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After extensive examination, scholars have determined that many dynamic systems exhibit intricate connections not only with their current and past states but also with the delay function itself. As a result, their focus shifts towards fractional neutral stochastic differential equations, which find applications in diverse fields such as biology, physics, signal processing, economics, and others. The fundamental principles of existence and uniqueness of solutions to differential equations, which guarantee the presence of a solution and its uniqueness for a specified equation, are pivotal in both the mathematical and physical realms. A crucial approach for analyzing complex systems of differential equations is the utilization of the averaging principle, which simplifies problems by approximating existing ones. Applying contraction mapping principles, we present results concerning the concepts of existence and uniqueness for the solutions of fractional neutral stochastic differential equations. Additionally, we present Ulam-type stability and the averaging principle results within the framework of Lp space. This exploration involved the utilization of Jensen’s, Gröenwall-Bellman’s, Hölder’s, Burkholder-Davis-Gundy’s inequalities, and the interval translation technique. Our findings are established within the context of the conformable fractional derivative, and we provide several examples to aid in comprehending the theoretical outcomes. [ABSTRACT FROM AUTHOR]

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

Hepatitis B is a worldwide viral infection that causes cirrhosis, hepatocellular cancer, the need for liver transplantation, and death. This work proposed a mathematical representation of Hepatitis B Virus (HBV) transmission traits emphasizing the significance of applied mathematics in comprehending how the disease spreads. The work used an updated Atangana-Baleanu fractional difference operator to create a fractional-order model of HBV. The qualitative assessment and wellposedness of the mathematical framework were looked at, and the global stability of equilibrium states as measured by the Volterra-type Lyapunov function was summarized. The exact answer was guaranteed to be unique using the Lipschitz condition. Additionally, there were various analyses of this new type of operator to support the operator’s efficacy. We observe that the explored discrete fractional operators will be χ²-increasing or decreasing in certain domains of the time scale Nj := j, j +1,... by looking at the fundamental characteristics of the proposed discrete fractional operators along with χ-monotonicity descriptions. For numerical simulations, solutions were constructed in the discrete generalized form of the Mittag-Leffler kernel, highlighting the impacts of the illness caused by numerous causes. The order of the fractional derivative had a significant influence on the dynamical process utilized to construct the HBV model. Researchers and policymakers can benefit from the suggested model’s ability to forecast infectious diseases such as HBV and take preventive action. [ABSTRACT FROM AUTHOR]

This study focused on introducing a fresh model of fractional operators incorporating multiple delays, termed fractional integro-differential Langevin equations with multiple delays. Additionally, the research evaluated the relative controllability of this model within finite-dimensional spaces. Employing fixed-point theory yields the desired outcomes, with the controllability assessment facilitated by Schauder's fixed point and the Grammian matrix defined through the Mittag-Leffer matrix function. Validation of the results was conducted through an application. [ABSTRACT FROM AUTHOR]

The aim of this work is to solve a numerical computation of the neutral fractional functional integro-differential equation based on a new approach to the Legendre wavelet method. The concept of fractional derivatives was examined in the sense of Caputo. The properties of the Legendre wavelet and function approximation were employed to determine the approximate solution of a given dynamical system. Moreover, the error estimations and convergence analysis of the truncated Legendre wavelet expansion for the proposed problem were discussed. The validity and applicability of this proposed technique to numerical computation were shown by illustrative examples. Eventually, the results of this technique demonstrate its great effectiveness and reliability. [ABSTRACT FROM AUTHOR]

The main objective of our research was to explore and develop a fractional-order derivative within the predator-prey framework. The framework includes prey refuge and selective nonlinear harvesting, where the harvesting progressively approaches a threshold value as the density of the harvested population advances. For memory effect, a non-integer order derivative is better than an integer-order derivative. The solutions to the fractional framework were shown to be existence, uniqueness, non-negativity, and boundedness. Matignon's condition was used for analysing local stability, and a suitable Lyapunov function provided global stability. While discussing the Hopf bifurcation's existence condition, we explored derivative order and refuge as bifurcation parameters. We aimed at redefining the predator-prey framework to incorporate fractional order, refuge, and harvesting. This kind of nonlinear harvesting is more realistic and reasonable than the model with constant yield harvesting and constant effort harvesting. The Adams-Bashforth-Moulton PECE algorithm in MATLAB software was used to simulate the proposed outcomes, investigate the impact on various factors, and analyse harvesting's effect on non-integer order predator-prey interactions. [ABSTRACT FROM AUTHOR]

One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results. [ABSTRACT FROM AUTHOR]

