Fractals

This paper focuses on parameter convergence and precise modeling for fractional-order nonlinear systems with functional uncertainties via using adaptive backstepping neural network control (ABNNC) and composite learning adaptive backstepping neural network control (CLABNNC). In the ABNNC design, a command filter is proposed, and the neural network approximation system is considered to deal with the unknown function, where an adaptation law is designed to ensure tracking errors converge to an arbitrarily small region near the origin under a strict persistent excitation condition that is too strict for the convergence of adaptive parameters. In order to relax this condition, a composite learning adaptation law is established by taking advantage of the tracking error and the prediction error to update the free parameter of the neural network system. The proposed CLABNNC method can not only ensure the convergence of tracking errors, but also achieve the accurate approximation of functional uncertainties under a weaker interval excitation condition. Finally, a numerical simulation example is put forward to demonstrate the effectiveness of our method.

The aim of this paper is two-fold. First, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the degenerate Bernoulli polynomials and the degenerate Frobenius–Euler polynomials. Second, we consider the degenerate Eulerian polynomials and deduce the generating function and a recurrence relation for them.

The zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive integers, which are also known as the conditional Poisson distributions or the positive Poisson distributions. Recently, as a natural extension of those distributions, Kim–Kim studied the zero-truncated degenerate Poisson distributions. In this paper, we introduce the $r$-truncated degenerate Poisson random variable with parameter $\alpha >0$, whose probability mass function is given by $p_{\lambda ,r}(i)=\frac{{(1)}_{i,\lambda }}{e_{\lambda }(\alpha )-e_{\lambda ,r}(\alpha )}\frac{{\alpha }^i}{i!},(i=r+1,r+2,r+3,\dots )$, and investigate various properties of this random variable.

The fractional dynamics of the lethal Nipah virus (NiV) is investigated in this research. The transmission patterns of the Nipah virus are studied quantitatively and qualitatively. Fixed-point results are utilized to study the NiV fractional model qualitatively. An efficient numerical method Predictor–Corrector (PC) is used to deduce the numerical results of the NiV model. The stability of the numerical approach is studied through nonlinear analysis. The numerical results are graphically presented in various fractional orders. The effects of transmission and infection rates on the evolution of the proposed model are presented through numerical simulations.

Fractional-order models describing neuronal dynamics often exhibit better compatibility with diverse neuronal firing patterns that can be observed experimentally. Due to the overarching significance of synchronization in neuronal dynamics, we here study synchronization in multiplex neuronal networks that are composed of fractional-order Hindmarsh–Rose neurons. We compute the average synchronization error numerically for different derivative orders in dependence on the strength of the links within and between network layers. We find that, in general, fractional-order models synchronize better than integer-order models. In particular, we show that the required interlayer and intralayer coupling strengths for interlayer or intralayer synchronization can be weaker if we reduce the derivative order of the model describing the neuronal dynamics. Furthermore, the dependence of the interlayer or intralayer synchronization on the intralayer or interlayer coupling strength vanishes with decreasing derivative order. To support these results analytically, we use the master stability function approach for the considered multiplex fractional-order neuronal networks, by means of which we obtain sufficient conditions for the interlayer and intralayer synchronizations that are in agreement with numerical results.

A nonclassical model known as the guava model with the conformable derivative describing the interaction of guava pests and natural enemies is studied in this paper. To this end, first the Adams–Bashforth–Moulton predictor–corrector (ABM-PC) scheme is adopted to numerically solve the guava model with the conformable derivative such that its performance is examined using the finite-difference (FD) method. The truncation error of the ABM-PC scheme is then presented in a detailed way. The effect of the order of the conformable derivative on the dynamical characteristics of guava pests and natural enemies is investigated by considering a series of graphical representations. In the end, based on the results given in this study, it is shown which day is more beneficial to harvest the guava.

Let $(M,g)$ be a smooth n-dimensional Riemannian manifold for $n\ge 3$. Consider the conformal perturbation $\tilde{g}=hg$ where $h$ is a smooth bounded positive function on $M$. Denote by $\tilde{\mathrm{\Delta }}$ the Laplace–Beltrami operator of manifold $(M,\tilde{g})$. In this paper, we derive the upper bounds of the heat kernels for ${(-\tilde{\mathrm{\Delta }})}^{\sigma }$ with $0<\sigma \le 1$. Moreover, we also investigate the gradient estimates of the heat kernel for $\tilde{\mathrm{\Delta }}$.

