and bottom-up methods. In this paper, dispersion modeling, emissions inventory, and sampling methods were discussed. Also, analytical methods involved in top-down source apportionment were mentioned. The two techniques are needed to evaluate pollutants and their sources. Based on these two approaches, pollution control strategy would be developed and decisions can be made in deciding the right approach to solve or reduce the pollution problems.

This paper discusses approaches to the numerical integration of the coupled nonlinear Schrödinger equations system in case of few-mode wave propagation. The wave propagation assumes the propagation of up to nine modes of light in an optical fiber. In this case, the light propagation is described by the non-linear coupled Schrödinger equation system, where propagation of each mode is described by own Schrödinger equation with other modes interactions. In this case, the non-linear coupled Schrödinger equation system solving becomes increasingly complex, because each mode affects the propagation of other modes. The suggested solution is based on the direct numerical integration approach, which is based on a finite-difference integration scheme. The well-known explicit finite-difference integration scheme approach fails, due to the non-stability of the computing scheme. Due to this fact, the combined explicit/implicit finite-difference integration scheme, based on the implicit Crank–Nicolson finite-difference scheme, is used. It allows ensuring the stability of the computing scheme. Moreover, this approach allows separating the whole equation system on the independent equation system for each wave mode at each integration step. Additionally, the algorithm of numerical solution refining at each step and the integration method with automatic integration step selection are used. The suggested approach has performance gains (or resolutions) up to three or more orders of magnitude in comparison with the split-step Fourier method due to the fact that there is no need to produce direct and inverse Fourier transforms at each integration step. The main advantage of the proposed method is the ability to calculate the propagation of an arbitrary number of modes in the fiber.

Due to the nonlinear stiffness and fractional-order damping characteristics of some real suspension systems, coupled with the presence of viscoelastic materials, fractional-order nonlinear suspension system modeling has become a focus of research in recent years. In this study, an electric oil and gas actuator based on fractional-order PID position feedback control is proposed, through which the damping coefficient of the suspension system is adjusted to realize the active control of the suspension. A fractional order PID algorithm is used to control the motor's rotational angle to realize the damping adjustment of the suspension system. In this process, the road roughness is collected by the sensors as the criterion of damping adjustment, and the particle swarm algorithm is utilized to find the optimal objective function under different road surface slopes, to obtain the optimal cornering value. According to the mathematical and physical model of the suspension system, the simulation model and the corresponding test platform of this type of suspension system are built. The simulation and experimental results show that the simulation results of the fractional-order nonlinear suspension model are closer to the actual experimental values than those of the traditional linear suspension model, and the accuracy of each performance index is improved by more than 18.5%. The designed active suspension system optimizes the body acceleration, suspension dynamic deflection, and tire dynamic load to 89.8%, 56.7%, and 73.4% of the passive suspension, respectively. Notably, the designed fractional order PID control circuit for the motor shows better control results compared to the conventional PID control circuit. This study provides an effective theoretical and empirical basis for the control and optimization of fractional-order nonlinear suspension systems.

For solving the fractional differential equations in computational fluid dynamics (CFD), it’s complicated and difficult by the Laplace and Fourier transforms. Based on the Caputo fractional derivative, the analytical solutions for unsteady unidirectional flows of a generalized Oldroyd-B fluid are deduced by the separation of variables method. Results show that the analytical solutions are given easily, and have good university. For some specific parameter values, the well-known analytical solutions for the generalized second grade fluid, the generalized Upper-Convected Maxwell (UCM) fluid as well as the ordinary Oldroyd-B fluid can be obtained.

This study develops a predictor-corrector algorithm for the numerical simulation of IVPs involving singular generalized fractional derivatives with Mittag-Leffler kernels. The proposed algorithm converts the considered IVP into a Volterra-type integral equation and then uses Trapezoidal rule to obtain approximate solutions. Numerical approximate solutions of some singular generalized fractional derivative with Mittag-Leffler kernels models have been presented to demonstrate the efficiency and accuracy of the proposed algorithm. The algorithm describes the influence of the fractional derivative parameters on the dynamics of the studied models. The suggested method is expected to be effectively employed in the field of simulating generalized fractional derivative models

In this paper, by means of the Banach fixed point theorem and the Krasnoselskii's fixed point theorem, we investigate the existence of solutions for some fractional neutral functional integro-differential equations involving infinite delay. This paper deals with the fractional equations in the sense of Caputo fractional derivative and in the Banach spaces. Our results generalize the previous works on this issue. Also, an analytical example is presented to illustrate our results.