In numerous domains, fractional stochastic delay differential equations are used to model various physical phenomena, and the study of well-posedness ensures that the mathematical models accurately represent physical systems, allowing for meaningful predictions and analysis. A fractional stochastic differential equation is considered well-posed if its solution satisfies the existence, uniqueness, and continuous dependency properties. We established the well-posedness and regularity of solutions of conformable fractional stochastic delay differential equations (CFrSDDEs) of order γ ∈ (1/2, 1) in Lp spaces with p ≥ 2, whose coefficients satisfied a standard Lipschitz condition. More specifically, we first demonstrated the existence and uniqueness of solutions; after that, we demonstrated the continuous dependency of solutions on both the initial values and fractional exponent γ. The second section was devoted to examining the regularity of time. As a result, we found that, for each π ∈ (0, γ-1/2), the solution to the considered problem has a F-Hölder continuous version. Lastly, two examples that highlighted our findings were provided. The two main elements of the proof were the Burkholder-Davis-Gundy inequality and the weighted norm. [ABSTRACT FROM AUTHOR]

Recurrent Neural Networks (RNNs), a type of machine learning technique, have recently drawn a lot of interest in numerous fields, including epidemiology. Implementing public health interventions in the field of epidemiology depends on efficient modeling and outbreak prediction. Because RNNs can capture sequential dependencies in data, they have become highly effective tools in this field. In this paper, the use of RNNs in epidemic modeling is examined, with a focus on the extent to which they can handle the inherent temporal dynamics in the spread of diseases. The mathematical representation of epidemics requires taking time-dependent variables into account, such as the rate at which infections spread and the long-term effects of interventions. The goal of this study is to use an intelligent computing solution based on RNNs to provide numerical performances and interpretations for the SEIR nonlinear system based on the propagation of the Zika virus (SEIRS-PZV) model. The four patient dynamics, namely susceptible patients S(y), exposed patients admitted in a hospital E(y), the fraction of infective individuals I(y), and recovered patients R(y), are represented by the epidemic version of the nonlinear system, or the SEIR model. SEIRS-PZV is represented by ordinary differential equations (ODEs), which are then solved by the Adams method using the Mathematica software to generate a dataset. The dataset was used as an output for the RNN to train the model and examine results such as regressions, correlations, error histograms, etc. For RNN, we used 100% to train the model with 15 hidden layers and a delay of 2 seconds. The input for the RNN is a time series sequence from 0 to 5, with a step size of 0.05. In the end, we compared the approximated solution with the exact solution by plotting them on the same graph and generating the absolute error plot for each of the 4 cases of SEIRS-PZV. Predictions made by the model appeared to be become more accurate when the mean squared error (MSE) decreased. An increased fit to the observed data was suggested by this decrease in the MSE, which suggested that the variance between the model's predicted values and the actual values was dropping. A minimal absolute error almost equal to zero was obtained, which further supports the usefulness of the suggested strategy. A small absolute error shows the degree to which the model's predictions matches the ground truth values, thus indicating the level of accuracy and precision for the model's output. [ABSTRACT FROM AUTHOR]

Alzheimer's disease stands as one of the most widespread neurodegenerative conditions associated with aging, giving rise to dementia and posing significant public health challenges. Mathematical models are considered as valuable tools to gain insights into the mechanisms underlying the onset, progression, and potential therapeutic approaches for AD. In this paper, we introduce a mathematical model for AD that employs the fractal fractional operator in the Caputo sense to characterize the temporal dynamics of key cell populations. This model encompasses essential elements, including amyloid- β ( A β ), neurons, astroglia and microglia. Using the fractal fractional operator, we have established the existence and uniqueness of solutions for the model under consideration, employing Leray-Schaefer's theorem and the Banach fixed-point methods. Utilizing functional techniques, we have analyzed the proposed model stability under the Ulam-Hyers condition. The suggested model has been numerically simulated by using a fractional Adams-Bashforth approach, which involves a two-step Lagrange polynomial. For numerical simulations, different ranges of fractional order values and fractal dimensions are considered. This new fractal fractional operator in the form of the Caputo derivative was determined to yield better results than an ordinary integer order. Various outcomes are shown graphically by for different fractal dimensions and arbitrary orders. [ABSTRACT FROM AUTHOR]

The generalized q-deformed sinh Gordon equation (GDSGE) serves as a significant nonlinear partial differential equation with profound applications in physics. This study investigates the GDSGE's mathematical and physical properties, examining its solutions and clarifying the essence of the q-deformation parameter. The Sardar sub-equation method (SSEM) and sine-Gordon expansion method (SGEM) are employed to solve this GDSGE. The synergistic application of these techniques improves our knowledge of the GDSGE and provides a thorough foundation for investigating different evolution models arising in various branches of mathematics and physics. A positive aspect of the proposed methods is that they offer a wide variety of solitons, including bright, singular, dark, combination dark-singular, combined dark-bright, and periodic singular solitons. Obtained solutions demonstrate the method's high degree of reliability, simplicity, and functionalization for various nonlinear equations. To better describe the physical characterization of solutions, a few 2D and 3D visualizations are generated by taking precise values for parameters using mathematical software, Mathematica. [ABSTRACT FROM AUTHOR]

