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We define a new class of linear canonical wavelet transform (LCWT) and study its properties like inner product relation, reconstruction formula and also characterize its range. We obtain Donoho-Stark’s uncertainty principle for the LCWT and give a lower bound for the measure of its essential support. We also give the Shapiro’s mean dispersion theorem for the proposed LCWT.

Accurate prediction of photovoltaic (PV) power generation is the key to daily dispatch management and safe and stable grid operation. In order to improve the accuracy of the prediction, a finite iterative PV power prediction model with long range dependence (LRD) characteristics was developed using fractional Lévy stable motion (fLsm) and applied to a real dataset collected in the DKASC photovoltaic system in Alice Springs, Australia. The LRD prediction model considers the influence of current and past trends in the stochastic series on the future trends. Firstly, the calculation of the maximum steps prediction was introduced based on the maximum Lyapunov. The maximum prediction steps could provide the prediction steps for subsequent prediction models. Secondly, the order stochastic differential equation (FSDE) which describes the fLsm can be obtained. The parameters of the FSDE were estimated by using a novel characteristic function method. The PV power forecasting model with the LRD characteristics was obtained by discretization of FSDE. By comparing statistical performance indicators such as root max error, mean square error with Conv-LSTM, BiLSTM, and GA-LSTM models, the performance of the proposed fLsm model has been demonstrated. The proposed methods can provide better theoretical support for the stable and safe operation of PV grid connection. They have high reference value for grid dispatching department.

Motivated by the recent research about non-integer order derivatives of Dirac delta function, we use a fractional Taylor series to explore a fractional distributional representation of gamma function. It proved significant to develop a novel fractional kinetic equation, which is an essential mathematical model to measure the degree of transformation in the chemical structure of stars (sun) as a key component of our environment. In addition to the classical solution of the formulated fractional kinetic equation, the distributional solution is also attained which is not possible without using fractional derivatives of Dirac delta function. Novel identities involving the gamma function are obtained using several fractional transforms. Convergence of new series is proved using the theory of distributions leading to further new applications.

We investigate the behavior of a complex three-strain model with a generalized incidence rate. The incidence rate is an essential aspect of the model as it determines the number of new infections emerging. The mathematical model comprises thirteen nonlinear ordinary differential equations with susceptible, exposed, symptomatic, asymptomatic and recovered compartments. The model is well-posed and verified through existence, positivity and boundedness. Eight equilibria comprise a disease-free equilibria and seven endemic equilibrium points following the existence of three strains. The basic reproduction numbers $ \mathfrak{R}_{01} $, $ \mathfrak{R}_{02} $ and $ \mathfrak{R}_{03} $ represent the dominance of strain 1, strain 2 and strain 3 in the environment for new strain emergence. The model establishes local stability at a disease-free equilibrium point. Numerical simulations endorse the impact of general incidence rates, including bi-linear, saturated, Beddington DeAngelis, non-monotone and Crowley Martin incidence rates.

Self-similarity is a common feature among mathematical fractals and various objects of our natural environment. Therefore, escape criteria are used to determine the dynamics of fractal patterns through various iterative techniques. Taking motivation from this fact, we generate and analyze fractals as an application of the proposed Mann iterative technique with h-convexity. By doing so, we develop an escape criterion for it. Using this established criterion, we set the algorithm for fractal generation. We use the complex function f(x)=xn+ct, with n≥2,c∈C and t∈R to generate and compare fractals using both the Mann iteration and Mann iteration with h-convexity. We generalize the Mann iterative scheme using the convexity parameter h(α)=α2 and provide the detailed representations of quadratic and cubic fractals. Our comparative analysis consistently proved that the Mann iteration with h-convexity significantly outperforms the standard Mann iteration scheme regarding speed and efficiency. It is noticeable that the average number of iterations required to perform the task using Mann iteration with h-convexity is significantly less than the classical Mann iteration scheme. Moreover, the relationship between the fractal patterns and the input parameters of the proposed iteration is extremely intricate.