The fractional derivatives are a generalization the derivatives of integer order and find applications in studying memory processes in various scientific fields. Numerical methods are used to solve and analyze fractional models of real world problems. In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic formula, whose generating function is the polylogarithmic function. In the paper, we prove the convergence of the approximation and derive an estimate for the error and order. We consider an application of the approximation for the construction of finite difference schemes for numerical solution of the two-term ordinary fractional differential equation and the time-fractional Black-Scholes equation for option pricing. The properties of the approximation are used to prove the convergence of the methods used for numerical solution of the fractional differential equations. The theoretical results for the error and order of the methods are illustrated by the results of numerical experiments.

In the present study, nonlinear vibration of a nanobeam resting on fractional order viscoelastic Winkler-Pasternak foundaion is studied using nonlocal elasticity theory. D'Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. Detailled parametric study is conducted, the effects of variation in different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations

Experimental studies on the development of processes of obtaining ultradispersed metal-ceramic materials and coatings Mo-ZrO₂, Mo-Al₂O₃, W-ZrO₂, W-Al₂O₃ CVD by deposition from the gas phase in CO₂ process using low-temperature nonequilibrium plasma were carried out.The structure, microhardness, phase composition, and high-temperature annealing behavior of the materials were studied. It is shown that the choice of parameters of the deposition process and temperature annealing can be controlled by the distribution function of ultradispersed particles in the material and their sizes.High-temperature X-ray studies of "molecular" two-phase alloys Mo- Al₂O₃ and Mo- ZrO₂show that, depending on the elastic and thermal properties of the matrix materials and dispersed particles, the molybdenum matrix can be in an all-round tensed or all-round compressed state.In the study of the electro-physical characteristics of the two-phase disperse-strengthened alloys Mo- ZrO₂, Mo- Al₂O₃, W- ZrO₂, W- Al₂O₃ it has been established that in the region of temperatures 1023...1273K anomalous changes of specific electrical resistance and instability of thermal eddy in magnitude are observed apparently connected with the presence of internal stresses in material caused by ultradisperse particles.Theoretical aspects of the problem of increasing the carriers density in nanocomposite materials are reviewedTheoretical analysis of the effect of the ensemble of microparticles on the change of electron density in ultradisperse metal-ceramic materials is carried out. It is shown that the average values of the electron density in the matrix with the introduction of 1% of particles with a radius of 50 A increase by more than two orders of magnitude.The possibilities of creating ultradisperse compositions using radiation-plasma methods are considered.

The present study was designed to develop a self-micellizing solid dispersion (SMSD) containing Thymoquinone (TQM), a phytonutrient obtained from Nigella sativa seeds, aiming to improve its biopharmaceutical and nephroprotective functions. The apparent solubility of TQM in polymer solutions was used to choose an appropriate amphiphilic polymer that could be used to make an SMSD system. Based on the apparent solubility, Soluplus® was chosen as an appropriate carrier, and mixing with TQM, SMSD-TQM with different loadings of TQM (5–15%) was made by solvent evaporation and freeze-drying techniques, respectively, and the formulations were optimized. The optimized SMSD-TQM was evaluated in terms of particle size distribution, morphology, release characteristics, pharmacokinetic behavior, and nephroprotective effects in a rat model of acute kidney injury. SMSD-TQM significantly improved the dissolution characteristics (97.8%) of TQM in water within 60 min. Oral administration of SMSD-TQM in rats exhibited a 4.9-fold higher systemic exposure than crystalline TQM. In a cisplatin-induced (6 mg/kg, i.p.) acute kidney-damaged rat model, oral SMSD-TQM (10 mg/kg) improved the nephroprotective effects of TQM based on the results of kidney biomarkers and histological abnormalities. These findings suggest that SMSD-TQM might be efficacious in enhancing the nephroprotective effect of TQM by overcoming biopharmaceutical limitations.