A class of Erdélyi–Kober fractional differential inequalities with a polynomial nonlinearity and a singular potential function is investigated in this paper. By mean of the test function method, we establish sufficient conditions for the nonexistence of global weak solutions. Some examples are provided to illustrate our obtained results. To the best of our knowledge, the issue of nonexistence of global solutions for fractional differential equations or inequalities, involving Erdélyi–Kober fractional derivatives, was never addressed in the literature.

This paper develops optimal family of fourth-order iterative techniques in order to find a single root and to generalize them for simultaneous finding of all roots of polynomial equation. Convergence study reveals that for single root finding methods, its optimal convergence order is 4, while for simultaneous methods, it is 12. Computational cost and numerical illustrations demonstrate that the newly developed family of methods outperformed the previous methods available in the literature.

This research paper aims to introduce an extension of the $k$-Hurwitz–Lerch $\zeta$-function of matrix arguments and interpret its several properties, like generating matrix relations, derivative formulae, Mellin transforms and integral representations. Further, we discuss the solution of the fractional kinetic equations (FKEs) involving an extended $k$-Hurwitz–Lerch $\zeta$-function of matrix arguments. In addition, the results investigated in view of an extended $k$-Hurwitz–Lerch $\zeta$-matrix function are rather general in nature and can easily give various known and novel FKEs.

The primary goal of this paper is to study a nonlinear fuzzy fractional dynamic system (FFDS) involving a time-dependent variational inequality. We use the monotone argument and Knaster–Kuratowski–Mazurkiewicz (KKM) theorem to prove that the variational system of FFDS is solvable and its solutions become a bounded, closed and convex set. Employing this result together with Bohnenblust–Karlin fixed point theorem and Filippov implicit function, we show the existence of a mild solution to FFDS.

The derangement number $d_n$ is the number of fixed point free permutations on a set of $n$ elements and the derangement polynomial $d_n(x$) is a natural extension of the derangement number $d_n$. The aim of this paper is to introduce $q$-derangement numbers and polynomials, which are $q$-analogs of the derangement numbers and polynomials, and to investigate their connection with some other $q$-special numbers and polynomials. In more detail, we derive explicit expressions and recurrence relations for the $q$-derangement numbers and polynomials. Further, we obtain some identities involving such polynomials and numbers and other special $q$-polynomials and numbers, which include $q$-Bell polynomials, $q$-analogs of Fubini polynomials and $q$-Stirling numbers of the second kind.

We study in this work a geometric inverse problem using the Kohn–Vogelius formulation method. Using this technique, the problem is reformulated as an optimization problem. The resolution is then carried out by the method of topological sensitivity. A real application is finally presented to show the effectiveness of this developed approach.

We study the uniqueness, the continuity in $L^2$ and the large time decay for the Leray solutions of the 3D incompressible Navier–Stokes equations with nonlinear exponential damping term $a(e^{b|u{|}^4}-1)u$, ($a,b>0$).

In this paper, a novel fuzzy sliding-mode backstepping control approach is put forward to tackle the tracking control issue of fractional-order chaotic nonlinear systems subject to uncertain control coefficient and input saturation. First, a kind of sliding surface with nonlinear term of exponential monotonic attenuation is introduced to facilitate the fast error convergence and the chaos effect suppression. Subsequently, fuzzy logic systems and Nussbaum gains are synthesized to cope with the entire uncertainties including unknown control coefficients. Furthermore, dynamic surface technique is utilized to circumvent the occurrence of “complexity explosion” during the overall backstepping control procedure. By means of the proposed control scheme, system state is guaranteed to maintain close to the sliding surface without any reaching phrase, and whereafter the tracking error converges to a sufficiently small residual set containing the origin. Finally, the comparison between the proposed control scheme and an alternative existing chaos control approach is given through numerical simulation, which further confirms the validity of the obtained results and the superiority of the proposed strategy.