In the present study, the effects of the strong Allee effect on the dynamics of the modified Leslie-Gower predator-prey model, in the presence of nonlinear prey-harvesting, have been investigated. In our findings, it is seen that the behaviors of the described mathematical model are positive and bounded for all future times. The conditions for the local stability and existence for various distinct equilibrium points have been determined. The present research concludes that system dynamics are vulnerable to initial conditions. In addition, the presence of several types of bifurcations (e.g., saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic bifurcation) has been investigated. The first Lyapunov coefficient has been evaluated to study the stability of the limit cycle that results from Hopf bifurcation. The presence of a homoclinic loop has been demonstrated by numerical simulation. Finally, possible phase drawings and parametric figures have been depicted to validate the outcomes.

Natural disasters present ongoing risks to human life and the global economy, with climate change and environmental factors exacerbating these threats. This article introduces an innovative approach to earthquake prediction and modeling, utilizing a combination of modern mathematical techniques, data-driven methodologies, and artificial intelligence (AI). By integrating the finite element method and wavelet collocation method, our approach enables the solution of complex partial differential equations involving fractional derivatives. This integration provides a more accurate prediction of earthquake behavior by leveraging AI analysis of the resulting numerical solutions.Our advanced modeling technique enables three-dimensional earthquake modeling, surpassing the limitations of traditional methods and offering unprecedented insights. The comprehensive AI approach shows great potential for improving our understanding of earthquake behavior, facilitating the design of earthquake-resistant structures, and effectively mitigating the devastating impacts of earthquakes.This research significantly contributes to the field of disaster risk reduction and offers practical applications for both researchers and practitioners. By combining mathematical rigor, data-driven insights, and the power of AI, our next-generation approach opens new avenues for enhancing our preparedness and response to seismic events.

The COVID-19 pandemic still gains the attention of many researchers worldwide. Over the past few months, China faced a new wave of this pandemic which increases the risk of its spread to the rest of the world. Therefore, there has become an urgent demand to know the expected behavior of this pandemic in the coming period. In this regard, there are many mathematical models from which we may obtain accurate predictions about the behavior of this pandemic. Such a target may be achieved via updating the mathematical models taking into account the memory effect in the fractional calculus. This paper generalizes the power-law growth model of the COVID-19. The generalized model is investigated using two different definitions in the fractional calculus, mainly, the Caputo fractional derivative and the conformable derivative. The solution of the first-model is determined in a closed series form and the convergence is addressed. At a specific condition, the series transforms to an exact form. In addition, the solution of the second-model is evaluated exactly. The results are applied on eight European countries to predict the behavior/variation of the infected cases. Moreover, some remarks are given about the validity of the results reported in the literature.

The ongoing study deals with various forms of solutions for the biological population model with a novel beta-time derivative operators. This model is very conducive to explain the enlargement of viruses, parasites and diseases. This configuration of the aforesaid classical scheme is scouted for its new solutions especially in soliton shape via two of the well known analytical strategies, namely: the extended Sinh-Gordon equation expansion method (EShGEEM) and the Expa function method. These soliton solutions suggest that these methods have widened the scope for generating solitary waves and other solutions of fractional differential equations. Different types of soliton solutions will be gained such as dark, bright and singular solitons solutions with certain conditions. Furthermore, the obtained results can also be used in describing the biological population model in some better way. The numerical solution for the model is obtained using the finite difference method. The numerical simulations of some selected results are also given through their physical explanations. To the best of our knowledge, No previous literature discussed this model through the application of the EShGEEM and the Expa function method and supported their new obtained results by numerical analysis.

To solve the fractional gas dynamic equation, this paper presents an effective algorithm using the collocation method and Müntz-Legendre (M-L) polynomials. The approach chooses a solution of a finite-dimensional space that satisfies the desired equation at a set of collocation points. The collocation points in this study are selected to be uniformly spaced meshes or the roots of shifted Legendre and Chebyshev polynomials. Müntz-Legendre polynomials have the interesting property that their fractional derivative is also a Müntz-Legendre polynomial. This property ensures that these bases do not face the problems associated with using the classical orthogonal polynomials when solving fractional equations using the collocation method. The numerical simulations illustrate the method’s effectiveness and accuracy.