This paper explored the existence and uniqueness of a neutral fractional impulsive dynamic equation over time scales that included nonlocal initial conditions and employed the Caputo-nabla derivative (C∇D). The establishment of existence and uniqueness relies on the fine fixed point theorem. Furthermore, a comparison was conducted between the fractional order C∇D and the Riemann-Liouville nabla derivative (RL∇D) over time scales. Theoretical findings were substantiated through a numerical methodology, and an illustrative graph using MATLAB was presented for the provided example. [ABSTRACT FROM AUTHOR]

In this article, we considered the nonlinear time-fractional Jaulent–Miodek model (FJMM), which is applied to modeling many applications in basic sciences and engineering, especially physical phenomena such as plasma physics, fluid dynamics, electromagnetic waves in nonlinear media, and many other applications. The Caputo fractional derivative (CFD) was applied to express the fractional operator in the mathematical formalism of the FJMM. We implemented the modified generalized Mittag-Leffler method (MGMLFM) to show the analytical approximate solution of FJMM, which is represented by a set of coupled nonlinear fractional partial differential equations (FPDEs) with suitable initial conditions. The suggested method produced convergent series solutions with easily computable components. To demonstrate the accuracy and efficiency of the MGMLFM, a comparison was made between the solutions obtained by MGMLFM and the known exact solutions in some tables. Also, the absolute error was compared with the absolute error provided by some of the other famous methods found in the literature. Our findings confirmed that the presented method is easy, simple, reliable, competitive, and did not require complex calculations. Thus, it can be extensively applied to solve more linear and nonlinear FPDEs that have applications in various areas such as mathematics, engineering, and physics. [ABSTRACT FROM AUTHOR]

In this article by means of shifted knots properties, we introduce a new type of coupled Bernstein operators for Bézier basis functions. First, we construct the operators based on shifted knots properties of Bézier basis functions then investigate the Korovkin's theorem, establish a local approximation theorem, and provide a convergence theorem for Lipschitz continuous functions and Peetre's K-functional. In addition, we also obtain an asymptotic formula of the type Voronovskaja. [ABSTRACT FROM AUTHOR]

The outcomes of a nonlinear fractional neutral dynamic equation with initial conditions on time scales are examined in this work using the Riemann-Liouville nabla (▽) derivative. The existence, uniqueness, and stability results for the solution are examined by using standard fixed point techniques. For the result illustration, an example is given along with the graph using MATLAB. [ABSTRACT FROM AUTHOR]

Artificial neural networks (ANNs) have transformed machine learning and computational intelligence by providing unprecedented powers in modeling complicated data and addressing a wide range of challenges. In the field of ANNs, back propagation is a key approach for training neural networks. However, obtaining optimum network efficiency while tackling over fitting and controlling uncertainty is a difficult task. The present study employs the Bayesian Regularization Method with Neural Network Backpropagation (BRM-NNB) technique to investigate the rumors spreading delay model. With their rapid spread, rumors have the potential to cause fear and even financial loss. Thus, we must take decisive actions to stop the rumor from spreading. Nowadays, rumors can spread through instant messaging, emails, or publishing, thanks to the development of the internet. In this research, an XY-SIR rumors spreading delay model (XY-SIR-RS-DM) is investigated in relation to the novel spreading pattern. Media networks can be categorized into susceptible and infected media, while friendship networks can be categorized into three groups: spreaders (S, I, and R), who actively disseminate rumors, those who are ignorant and those who have no desire to do so. To estimate the solution of the suggested model, the Bayesian regularization method with neural network back propagation (BRM-NNB) is used. The data set is generated by applying the explicit Runge-Kutta method. The computing BRM-NNB strategy is implemented for three different performances, where the training, testing, and verification data are reported as 80%, 15%, and 5%, respectively, with 10 hidden neurons. To verify the validity of the developed artificial intelligence (AI) approach represented by the BRM-NNB, outcome comparisons are presented. The result is compatible with obtaining a minimal absolute error that is nearly equal to zero, thereby proving the efficacy of the proposed method. [ABSTRACT FROM AUTHOR]

Among women of reproductive age, PCOS (polycystic ovarian syndrome) is one of the most prevalent endocrine illnesses. In addition to decreasing female fertility, this condition raises the risk of cardiovascular disease, diabetes, dyslipidemia, obesity, psychiatric disorders and other illnesses. In this paper, we constructed a fractional order model for polycystic ovarian syndrome by using a novel approach with the memory effect of a fractional operator. The study population was divided into four groups for this reason: Women who are at risk for infertility, PCOS sufferers, infertile women receiving therapy (gonadotropin and clomiphene citrate), and improved infertile women. We derived the basic reproductive number, and by utilizing the Jacobian matrix and the Routh-Hurwitz stability criterion, it can be shown that the free and endemic equilibrium points are both locally stable. Using a two-step Lagrange polynomial, solutions were generated in the generalized form of the power law kernel in order to explore the influence of the fractional operator with numerical simulations, which shows the impact of the sickness on women due to the effect of different parameters involved. [ABSTRACT FROM AUTHOR]