A general integer-valued time-series model with a conditional variance proportional to the conditional mean is proposed. Specifically, the conditional distribution is a Poisson mixture with a dependent mixing sequence, which results in a negative binomial distribution with a linear conditional variance-to-mean relationship. In addition, the conditional mean is specified as a general parametric function of past observations. We first propose stationarity, ergodicity, and finite moment conditions for the model. Furthermore, the parameters are estimated using the Poisson quasi-maximum likelihood estimate, whose asymptotic properties are studied under weak conditions. Illustrations of the proposed methodology on simulated and actual time series of counts are given.

The purpose of the current investigation is to solve the fifth kind of induction motor model using an advanced computational scheme by operating the artificial neural networks (ANNs), global scheme as genetic algorithm (GA) along with the rapid local search sequential quadratic programming technique (SQPT), i.e. ANN-GA-SQPT. ANNs are implemented to discretize the fifth kind of induction motor model to express the merit function based on the mean square error. The numerical presentation of the proposed ANN-GA-SQPT is pragmatic for three different problems based on the fifth kind of induction motor model to authenticate the efficacy, consistency and importance of the proposed ANN-GA-SQPT. Moreover, statistical representations are provided in order to check the precision, convergence and accuracy of the present ANN-GA-SQPT.

The aim of this study is to perform the numerical investigations of a fractional nonlinear dengue model using artificial neuron networks (ANNs) along with the Levenberg–Marquardt backpropagation (LMB), i.e. ANNs. The fractional nonlinear dengue model is divided into five classes. The stochastic-based ANNs-LMB scheme is pragmatic on three variants of authentication, training and testing. The data magnitudes for three different variations based on the fractional nonlinear dengue model are selected as 80% for training, 10% for both testing and validation. The numerical procedures of the fractional nonlinear dengue model will be performed through ANNs-LMB and comparative investigations using the reference values that are calculated on the basis of Adams–Bashforth–Moulton scheme. The solution of the fractional nonlinear dengue model is obtained through the ANNs-LMB to reduce the mean square error (MSE). To authenticate the capability and efficiency of the proposed ANNs-LMB, the obtained numerical measures of correlation, MSE results, regression and error histograms (EHs) are provided.

Although the passivity of integer-order systems has been extensively analyzed, the research outcomes on the passivity of fractional-order nonlinear systems (FONSs) are scarce. This paper presents some theoretical results on passivity and passivation of FONSs. Based on the definition of the passivity of FONSs, and by using the Lyapunov stability theory and the linear matrix inequality (LMI) method, some conditions are derived to assure the FONSs is passive, which enrich the existing theoretical knowledge about the passivity of FONSs. Moreover, an observer-based output passive control is established to ensure that the corresponding closed-loop system is passive by means of LMI technique and matrix singular value decomposition (SVD). Ultimately, the practicality of our yielded results is revealed by two numerical simulations.

Dynamics of chemical reactor systems are found with highly nonlinear behavior. Computational modeling of a fractional-order chemical reactor system and investigating nonlinear dynamical changes and its control are the main focus of this research work. Chaos theory is a blooming fertile field in recent years, which is used widely to quantify nonlinear behaviors such as quasi-oscillations, bi-stability and bifurcation. The work starts from deriving state-space model of the system with first-order differential equations. There are six equilibrium points and the Jacobian matrix is derived for investigating the stability of the equilibrium points. Eigenvalues of each equilibrium point are calculated. Based on the sign of the real part of the eigenvalues and the existence of imaginary part, we found two equilibrium points behave as saddle spirals and the remaining four equilibrium points are saddle nodes. The stability of the system for different parameter values is investigated and presented. The influence of parameters in the system dynamics is discussed and significant parameter values are highlighted for further study. We considered Caputo’s definition for formulating the fractional-order (FO) model of the system based on the advantages highlighted in various literatures. The stable and unstable regions are portrayed with parameter variations. The results clarified that the analysis can be refined using fractional-order treatment of chaotic systems. We proceeded with our investigation towards obtaining different oscillations, particularly chaotic oscillations. The challenges lie in finding the proper fractional order to handle the system. We showed the bifurcation diagram for a range of fractional-order values and clarified the transitions from periodic oscillations to chaotic behavior and period-doubling bifurcations. The phase portraits are presented to show the limit cycle oscillations for fractional-order 0.95, period-doubling during 0.98, and chaotic oscillations for higher values. We proceeded with our investigation with fractional-order as 0.99. Bifurcation plots for parameter variation are obtained. Chaotic regions, periodic oscillations, period-halving and period-doubling are observed and the influences are discussed. We emphasize the intricate properties which are not addressed during the integer-order treatment of the system and nail the importance of fractional-order treatment. An adaptive sliding mode (ASM) controller is derived and implemented to control the system precisely. The effectiveness is shown by providing simulation results of the system with parameter estimation and controlled state time history plots. The work can be extended to verify the simulated results with equivalent electronic circuits.