In this paper, the fundamental properties of fractional calculus are discussed with the aim of extending the definition of fractional operators by using wavelets. The Haar wavelet fractional operator is defined, in a more general form, independently on the kernel of the fractional integral.

Engineering applications of the fractional Weibull distribution (fWd) are quite limited because a corresponding stochastic process is not yet constituted and completely analyzed of fundamental properties. In order to fill this gap, the fractional Weibull process (fWp) is defined in this paper with the realization algorithm. The self-similarity property as well as long range dependence (LRD) are proven for the future research. The simulation is conducted by the actual data. The fWd is utilized to fit the actual probability distribution and the corresponding process is generated to reflect the stochasticity of the data. The random walk based on the fWp expands the simulation to the planar space.

In this paper, we derive some classical and fractional properties of the rRs matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss hypergeometric matrix functions. We will also show the relationship with other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials.

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an application of two power methods. We obtain a special solution with the homotopy perturbation method (HPM) combined with the ZZ transformation scheme; then we present the problem and study the existence of the solution, and also we apply this new method to solving the fractional SIR epidemic with the ABC operator. The solutions show up as infinite series. The behavior of the numerical solutions of this model, represented by series of the evolution in the time fractional epidemic, is compared with the Adomian decomposition method and the Laplace–Adomian decomposition method. The results showed an increase in the number of immunized persons compared to the results obtained via those two methods.

In the challenge of Artificial Intelligence in processing semantically evaluable information, the application of deep learning techniques depends not only on the algorithms, but also on the principles that explain how they work. The malfunction of a machine learning system, ML, can occur due lack of knowledge of the algorithm intended behavior. The difficulty in debugging ML can be overcome by using strategies based on the universal structure of language that overlaps in the cognitive architecture of biological and intelligent systems. The appropriate choice of an algorithm inspired by the functioning of human language offers the computational scientist methodological strategies to clarify its performance analysis to optimize the interpretative activity under the good instrumentation of the system and to reach the performance level of an application considered safe. Neurolinguistic principles that link interpretation to language and cognition; the semantic dimension that arises not only from the linguistic system, but also from the context in which the information is produced; and the theoretical bases for understanding language as a 'form' (process) and not as a substance (set of signs) provide the groundwork for the intelligent systems’ improvement so that they have universal consistency and lessen the effects of the ‘curse of dimensionality’ or of the bias in the interpretation by the system. Semantics and statistics are considered to understand universal consistency as opposed to ideal consistency when evaluating a data set, since training alone is not sufficient to avoid data manipulation. We conclude that the 'key' for a good information classifier to achieve an acceptable performance of neural networks is in the dynamic aspect of language (language as a form / process) that: Guides the apprehension of how neural networks have access to weights (values); replicates this for intelligent systems making them invariant to many input transformations and guarantees an infinite amount of finite sample information, avoiding semantic distortion.

In this paper, an adaptive remaining useful life prediction model is proposed for electric vehicle lithium batteries. Capacity degradation of the electric car lithium batteries is modeled by the multi-fractal Weibull motion. The varying degree of long-range dependence and the 1/f characteristics in the frequency domain are also analyzed. The age and state-dependent degradation model is derived, with the associated adaptive drift and diffusion coefficients. The adaptive mechanism considers the quantitative relations between the drift and diffusion coefficients. The unit-to-unit variability is considered a random variable. To facilitate the application, the convergence of the RUL prediction model is proved. Replacement of the lithium battery in the electric car is recommended according to the remaining useful life prediction results. The effectiveness of the proposed model is shown in the case study.

Abstract The well-known novel virus (COVID-19) is a new strain of coronavirus family, declared by the World Health Organization (WHO) as a dangerous epidemic. More than 3.5 million positive cases and 250 thousand deaths (up to May 5, 2020) caused by COVID-19 and has affected more than 280 countries over the world. Therefore studying the prediction of this virus spreading in further attracts a major public attention. In the Arab Emirates (UAE), up to the same date, there are 14,730 positive cases and 137 deaths according to national authorities. In this work, we study a dynamical model based on the fractional derivatives of nonlinear equations that describe the outbreak of COVID-19 according to the available infection data announced and approved by the national committee in the press. We simulate the available total cases reported based on Riesz wavelets generated by some refinable functions, namely the smoothed pseudosplines of types I and II with high vanishing moments. Based on these data, we also consider the formulation of the pandemic model using the Caputo fractional derivative. Then we numerically solve the nonlinear system that describes the dynamics of COVID-19 with given resources based on the collocation Riesz wavelet system constructed. We present graphical illustrations of the numerical solutions with parameters of the model handled under different situations. We anticipate that these results will contribute to the ongoing research to reduce the spreading of the virus and infection cases.

