In this research, the numerical investigations of the fractional order immune diabetes regulation system by using a competent Bayesian regularization neural network procedure have been provided. The fractional order derivatives are used to get better results in comparison with the integer order. The division of the mathematical system is performed in resting and activated macrophages, and the antigen, autolytic, and beta cells. The data generalization is accessible by using the traditional Adam scheme in order to decrease the mean square error, while the data is separated into testing 16%, training 70%, and substantiation 14%. The designed neural network structure is updated by using the optimization tests through Bayesian regularization, a single layer sigmoid activation function, and twenty-five neurons. As conventional modeling schemes depend on shortening traditions or linear calculations, while the stochastic BRNN can perform complicated data patterns and deliver precise calculations of system performance. The correctness of the designed optimizer is obtained through the overlapping of the outcomes and lesser absolute error for each class of the model. Moreover, few curves based on state transitions, regression, error histograms provide the competences of the proposed solver.

The purpose of this work is to solve the fractional-order model of chaotic virotherapy dynamics by executing a neural network scheme. The chaotic virotherapy dynamics is divided into four categories: uninfected tumor cells, infected tumor cells, immune cells, and virus-free cells. The implementation of fractional derivatives permits the unification of memory properties and long-range dependencies in mathematical systems. The designed computing structure is performed by using a single layer and applying a radial basis activation function in the hidden layer with 12 neurons. Optimization is performed using the resilient backpropagation algorithm, which is one of the reliable schemes for the nonlinear models. The Adam optimizer is used to generate the dataset, which is divided into training as 77%, testing as 12%, and validation as 11%. The results of the fractional-order model of chaotic virotherapy dynamics are presented for solving three variations of the fractional-order values, while the accuracy of the results is validated through solution matching and best/optimal training values. In addition, the consistency of the designed solver is observed through the different states of transition, the regression coefficient, and the error histogram.

This article explores the chaotic dynamics of a hyperchaotic Lorenz system within the framework of fractional-order calculus, employing the Caputo derivative. The equilibrium point stability is analyzed using local stability theory, offering deeper insights into its dynamic behavior. Key analytical tools, such as phase portraits, time series, bifurcation diagrams, Lyapunov exponents, 0-1 tests, and Poincaré maps, are used further to investigate the dynamic properties of the new system. The system can produce periodic attractors of varying periods and chaotic attractors of diverse forms. A backstepping control strategy is proposed to achieve synchronization of a fractional-order hyperchaotic Lorenz system. Additionally, the effectiveness of the designed Radial Basis Function Neural Network (RBFNN) is validated through Root Mean Square Error (RMSE) metrics and extensive error analysis. This research introduces an innovative methodology integrating artificial intelligence to model and analyze fractional-order hyperchaotic dynamical systems, providing significant insights for future advancements in this field.

In this paper, we explore advanced generalizations of Jensen's inequality specifically for geometrically-arithmetic (GA)-convex functions. Our primary objective is to extend the classical results associated with convex functions to a broader category, thus contributing to the deeper understanding of GA-convexity. To achieve this, we introduce a new function, denoted by Phi, which is directly related to a given GA-convex function phi. This function Phi plays a pivotal role in deriving more intricate inequalities, particularly those that involve both Jensen's inequality and the Jensen-Mercer inequality. These inequalities provide new insights into the behavior of GA-convex functions, offering stronger and more generalized versions of the classical results. Moreover, we demonstrate the utility of these new inequalities by presenting their applications in the context of mean theory. In this setting, the generalized inequalities help to establish novel relationships among various types of means, enriching the existing body of knowledge in this domain. Our findings have the potential to advance research in mathematical inequalities, particularly in the study of convexity and its applications to different areas of mathematical analysis.

ABSTRACT A fractional order mathematical model of the saturated rate of incidence is solved numerically by applying a neural network scheme. The mathematical fractional order saturated rate of incidence system is classified into four classes of individuals: susceptible, infected, vaccinated, and recovered. The solution of the model is presented in three cases based on the fractional order values of the saturated rate of incidence system to authenticate better performance closer to 0 or 1. The solutions of the model are presented by designing a neural network construction containing Levenberg–Marquardt Backpropagation neural network, a single hidden layer with 10 neurons, and a sigmoid activation function. An Adam numerical scheme is used to get the dataset, which is divided into training, testing, and validation with the ratio of 70%, 20%, and 10%. The capability of the proposed neural network is perceived by the overlapping of the outcomes, small absolute error, and best training values. To validate the solver's reliability, different tests like regression, transition state, and error histogram have also been performed.