In this paper, some important objectives have been achieved, which are as follows: First, we present a method of the inverse for a class of non-singular block matrices and some associated properties. Also, the accuracy of a new method is verified with some illustrated examples by applying the MATLAB lines. Second, applying a class of block matrices, we give the exact solution for fractional matrix differential equation systems using the Laplace fractional transformation method. Finally, illustrative examples and individual cases are also presented and discussed to demonstrate our new approach.

In this paper, the state-constrained adaptive control problem for a class of uncertain fractional-order nonlinear systems with unknown external disturbances is studied. Using the fuzzy approximation and backstepping control layout, an adaptive disturbance-observer-based control method is proposed, which can effectively estimate the unknown external disturbances through the designed disturbance observers. As we know, the standard backstepping control has inherent computational complexity. Therefore, a design approach of fractional-order command filter is introduced, which takes advantage of fractional-order command filter to pre-estimate the virtual input signal and its fractional derivative. In addition, barrier Lyapunov functions are exploited to cope with the fractional states-constrained control problem. The combination of barrier Lyapunov function technique and backstepping design not only ensures excellent tracking performance, but also enhances the robustness of the system. Furthermore, based on the fractional Lyapunov approach, the relevant stability analysis is established, which demonstrates that all states of the system remain within their constrained scope, and all signals of the closed-loop system are bounded. Finally, the effectiveness of the proposed scheme is verified by numerical simulation examples of RLC circuit and horizontal platform system.

Asymmetrical models like the generalized and q-generalized extreme value distributions have been widely used to describe a variety of random events observed in studies including survival, finance, and dependability. The q-analogues, with the extra parameter q, provide for more modelling freedom. In this paper, Shannon entropy, Varentropy and Varexentropy measures are computed in closed form in cases of generalized and q-generalized extreme value distributions. Maximum likelihood estimation method is applied for computing the values of Shannon entropy, Varentropy and Varexentropy measures.

This paper shows how to approximate the solution to a nonlinear conformable space-time fractional partial differential equations. The proposed method is based on the Cubic B-spline polynomials and Galerkin method. Two test problems show that the approach we use to approximate the proposed equation is accurate and efficient. We apply the Von Neumann approach to show that stability requires some conditions.

The combined effect of the solar radiation pressure and the oblateness of the planets on the position and stability of the Lagrangian points within the restricted elliptic three-body problem was studied. We found that the solar radiation pressure and planet oblateness slightly changed the location of the Lagrangian points. New formulas for the location of the collinear points using the Lagrange inversion method were introduced. For large values of the force ratio B; the ratio between the solar radiation force and the gravitational attraction force due to the Sun; we found that the collinear point $L_3$ is stable.

Due to the rapid development of algorithms and techniques that are used to deal with mixed integer nonlinear programming (MINLP) problems, many global MINLP solvers were introduced. In this paper, computational experiments were done to compare between the performances of five of these solvers. Some of these solvers do not support trigonometric functions. Therefore, piecewise linear approximation (PLA) is applied to problems having these function so the solvers can deal with these problems. Additional computational tests were performed on to show how PLA can be useful, even to some powerful global solvers.

In this paper, a system of nonlinear delay integral equations related to the spread of certain infectious diseases is investigated. Namely, using Perov’s fixed-point theorem, we obtain sufficient conditions for which the system admits a unique positive solution, and provide a numerical algorithm that converges to this solution. Moreover, we establish some comparison theorems, and study the data dependence of solutions.