Abstract Fractional calculus (FC) is useful in studying physical phenomena with memory effect. In this paper, a fractional form of Ambartsumian equation is considered utilizing the Caputo fractional derivative. The Heaviside expansion formula in classical calculus (CC) is extended/developed in view of FC. Then, the extended Heaviside expansion formula is applied to obtain the exact solution in a simplest form. Several theorems and lemmas are proved to facilitate the evaluation of the inverse Laplace transform of specific expressions in fractional forms. The exact solution is established in terms of a one-parameter Mittag-Leffler function which is provided for the first time for the Ambartsumian equation in FC. The present solution reduces to the corresponding one in the relevant literature as the fractional order tends to one. Moreover, the convergence of the obtained solution is theoretically proved. Comparisons with another approach in the literature are performed. The advantage of the present analysis over the existing one in the relevant literature is discussed and analyzed.

A novel idea is proposed to forecast lost load and time interval between power outages based on the similarity between the series generated by a stochastic model and the actual series. The lost load obeys a power-law distribution, which can be described as a stochastic process with long-range dependence (LRD). Fractional Brownian motion (fBm) is a stochastic model with LRD, we discretize the stochastic differential equation (SDE) driven by fBm and the difference iterative equation is constructed for forecasting. By calculating the Hurst exponent of lost load series proves that it has LRD characteristics. The stochastics series produced by fBm has a strong similarity with the actual series when the Hurst exponent is taken into fBm model. Moreover, the forecasting accuracy is enriched by considering the appropriate sample size and forecasting step size. The same process for the analysis of lost load can be applied to forecast the time interval between power outages. The efficiency of the proposed model is demonstrated by a case study of the medium voltage power grid in Shanghai compared with other approaches. Finally, the value at risk (VaR) and the conditional value at risk (CVaR) as two system-level indices for assessment of future power outages.

Short-term power load forecasting plays a key role in power supply systems. Many methods have been used in short-term power load forecasting during the past years. A new short-term power load forecasting method is proposed in this study. First, the study represents a Fractional Auto-regressive Integrated Moving Average (FARIMA) model based on long-range dependence (LRD). The LRD model is governed by the Hurst exponent, which shows whether the model exhibits the LRD or not. Then, the study employs Cuckoo Search (CS) algorithm based on two parameters dynamic adjustment for parameter optimization of the forecasting model. As test problem, we use the real power consumption data, and test it for different forecasting models. Our results indicate that the FARIMA model and the improved optimization algorithm show relatively high accuracy and effectiveness in forecasting short-term power load.

Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B-spline tight framelets and use it to introduce our numerical scheme. The method provides a robust solution for the given WSVFIE by using the resulting matrices based on these biorthogonal wavelet. We demonstrate the validity and accuracy of the proposed method by some numerical examples.

As neuroimaging scanners grow in popularity in hospitals and institutes, the tasks of radiologists are increasing. Emotion, fatigue, and other factors may influence the manual interpretation of results. This manual interpretation suffers from inter- and intra-radiologist variance. Computer-aided medical diagnosis (CAMD) are procedures in medicine that assist radiologists and doctors in the interpretation of medical images, which may come from CT, X-ray, ultrasound, thermography, MRI, PET, SPECT, etc. In practice, CAMD can help radiologists to interpret medical images within seconds. Conventional CAMD tools are built on top of handcrafted features. Recent progress on deep learning opens a new era in which features can be automatically built from a large amount of data. Many important medical projects were launched during the last decade (Human brain project, Blue brain project, Brain Initiative, etc.) that provide massive amounts of data. This emerging big medical data can support the use of deep learning.