Purpose: The purpose of this work is to provide a reliable neural network process for the spreading virus in computers with kill signals. The mathematical model shows susceptible, exposed, infected individuals to form the virus inactive, and kill signals classes. Method: A structure of deep neural network (DNN) is designed by using two different hidden layers having radial basis activation functions in both layers, optimization through the Bayesian regularization, twenty and thirty numbers of neurons in primary and secondary hidden layers for the spreading virus in computers with kill signals. The stochastic DNN framework is presented to solve the spreading virus in computers with kill signals by selecting the data for training as 70 %, and 15 %, 15 % for both validation and testing. Results: The accuracy of the scheme is observed through the overlapping of the solutions along with negligible absolute error for solving the model. The consistency of the solver is observed through the process of error histogram, regression, and state transition. Novelty: The proposed DNN structure having radial basis activation function has never been applied for the spreading virus in computers with kill signals.

The purpose of this study is to find the numerical solutions of the delay Parkinson's disease model by employing a computing neural network framework. The disease model is divided into five components: healthy brain neurons, infected brain neurons, activated microglia cells, extracellular α-synuclein, and the activated T-cell population. A two-layered neural network scheme using radial basis functions in both hidden layers, and twelve and twenty neurons in layer-1 and layer-2 for solving the Parkinson's disease model. A dataset is obtained using the implicit Runge-Kutta method, which is trained by the Bayesian regularization taking reasonable percentages of training, testing and validation. The objective of this research is to minimize the mean square error using the proposed two-layered neural network structure. The absolute error values ranged from 10-07 to 10-09 confirming the accuracy of the designed technique. In addition, the optimal training between 10-11 to 10-13, along with error histograms, regression analysis, and state transition plots, validates the accuracy of the proposed scheme.

The present investigations provide the solutions to the human liver system by a neuro computing stochastic structure. The dynamics of the human liver system have two classes: blood and liver. The tests of optimization are performed by the Levenberg-Marquardt backpropagation neural network (LMBNN) structure using a single hidden layer, thirteen neurons and a log-sigmoid activation function for solving the human liver model. The reference database results are obtained through the Adam solver, which is used to lessen the mean square error in the sense of training, authorization, and testing. The correctness of the designed LMBNN procedure is presented through the assessment of reference and accomplished results, and best authentication achieved as 10− 11, 10− 13, and 10− 14, while the negligible absolute errors as 10− 06 to 10− 09 provide the authenticity of the designed LMBNN. The accuracy of the solver using different statistical analysis including regression performances 1, state transition, and error histogram is also presented.

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by α, these operators offer a unique flexibility: setting α=1 recovers the classical inequalities for harmonically convex functions, while setting α=0 yields inequalities for differentiable harmonically convex functions. This framework allows us to unify classical and fractional cases within a single operator. To validate the theoretical results, we provide several illustrative examples supported by graphical representations, marking the first use of such visualizations for inequalities derived via P.C.H operators. Additionally, we demonstrate practical applications of the results by deriving new fractional-order recurrence relations for the modified Bessel function of type-1, which are useful in mathematical modeling, engineering, and physics. The findings contribute to the growing body of research in fractional inequalities and harmonic convexity, paving the way for further exploration of generalized convexities and higher-order fractional operators.

In this paper, we give extensions of Jensen-Mercer inequality for functions whose derivatives in the absolute values are uniformly convex considering the class of k-fractional integral operators.