Mathematical modeling can be a powerful tool to predict disease spread in large populations as well as to understand different factors which can impact it such as social distancing and vaccinations. This study aimed to describe the spread the coronavirus disease 2019 (COVID-19) pandemic in Saudi Arabia using a simple discrete variant of the Gompertz model. Unlike time-continuous models which are based on differential equations, this model treats time as a discrete variable and is then represented by a first-order difference equation. Using this model, we performed a short-term prediction of the number of cumulative cases of COVID-19 in the country and we show that the results match the confirmed reports.

There are three over-determined boundary value problems for the inhomogeneous Cauchy–Riemann equation, the Dirichlet problem, the Neumann problem and the Robin problem. These problems have found their significant importance and applications in diverse fields of science, for example, applied mathematics, physics, engineering and medicine. In this paper, we explicitly investigate the solvability of these three basic boundary value problems for the Cauchy–Riemann operator on an isosceles orthogonal triangle. The solvability conditions are explicitly obtained. The plane parqueting reflection principle is used. Cauchy–Pompeiu-type formulae for the triangle are established. The boundary behavior of a related integral operator of the Schwarz type is studied in detail. Then, the boundary values of solutions at the corner points are found.

This paper addresses the numerical study of variable-order fractional differential equation based on finite-difference method. We utilize the implicit numerical scheme to find out the solution of two-dimensional variable-order fractional modified sub-diffusion equation. The discretized form of the variable-order Riemann–Liouville differential operator is used for the fractional variable-order differential operator. The theoretical analysis including for stability and convergence is made by the von Neumann method. The analysis confirmed that the proposed scheme is unconditionally stable and convergent. Numerical simulation results are given to validate the theoretical analysis as well as demonstrate the accuracy and efficiency of the implicit scheme.

In human–machine interaction, non-verbal communication through the recognition of facial expressions is a process that operates on a time scale of the order of milliseconds, bringing a new dimension to how machines can affect modern social life. The ambiguity at this time scale is significant, necessitating that humans and machines rely on rich cognitive abilities rather than slow symbolic conclusions, making it necessary for humans and machines to rely on rich perceptual skills rather than sluggish figurative conclusions, especially in critical applications such as elderly care and supportive procedures for people with mobility or communication problems. This research field includes interdisciplinary contributions and specialized support from cognitive areas of psychology, sociology, linguistics, industrial design, and informatics. This paper presents an innovative emotion recognition system through dynamic facial analysis for optimal decision-making in non-verbal communication. It is an innovative model of artificial vision, for the recognition of emotions in the human–machine interaction, combining the look and the basic facial expressions of a person. This approach implements and proposes for the first time in the literature the application of a Common Space Variational Recurrent Deep Embedding (CSVRdE) intelligent learning system. The suggested technique simplifies the process of training customized extraction functions for appropriate image transformations in complicated neural network architectures, resulting in increased learning consistency, superior prediction reliability, and outstanding classification efficiency. Specifically, the suggested technique produces extremely accurate findings without reoccurring difficulties of unknown origin since all characteristics in the dataset are effectively managed.

To enhance the accuracy results of the change detection of buildings utilizing high-resolution remote sensing (HRRS) images, a novel method was proposed by combining both tensor and deep belief network (DBN). To optimize the description of the essential characteristics for changes in buildings, a tensor-based structure covering time-space-spectrum-shadow features integrated into the model (TSSS-Cube) is proposed. The changes occurring as a combination of shadow and spectral features and spatio-temporal autocorrelation at each pixel are represented by a third-order tensor to maintain the structural information and the constraint integrity between them. Then, a restricted Boltzmann machine (TC-RBM) that can be directly used to process TSSS-Cube data is designed, and the support tensor machine (STM) is used to replace the conventional backpropagation neural network at the top of the DBN to construct a multi-tensor deep belief network (MTR-DBN) composing of multi-layer TC-RBMs and an STM classifier. Finally, the multi-layer TC-RBMs in MTR-DBN are trained layer by layer, and the global parameters of the MTR-DBN are optimized by combining a limited number of labeled data and fine-tuning the STM classifier. The implementation of both supervised and unsupervised learning methods comprehensively provides advantages to increase the accuracy result of the MTR-DBN network to detect changes. Three representative different sub-regions are selected from the whole original experimental area respectively for building change detection experiments, and a dataset composed of double-temporal HRRS images in 2012 and 2016 is used as the related experimental dataset. The experimental results show that both a change detection accuracy result with a higher average and better detection efficiency is attained by the proposed method called the MTR-DBN when compared with other similar methods.