In this paper, a suitable Lyapunov-Krasovskii functional together with the inequality analysis technique for neutral-type system, and sufficient exponential stability conditions are proposed in the form of linear matrix inequalities (LMIs). Firstly, the system model of leakage delay and semi-Markovian process for neutral-type neural network is established, where the sojourn-time follows Weibull distribution, which is time-varying instead of being constant in the transition rate. Secondly, through combinations of stability analysis method, Lyapunov-Krasovskii functional and inequality analysis technique, some sufficient conditions have been achieved to undertake the exponential stabilization of given neural network system. Moreover, time delays of the system contain leakage and mixed delays, which are more general and appropriate. Finally, in order to show the many advantages of our model, the obtained results will be compared with previous works based on methods different from semi-Markovian jump or leakage delay.

In this paper, we study on the conformable (2+1)-dimensional Ablowitz-KaupNewell-Segur equation in order to show the existence of complex combined dark-bright soliton solutions. To this purpose an effective method which is the sine-Gordon expansion method is used. The 2D and 3D surfaces under some suitable values of parameters are also plotted.

Because of noise interference, improper exposure, and the over thickness of human tissues, the detailed information of DR (digital radiography) images can be masked, including unclear edges and reduced contrast. An image-enhancement algorithm based on wavelet multiscale decomposition is proposed to address the shortcomings of existing single-scale image-enhancement algorithms. The proposed algorithm is based on Shannon–Cosine wavelets by taking advantage of the interpolation, smoothness, tight support, and normalization properties. Next a multiscale interpolation wavelet operator is constructed to divide the image into several sub-images from high frequency to low frequency, and to perform different multi-scale wavelet transforms on the detailed image of each channel. So that the most subtle and diagnostically useful information in the image can be effectively enhanced. Moreover, the image will not be over-enhanced and combined with the high contrast sensitivity of the human eye’s visual system in smooth regions, different attenuation coefficients are used for different regions to achieve the purpose of suppressing noise while enhancing details. The results obtained by some simulations show that this method can effectively eliminate the noise in the DR image, and the enhanced DR image detail information is clearer than before while having high effectiveness and robustness.

The semantic web invests in systems that work collaboratively. In this article we show that the collaborative way is not enough, because the system must ‘understand’ the data resources that are provided to it, to organize them in the direction indicated by the system’s core, the algorithm. In order for intelligent systems to imitate human cognition, in addition to technical skills to model algorithms, we show that the specialist needs a good knowledge of the principles that explain how human language constructs concepts. The content of this article focuses on the principles of the conceptual formation of language, pointing to aspects related to the environment, to logical reasoning and to the recursive process. We used the strategy of superimposing the dynamics of human cognition and intelligent systems to open new frontiers regarding the formation of concepts by human cognition. The dynamic aspect of the recursion of the human linguistic process integrates visual, auditory, tactile input stimuli, among others, to the central nervous system, where meaning is constructed. We conclude that the human linguistic process involves axiomatic (contextual/biological) and logical principles, and that the dynamics of the relationship between them takes place through recursive structures, which guarantee the construction of meanings through long-range correlation under scale invariance. Recursion and cognition are, therefore, interdependent elements of the linguistic process, making it a set of sui generis structures that evidence that the essence of language, whether natural or artificial, is a form and not a substance.

We propose a numerical scheme based on the Galerkin method for solving the time-fractional partial differential equations. To this end, after introducing the Chebyshev cardinal functions (CCFs), using the relation between fractional integral and derivative, we represent the Caputo fractional derivative based on these bases and obtain an operational matrix. Applying the Galerkin method and using the operational matrix for the Caputo fractional derivative, the desired equation reduces to a system of linear algebraic equations. By solving this system, the unknown solution is obtained. The convergence analysis for this method is investigated, and some numerical simulations show the accuracy and ability of the technique.

This paper is devoted to an innovative and efficient technique for solving space–time fractional differential equations (STFPDEs). To this end, we apply the Tau method such that the bases used are interpolating scaling functions (ISFs). The operational metrics for the derivative operator and fractional integration operator are used to introduce the operational matrix for the Caputo fractional derivative. Due to some characteristics of ISFs, such as interpolation, computation costs can be significantly reduced. We investigate the convergence of the technique, and some numerical implementations show that the method is effective for solving such equations.