Nigella sativa's thymoquinone (TQM), a water-insoluble phytonutrient exhibits nephroprotective effects. This study intends to develop a self-micellizing solid dispersion (SMSD) of TQM for better biopharmaceutical and nephroprotective performance. Soluplus®-based SMSD of TQM was created and tested for physicochemical properties, solubility, and pharmacokinetics in rats. Plasma creatinine, blood urea nitrogen (BUN), nephrotoxic indicators, and oxidative stress biomarkers were also tested. During SMSD preparation, TQM was found amorphous, boosting solubility. Minimal band changes between TQM and Soluplus® indicate insignificant drug-carrier interactions. SMSD-TQM generated fine micelles in water, improving TQM's solubility by 97.8% in 60 min. SMSD-TQM was 4.9 times more bioavailable orally in rats than crystalline TQM. In a rat model of acute renal damage by cisplatin (6 mg/kg, i.p.), SMSD-TQM (10 mg-TQM/kg, p.o.) reduced blood creatinine and BUN by 56% and 63.2%, respectively. These findings imply that SMSD-TQM may be a potent dosage option for enhancing TQM's nutrient value.

Road vehicles are a large contributor to Nitrogen Oxides (NOx) pollution. The routine road-side monitoring stations, however, may underrepresent the severity of personal exposure in urban areas, because long-term average readings cannot capture the effects of momentary, high peaks of air pollution. While numerical modelling tools historically have been used to propose an improved distribution of monitoring stations, ultra-high resolution Computational Fluid Dynamics models can further assist the relevant stakeholders in understanding the important details of pollutant dispersion and exposure at local level. This study deploys a 10 cm-resolution CFD model to evaluate actual high peaks of personal exposure to NOx from traffic, by tracking the gases emitted from the tailpipe of moving vehicles being dispersed towards the roadside. The investigation shows that a set of four Euro 5-rated diesel vehicles travelling at constant speed may generate momentary road-side concentrations of NOx as high as 1.25 mg/m3, with 25% expected increase for doubling the number of vehicles and approximately 50% reduction when considering Euro 6-rated vehicles. The paper demonstrates how the numerical tool can be used to identify the impact of measures to reduce personal exposure, such as protective urban furniture, as traffic patterns and environmental conditions change.

The antioxidant potential (capacity and activity) of aqueous fullerene dispersions (AFD) of non-functionalized C₆₀, C₇₀, and Gd@C₈₂ endofullerene (in micromolar concentration range) was estimated based on chemiluminescence measurements of the model of luminol and generation of organic radicals by 2,2’-azobis(2-amidinopropane) dihydrochloride (ABAP). The antioxidant capacity was estimated by the TRAP method, from the concentration of half-suppression, and from the suppression area in the initial period. All three approaches agree and show that the antioxidant capacity of AFDs increased in the row Gd@C₈₂ < C₇₀ < C₆₀. Mathematical modeling of the long-term kinetics data was used for antioxidant activity estimation. The effect of C₆₀ and C₇₀ is found to be quenching of the excited product of luminol with ABAP-generated radical and not an actual antioxidant effect; quenching constants differ insignificantly. Apart from quenching with a similar constant, the AFD of Gd@C₈₂ exhibits actual antioxidant action. The antioxidant activity in Gd@C₈₂ is 300-fold higher than quenching constants.

The discovery of a growing number of exoplanets and even extrasolar systems supports the scientific consensus that it is possible to find other signs of life in the universe. The present work proposes for the first time, an explicit mechanism inspired by the dynamics of biological dispersion, widely used in ecology and epidemiology, to study the dispersion of biogenic units, interpreted as complex organic molecules, between rocky or water exoplanets (habitats) located inside star clusters. The results of the dynamic simulation suggest that for clusters with populations lower than 4 M+/ly³ it is not possible to obtain biogenic worlds after 5 Gyr. Above this population size, biogenic dispersion seems to follow a power law, the larger the density of worlds lesser will be the impact rate (β ) value to obtain at least one viable biogenic Carrier habitat after 5 Gyr. Finally, when we investigate scenarios by varying β, a well-defined set of density intervals can be defined in accordance to its characteristic β value, suggesting that biogenic dispersion has a behavior of “minimal infective dose” of “minimal biogenic effective” events by interval i.e. once this dose has been achieved, doesn’t matter if additional biogenic impact events occur on the habitat.