Using mathematical models to describe the dynamics of infectious-diseases transmission in large communities can help epidemiological scientists to understand different factors affecting epidemics as well as health authorities to decide measures effective for infection prevention. In this study, we use a discrete version of the Generalized Logistic Model (GLM) to describe the spread of the coronavirus disease 2019 (COVID-19) pandemic in Saudi Arabia. We assume that we are operating in discrete time so that the model is represented by a first-order difference equation, unlike time-continuous models, which employ differential equations. Using this model, we forecast COVID-19 spread in Saudi Arabia and we show that the short-term predicted number of cumulative cases is in agreement with the confirmed reports.

Recently, Kim–Kim introduced the $\lambda$-umbral calculus, in which the $\lambda$-Sheffer sequences occupy the central position. In this paper, we introduce the fully degenerate Bell and the fully degenerate Dowling polynomials, and investigate some properties and identities relating to those polynomials with the help of $\lambda$-umbral calculus. Here, we note that the fully degenerate Bell poynomials and the fully degenerate Dowling polynomials are, respectively, degenerate versions of the Bell polynomials and the Dowling polynomials, of which the latters are the natural extension of the Whitney numbers of the second kind.

In this paper, we define the multi-Stirling numbers of the first kind by means of the multiple logarithm and as a generalization of the Stirling numbers of the first kind. Then we introduce two additional special numbers by using the multiple logarithm, namely the modified multi-Bernoulli numbers as a generalization of the higher-order Bernoulli numbers and the multi-Lah numbers of type 2 as a generalization of the Lah numbers. For both of these special numbers, we derive explicit expressions for them which involve the multi-Stirling numbers of the first kind.

In this paper, we study the question of the existence and nonexistence of solutions for some fractional equations with variable exponents. This paper generalizes some analog results in the classical fractional one. As we know, there are no previous results on the nonexistence of solutions for nonlinear equations with fractional $p(\cdot ,\cdot )$-Laplacian.

In this paper, we construct a $(p,k)$-hypergeometric function by using the Hadamard product, which we call the generalized $(p,k)$-hypergeometric function. Several properties, namely, convergence properties, derivative formulas, integral representations and differential equations are indicated of this function. The latter function is a generalization of the usual hypergeometric function, the k-analogue of hypergeometric function and other hypergeometric functions are recently presented. As an application, we obtain the solution of the generalized fractional kinetic equations involving of the generalized $(p,k)$-hypergeometric function.

Although the therapy of chronic myelogenous leukemia (CML) has progressed because of imatinib (IM) and other tyrosine kinase inhibitors (TKIs), the majority of patients still do not recover. To better regulate the remaining leukemic cell population, TKI combo therapy may be improved with a deeper understanding of the underlying mechanisms. We employed a mathematical system which incorporated the intricate phenomena of immune system to CML. We use a fractional derivative framework in this work to understand the dynamics of CML. Additionally, in our work, we concentrate on the qualitative characterization and dynamical behavior of CML interactions. For the proposed model, we examine the singularity and existence using fixed point theorems by Banach and Schaefer. We provide the necessary criteria for our suggested fractional model’s Ulam–Hyers stability. The influence of the factors on the dynamics of CML is highlighted by closely examining the solution paths by using a numerical scheme. To be more precise, we emphasized how the suggested system’s dynamic and chaotic behavior varied depending on the fractional order and other system factors. Policymakers are advised to consider the most crucial elements of CML dynamics. In order to inform policymakers and health authorities about the systems essential for control and treatment, it is crucial to investigate the dynamic characteristics of CML disease.

In this work, we present the dynamics of the one dimension fractional-order Rulkov map of biological neurons. The one-dimensional neuron map shows all the dynamical behaviors observed in the real-time experiment. The integer order one-dimensional Rulkov map exhibits chaotic dynamics in the presence of time-dependent external stimuli like periodic sinusoidal force or random Gaussian process. When we construct a large complex network of neurons, the higher system dimension, as well as the external forcing, is always an obstacle. Interestingly, our study shows even with constant external stimuli, the fractional-order one-dimensional neuron shows a rich variety of complex dynamics including chaotic dynamics. We present our results based on the Lyapunov exponent of the fractional-order systems.