Abstract The fractional sub-diffusion equation, which is obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order ϑ with 0

In the present paper, we investigate the influence of the choice of continuous linear operator for obtaining the approximate periodic solutions of ordinary second-order differential equations. In most of these problems, the periods are unknown, and the determination of these periods and periodic solutions is a difficult issue. So, a new computational method is proposed based on the symmetric operator, namely the reproducing kernel Hilbert space (RKHS) method to obtain the interval of these solutions. This operator, as a consequence of the symmetric inner product, is a symmetric operator and it will be used to show the influence on periodic solutions. The high efficiency of the proposed strategy is presented along with some illustrative examples which demonstrate their periodic interval dealing with the choice of an appropriate continuous linear operator.

In this paper, we study the existence of solutions for a multiplied system of fractional differential equations with nonlocal integro multi-point boundary conditions by using the p-Laplacian operator and the φ-Hilfer derivatives. The presented results are obtained by the fixed point theorems of Krasnoselskii. An illustrative example is presented at the end to show the applicability of the obtained results. To the best of our knowledge, this is the first time where such a problem is considered.

In this paper a review on harmonic wavelets and their fractional generalization, within the local fractional calculus, will be discussed. The main properties of harmonic wavelets and fractional harmonic wavelets will be given, by taking into account of their characteristic features in the Fourier domain. It will be shown that the local fractional derivatives of fractional wavelets have a very simple expression thus opening new frontiers in the solution of fractional differential problems This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

In quantum key distribution experiments, ground motion is usually used to simulate satellite-based motion. The posture fluctuation of the platform affects the normal operation of the acquisition, tracking, and pointing (abbreviated as ATP) system seriously. To achieve the verification of the ground motion platform, the ATP parameters of the ground simulation motion system cannot be designed only according to the satellite-based ATP parameters. To solve this problem, a set of initial pointing system and inertial stabilization system is added to the simulation ATP system. This provides a technical solution for the ground simulation ATP system similar to the satellite-based motion platform. In the meanwhile, a tracking control strategy based on the identification method is proposed by establishing identification symbols. Compared with traditional proportion, integral, and differential (abbreviated as PID) control, this method overcomes the shortcoming of tentative modification of the control parameters and improves the stability and adaptability of the tracking control process. Tracking accuracy of ±0.1° is achieved under heavy-load conditions. This guarantees the success of the quantum key distribution (QKD) verification test of the ground motion platform.

The fractional calculus of special functions has significant importance and applications in various fields of science and engineering. Here, we aim to find the fractional integral and differential formulas of the extended hypergeometric-type functions by using the Marichev–Saigo–Maeda operators. All the outcomes presented here are of general attractiveness and can yield a number of previous works as special cases due to the high degree of symmetry of the involved functions.

Gradual degradation of the bearing vibration signal is usually studied as a nonstationary stochastic time series. Roller bearings are working at high speed in a heavy load environment so that the combination of bearing faults gradually degraded during the rotation might lead to unpredicted catastrophic accidents. The degradation process has the property of long-range dependence (LRD), so that the fractional Brownian motion (fBm) is taken into account for a prediction model. Because of the dramatic changes in the bearing degradation process, the Hurst exponent that describes the fBm will change during the degradation process. A priori Hurst value of the conventional fBm in the prediction is fixed, thus inducing a minor accuracy of the prediction. To avoid this problem, we propose an improved prediction method. Based on the following steps, at the initial data processing, a skip-over factor is selected as the characteristics parameter of the bearing degradation process. A multifractional Brownian motion (mfBm) replaces the fBm for the degradation modeling. We will show that also our mfBm has the same property of long-range dependence as the fBm. Moreover, a time-varying Hurst exponent H(t) is taken to replace the constant H in fBm. Finally, we apply the quantum-behaved partial swarm optimization (QPSO) to optimize H(t) for a finite interval. Some tests and corresponding experimental results will show that our model QPSO + mfBm have a much better performance on the prediction effect than fBm.