The geometry of mesoscopic inhomogeneities plays an important role in determining the macroscopic propagation behaviors of elastic waves in a heterogeneous medium. Non-equiaxed inhomogeneities can lead to anisotropic wave velocity and attenuation. Developing an accurate scattering theory to describe the quantitative relation between the microstructure features and wave propagation parameters is of fundamental importance for seismology and ultrasonic nondestructive characterization. This work presents a multiple scattering theory for strongly scattering elastic media with general tri-axial heterogeneities. A closed analytical expression of the shape-dependent singularity of the anisotropic Green’s tensor for the homogeneous reference medium is derived by introducing a proper non-orthogonal ellipsoidal coordinate. Renormalized Dyson’s equation for the coherent wave field is then derived with the help of Feynman’s diagram technique and the first-order-smoothing approximation. The exact dispersion curves and the inverse Q-factors of coherent waves in several representative medium models for the heterogeneous lithosphere are calculated numerically. Numerical results for small-scale heterogeneities with the aspect ratio varying from 1 to 7 show satisfactory agreement with those obtained from real earthquakes. The results for velocity dispersion give rise to a novel explanation to the formation mechanism of different seismic phases. The new model has potential applications in seismology and ultrasonic microstructure characterization.

The application of naturally ventilated pig buildings (NVPBs) with outdoor exercise yards is on the rise mainly due to animal welfare considerations, while the issue of emissions from the buildings to the surrounding environment is important. Since air pollutants are mainly transported by airflow, the knowledge on the airflow characteristics downwind the building is required. The objective of this research was to investigate airflow properties downwind of a NVPB with a roofed outdoor exercise yard for roof slopes of 5°, 15°, and 25°. Air velocities downwind a 1:50 scaled NVPB model were measured using a Laser Doppler Anemometer in a large boundary layer wind tunnel. A region with reduced mean air velocities was found along the downwind side of the building with a distance up to 0.5 m (i.e. 3.8 times building height), in which the emission concentration might be high. It was found that a smaller roof slope (i.e. 5° slope) resulted in a higher and shorter wake zone and thus a shorter air pollutant dispersion distance. It was concluded that a smaller roof slope could contribute to the dilution of air pollutants and a lower air pollutant concentration near the ground.

Longitudinal dispersion coefficient (LDC) plays an essential role in modeling the transport of pollution and sediment in the natural rivers. As a result of transportation processes, the concentration of pollution changes along the river. Different studies have been conducted to provide simple equations for estimating LDC. In this study, Support Vector Regression (SVR), Gaussian Process Regression (GPR), M5P and Random Forest (RF) examined to predict the LDC in the natural streams. The hydraulic and geometric features of different rivers gathered for developing the mentioned models for LDC estimation and various statistical criteria were utilized to scrutinize of the models. Furthermore, the Taylor chart was used to evaluate the models and achieved results showed that among machine learning models, M5P displayed the superior performance with CC of 0.823, SI of 0.812, NS of 0.577 and WI of 0.879. As well, S-D model with CC of 0.795 SI of 0.827, NS of 0.558 and WI of 0.890 had more precise results than other empirical models. The results indicates that the developed M5P model with simple formulations was superior to other machine learning models and empirical models and therefore, it can be used as a proper tool for estimating LDC in natural rivers.

Crystallization behavior of hexadecane (C₁₆H₃₄), octadecane (C₁₈H₃₈), eicosane (C₂₀H₄₂), and docosane (C₂₂H₄₆) dispersions of similar mean droplet diameter (x_50.2 ≈ 15 µm) was investigated in quiescent systems and compared to crystallization under mechanical stress. In quiescent systems, the required supercooling decreased with increasing chain length of the alkanes to initiate crystallization. Crystallization of alkane dispersions under mechanical stress resulted in similar onset crystallization supercooling, as during quiescent crystallization. Increase of mechanical stress did not affect the onset crystallization supercooling within alkane dispersions.