In this paper, we investigate a BVP for a class impulsive fractional partial differential equations. We propose a new topological approach to prove the existence of at least one classical solution and at least two nonnegative classical solutions. The arguments are based upon recent theoretical results.

In this work, we construct a new family of inverse iterative numerical technique for extracting all roots of nonlinear equation simultaneously. Convergence analysis verifies that the proposed family of methods has local 10th-order convergence. Among the test models investigated are blood rheology, a fractional nonlinear equation model, fluid permeability in biogels, and beam localization models. In comparison to other methods, the family of inverse simultaneous iterative techniques gets initial estimations to exact roots within a given tolerance while using less function evaluations in each iterative step. Numerical results, basins of attraction for fractional nonlinear equation, residual graphs are presented in detail for the simultaneous iterative techniques. The newly developed simultaneous iterative techniques were thoroughly investigated and proven to be efficient, robust, and authentic in their domain.

In this research work, we offer an epidemic model for monkeypox virus infection with the help of non-integer derivative as well as classical ones. The model takes into account every potential connection that can aid in the spread of infection among the people. We look into the model’s endemic equilibrium, disease-free equilibrium, and reproduction number ${{ℛ}}_0$. In addition to this, we concentrated on the qualitative analysis and dynamic behavior of the monkeypox virus. Through fixed point theorem, Banach’s and Schaefer’s are applied to study the existence and uniqueness of the solution of the suggested system of the monkeypox virus infection. We provide the necessary criteria for the recommended fractional system’s Ulam–Hyers stability. Furthermore, a numerical approach is used to study the solution routes and emphasize how the parameters affect the dynamics of the monkey pox virus. The most crucial features of the dynamics of the monkeypox virus are noticed and suggested to decision-makers.

This study is conducted to solve a nonlinear biological prey–predator system (NBPPS) using a novel design of the Levenberg–Marquardt backpropagation approach (LMBA). The LMBA-based supervised neural networks (SNNs) deal with three kinds of sample data, training, validation, and testing. The percentages for these data to solve three different cases of the NBPPS are selected: for training 75%, validation 10%, and testing 15%, respectively. The numerical performances of the Adams method are used for the reference dataset to solve the NBPPS. The obtained form of the numerical solutions of the NBPPS based on the SNNs along with LMBA is used to reduce the functions of mean square error (MSE). For the correctness, competence, and effectiveness of the proposed SNNs along with LMBA, the numerical procedures are proficient based on the proportional schemes and analyses in terms of MSE results, correlation, error histograms, and regression.

The creation of language models is a vital object of natural language processing because it creates a possible distribution in a sequence of words and offers a unique representation of each occurrence of a word. Therefore, an intelligent language model can differentiate the subtle nuances of language and model its syntax and semantics. It also allows the acquisition of high-level representations. In addition, it serves as an initial model with helpful information, which can be transferred to various other word processing methods for a deeper and more complete understanding of the language. This work proposes an innovative Long Short-Term Memory (LSTM) neural network whose implementation is based on the use of an attention mechanism as a measure of alignment of the output with the input sequence and a corresponding coverage optimization mechanism. The optimization mechanism informs the model about the possible outcomes at each step which have already been produced in previous actions. The adaptation and evaluation of the system’s design parameters were tested in multidimensional and challenging data sets We were considering parametric and heuristic procedures for finding the optimal combination of hyper-parameters. Experimental results demonstrate the quality of the proposed system, revealing essential directions for the design of similar systems.

The aim of this paper is to make inference about a general class of time series models including fractional Brownian motion. The spectral of these processes is supported on lines not parallel to the diagonal $T_1(x)=x$, $T_j(x)={\alpha }_{j}x\pm {\beta }_j$, $j=2,\dots ,m$, in spectral square $[0,2\pi )\times [0,2\pi )$, and this class includes stationary, cyclostationary, almost cyclostationary time series and specially fractional Brownian motions. First, the periodogram of these processes is defined and auxiliary operator is applied to explore the distribution of the periodogram. Then the asymptotical estimation for the spectral density function is proposed and asymptotical Wishart function is found. Finally, the validity of the theoretical results is studied using simulated data sets.