In this paper, we analyzed and found the solution for a suitable nonlinear fractional dynamical system that describes coronavirus (2019-nCoV) using a novel computational method. A compartmental model with four compartments, namely, susceptible, infected, reported and unreported, was adopted and modified to a new model incorporating fractional operators. In particular, by using a modified predictor–corrector method, we captured the nature of the obtained solution for different arbitrary orders. We investigated the influence of the fractional operator to present and discuss some interesting properties of the novel coronavirus infection.

This paper re-investigates the mathematical transport model of chlorine used as a water treatment model, when a variable order partial derivative is incorporated for describing the chlorine transport system. This model was introduced in the literature and governed by a fractional partial differential equation (FPDE) with prescribed boundary conditions. The obtained solution in the literature was based on implementing the Laplace transform (LT) combined with the method of residues and expressed in terms of regular exponential functions. However, the present analysis avoids such a method of residues, and thus a new analytical solution is introduced in this paper via Mittag-Leffler functions. Therefore, an effective approach is developed in this paper to solve the chlorine transport model with non-integer order derivative. In addition, our results are compared with several studies in the literature in case of integer-order derivative and the differences in results are explained.

This article makes a qualitative analysis of what constitutes the academic work in its overlapping elements and processes. There is a joint analysis of theory and facts based on discourse analysis theory to observe intrinsic aspects that put creativity as an essential product of academic work, and extrinsic aspects necessary to establish trust and dignity as necessary assumptions to achieve its goal. We hope this approach become an available tool to protect creativity against abuses of power and to show how one builds a balanced academic work within an ethical dimension, preventing the construction of a fallacious product.

Medical Imaging is a technique to create visual representations of the interior of the body, with the aim of making accurate diagnoses and optimized treatments. Many medical imaging techniques are widely used to produce images, such as computer tomography (CT), ultrasound (US), positron emission tomography (PET), single photon emission computed tomography (SPECT), and magnetic resonance imaging (MRI)/functional MRI (fMRI).

The aim of this work is to consider bicomplex Bernstein polynomials attached to analytic functions on a compact C2-disk and to present some approximation properties extending known approximation results for the complex Bernstein polynomials. Furthermore, we obtain and present quantitative estimate inequalities and the Voronovskaja-type result for analytic functions by bicomplex Bernstein polynomials.

Here, we ascertain generalized integral formulas concerning the product of the generalized Mittag-Leffler function. These integral formulas are described in the form of the generalized Lauricella series. Some special cases are also presented in terms of the Wright hypergeometric function.

This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

Analysis of the fetal heart rate during pregnancy is essential for monitoring the proper development of the fetus. Current fetal heart monitoring techniques lack the accuracy in fetal heart rate monitoring and features acquisition, resulting in diagnostic medical issues. The challenge lies in the extraction of the fetal ECG from the mother ECG during pregnancy. This approach has the advantage of being a reliable and non-invasive technique. In the present paper, a wavelet/multiwavelet method is proposed to perfectly extract the fetal ECG parameters from the abdominal mother ECG. In a first step, due to the wavelet/mutiwavelet processing, a denoising procedure is applied to separate the noised parts from the denoised ones. The denoised signal is assumed to be a mixture of both the MECG and the FECG. One of the well-known measures of accuracy in information processing is the concept of entropy. In the present work, a wavelet/multiwavelet Shannon-type entropy is constructed and applied to evaluate the order/disorder of the extracted FECG signal. The experimental results apply to a recent class of Clifford wavelets constructed in Arfaoui, et al. J. Math. Imaging Vis. 2020, 62, 73–97, and Arfaoui, et al. Acta Appl. Math. 2020, 170, 1–35. Additionally, classical Haar–Faber–Schauder wavelets are applied for the purpose of comparison. Two main well-known databases have been applied, the DAISY database and the CinC Challenge 2013 database. The achieved accuracy over the test databases resulted in Se = 100%, PPV = 100% for FECG extraction and peak detection.