Fractional calculus is defined by expanded integer order integration and differentiation. In this paper, multiple mathematical models of a nonlinear dual tank device are precisely formulated by fractional calculus. Using the accurate model, multivariable Fractional-Order Controller is designed for the nonlinear device. The merits of the fractional-order design include: 1) Control of multivariable nonlinearities, 2) Compensation of uncertainties, 3) Elimination of coupling effects. Simulations and experiments are conducted to verify the precision of the fractional order models and the effectiveness of the multivariable fractional-order control design.

The main goal of the paper is to use a generalized proportional Riemann-Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved such as the fractional derivative of a squared function. Also some comparison results for GPRLFD are provided. Two types of equilibrium of the BAM model with GPRLFD are defined. In connection with the applied fractional derivative and its singularity at the initial time the Mittag-Leffler exponential stability in time of the equilibrium is introduced and studied. An example is given illustrating the meaning of the equilibrium as well as its stability properties.

Let g be a strictly increasing function on $\left(a,b\right),$ having a continuous derivative g′ on $\left(a,b\right).$ For the Lebesgue integrable function $f:\left(a,b\right)\to \mathbb{C}$ , we define the k-g-left-sided fractional integral of f by $S_{k,g,a+}f\left(x\right)={\int }_a^{x}k\left(g\left(x\right)-g\left(t\right)\right)g^{\prime }\left(t\right)f\left(t\right)dt,x\in \left(a,b\right]$ and the k-g-right-sided fractional integral of f by $S_{k,g,b-}f\left(x\right)={\int }_x^{b}k\left(g\left(t\right)-g\left(x\right)\right)g^{\prime }\left(t\right)f\left(t\right)dt,x\in [a,b),$ where the kernel k is defined either on $\left(0,\infty \right)$ or on $\left[0,\infty \right)$ with complex values and integrable on any finite subinterval. In this paper we establish some Ostrowski and trapezoid type inequalities for the k-g-fractional integrals of functions of bounded variation. Applications for mid-point and trapezoid inequalities are provided as well. Some examples for a general exponential fractional integral are also given.

Refractive index measurements have been an important task for a long time because that index plays an essential role in describing the optical properties of a material. Many methods have been developed to perform that task. Some of them use interferometry to achieve high precision, however these configurations are complicated. Some measure the critical angle using simple structures, but the accuracy is not satisfactory because it is difficult to judge the exact critical angle with intensity variations. Here we propose several new schemes based on measuring the polarization change in the total internal reflection. The proposed method has the merits of simple structure and easy incident angle determination that gives the maximum phase change. Additionally, it is possible to find the material dispersion by measuring the wavelength dependence of the polarization ellipticity. Some useful formulas relating the refractive index with the maximum phase change are obtained. This work can provide valuable alternatives for index measurement.

This paper aims to search for lump waves in a spatial symmetric (2+1)-dimensional dispersive wave model. Through an ansatz on positive quadratic functions, we conduct symbolic computations with Maple to generate lump waves for the proposed nonlinear model. A line of critical points of the lump waves is computed, whose two spatial coordinates travel at constant speeds. The corresponding maximum and minimum values are evaluated in terms of the wave numbers, and interestingly, all those extreme values do not change with time, either. The last section is the conclusion.

Guided ultrasonic waves are suitable for use in the context of structural health monitoring of thin-walled, plate-like structures. Hence, observing the wave propagation in the plates can provide an indication of whether damage has occurred in the structure. In this work, the wave propagation in fiber metal laminate consisting of thin steel foils and layers of carbon fiber-reinforced polymer is studied, focusing on the main propagation characteristics like dispersion diagrams and displacement fields. For this purpose, the dispersion diagrams derived from the analytical framework and numerical simulations are first determined and compared to each other. Next, the displacement fields are computed using the global matrix method for two excitation frequencies. The results derived from the analytical framework is used to validate numerically determined displacement fields based on a 2D and a 3D modeling approach. For both investigations the results of the analytical treatment and the numerical simulation show good agreement. Furthermore, the displacement field reveals the typical and well-known characteristics of the propagation of guided waves in thin-walled structures. Since the use of full 3D models involves a very high computational cost, this work also successfully investigates the possibility for model order reduction to decrease the computational time and costs of the simulation without the loss of accuracy.