Phytoplankton movement patterns and swimming behavior are important and basic topics in aquatic biology. Heavy tail distribution exists in diverse taxa and shows theoretical advantages in environments. The fat tails in the movement patterns and swimming behavior of phytoplankton in response to the food supply were studied. The log-normal distribution was used for fitting the probability density values of the movement data of Oxyrrhis marina. Results showed that obvious fat tails exist in the movement patterns of O. marina without and with positive stimulations of food supply. The algal cells tended to show a more chaotic and disorderly movement, with shorter and neat steps after adding the food source. At the same time, the randomness of turning rate, path curvature and swimming speed increased in O. marina cells with food supply. Generally, the responses of phytoplankton movement were stronger when supplied with direct prey cells rather than the cell-free filtrate. The scale-free random movements are considered to benefit the adaption of the entire phytoplankton population to varied environmental conditions. Inferentially, the movement pattern of O. marina should also have the characteristics of long-range dependence, local self-similarity and a system of fractional order.

This paper considers boundary value problem (BVP) for nonlinear first-order differential problems with multipoint and integral boundary conditions. A suitable Green function was constructed for the first time in order to reduce this problem into a corresponding integral equation. So that by using the Banach contraction mapping principle (BCMP) and Schaefer’s fixed point theorem (SFPT) on the integral equation, we can show that the solution of the multipoint problem exists and it is unique.

This paper is devoted to shedding some light on the advantages of using tight frame systems for solving some types of fractional Volterra integral equations (FVIEs) involved by the Caputo fractional order derivative. A tight frame or simply framelet, is a generalization of an orthonormal basis. A lot of applications are modeled by non-negative functions; taking this into account in this paper, we consider framelet systems generated using some refinable non-negative functions, namely, B-splines. The FVIEs we considered were reduced to a set of linear system of equations and were solved numerically based on a collocation discretization technique. We present many important examples of FVIEs for which accurate and efficient numerical solutions have been accomplished and the numerical results converge very rapidly to the exact ones.

The main goal of this paper is to define a simple but effective method for approximating solutions of multi-order fractional differential equations relying on Caputo fractional derivative and under supplementary conditions. Our basis functions are based on some original generalization of the Bessel polynomials, which satisfy many properties shared by the classical orthogonal polynomials as given by Hermit, Laguerre, and Jacobi. The main advantages of our polynomials are two-fold: All the coefficients are positive and any collocation matrix of Bessel polynomials at positive points is strictly totally positive. By expanding the unknowns in a (truncated) series of basis functions at the collocation points, the solution of governing differential equation can be easily converted into the solution of a system of algebraic equations, thus reducing the computational complexities considerably. Several practical test problems also with some symmetries are given to show the validity and utility of the proposed technique. Comparisons with available exact solutions as well as with several alternative algorithms are also carried out. The main feature of our approach is the good performance both in terms of accuracy and simplicity for obtaining an approximation to the solution of differential equations of fractional order.

In this paper, a wind speed prediction method was proposed based on the maximum Lyapunov exponent (Le) and the fractional Levy stable motion (fLsm) iterative prediction model. First, the calculation of the maximum prediction steps was introduced based on the maximum Le. The maximum prediction steps could provide the prediction steps for subsequent prediction models. Secondly, the fLsm iterative prediction model was established by stochastic differential. Meanwhile, the parameters of the fLsm iterative prediction model were obtained by rescaled range analysis and novel characteristic function methods, thereby obtaining a wind speed prediction model. Finally, in order to reduce the error in the parameter estimation of the prediction model, we adopted the method of weighted wind speed data. The wind speed prediction model in this paper was compared with GA-BP neural network and the results of wind speed prediction proved the effectiveness of the method that is proposed in this paper. In particular, fLsm has long-range dependence (LRD) characteristics and identified LRD by estimating self-similarity index H and characteristic index α. Compared with fractional Brownian motion, fLsm can describe the LRD process more flexibly. However, the two parameters are not independent because the LRD condition relates them by αH > 1.

In this paper, local fractional cylindrical wave solutions on Signorini hyperelastic materials are studied. In particular, we focus on the so-called Signorini potential. Cantor-type cylindrical coordinates are used to analyze, both from dynamical and geometrical point of view, wave solutions, so that the nonlinear fundamental equations of the fractional model are explicitly given. In the special case of linear approximation we explicitly compute the fractional wave profile.