The study addresses the need for a comprehensive mathematical framework for multiplicative (geometric) P-convex functions, specifically in the context of Pcap (Proportional Caputo-Hybrid) operators. The research identifies a significant gap in existing literature concerning the formulation of specific inequalities and their applications to this class of functions. This study aims to fill the gap by developing and presenting new HrHd (Hermite-Hadamard) type inequalities tailored to multiplicative P-convex functions using Pcap operators. The lack of a detailed understanding and established results in this area underscores the importance of the study. The main contributions include the derivation of novel inequalities that extend the traditional concepts of convexity into a multiplicative framework. Additionally, a new Newton's type identity applicable to multiplicatively P-differentiable functions is introduced, which provides fresh insights and tools for analysis in this domain. The practical implications of the findings are demonstrated through applications to special means and type-1 modified Bessel functions. These applications not only validate the theoretical results, but also highlight their versatility and relevance in broader mathematical contexts. Research significantly advances the theoretical understanding of multiplicative P-convex functions, offering new analytical tools and frameworks. This advancement has potential implications for various mathematical and applied fields, including optimization and numerical analysis. The study suggests that future research could explore additional applications and extend the theoretical framework to other types of functions and operators, thereby broadening the scope and impact of the findings in this emerging area of study.

Aspects of both hybrid and fractional calculus are combined in the (Proportional Caputo-Hybrid) P-cap operators, which are helpful in solving differential equations with non-integer orders and modeling a variety of complicated phenomena in science and engineering. In this paper, we establish the P-cap operators via multiplicative calculus which are termed as multiplicative Pcap operators. we initially formulate two H.H (Hermite-Hadamard)-type inequalities applicable to multiplicative (geometric) convex function via multiplicative P-cap operators. Subsequently, by leveraging certain characteristics of multiplicative convex functions, we present novel inequalities related to multiplicative convex function via multiplicative P-cap operators also demonstrating two novel identities applicable to multiplicatively differentiable functions. By leveraging these identities, we then establish inequalities of trapezoid and midpoint types specifically designed for multiplicatively convex functions. Additionally, we explore applications of these findings to special functions and special means.

The aim of this work is to provide the numerical results of the typhoid fever disease system by applying an artificial neural network. The nonlinear typhoid fever disease system is considered into susceptible, exposed, infected, and recovered. The typhoid fever disease system is one of the nonlinear models and numerical results of the system are accomplished via stochastic computing scheme. The optimization is performed by using the Levenberg-Marquardt backpropagation (LMQBP) neural network for solving the nonlinear typhoid fever disease system. An explicit Runge-Kutta solver implemented to calculate the dataset, which is used to lessen the mean square error by data separating into testing (10

In this study, we employ proportional Caputo-Hybrid (PCH) operators to establish HermiteHadamard (HH) type inequalities for multiplicative harmonically convex functions. A key advantage of these fractional operators lies in their flexibility, allowing the recovery of various forms of inequalities. Specifically, traditional HH-type inequalities for multiplicative harmonically convex functions emerge when the parameter alpha(o) is set to 1, while for multiplicatively differentiable harmonic convex functions, they appear when alpha(o) = 0. To support our findings, we present graphical illustrations based on concrete examples. Additionally, we explore applications to special functions, leading to novel multiplicative fractional order recurrence relations. A promising avenue for future research involves extending these inequalities to interval calculus, where functions take interval values rather than precise numbers, broadening their applicability to uncertainty analysis, numerical approximations, and fractional differential equations.

The purpose of the current study is to design a feed forward Gudermannian neural networks for a singular Lane–Emden model based Neumann, Neumann-Robin and Dirichlet boundary conditions arising in numerous physical systems. The procedure based on the GNNs is exploited through the optimization of global and local search methods, i.e., genetic algorithm and active-set technique. A fitness function is constructed using the model and its boundary conditions, while the efficiency of the scheme is observed through the optimization with the hybridization of global and local search schemes. Four different nonlinear variants of the singular Lane–Emden model based Neumann, Neumann-Robin and Dirichlet boundary conditions have been numerically presented to validate the efficiency and accuracy of the model. The comparison of the obtained and exact results is used to verify the validity of the designed procedure. Moreover, different statistical measures have been implemented to certify the reliability of the proposed technique.

Aspects of both hybrid and fractional calculus are combined in the proportional Caputo-hybrid (PCH) operators, which are helpful in solving differential equations with non-integer orders and modeling a variety of complicated phenomena in science and engineering. In this paper, we establish the PCH operators via multiplicative calculus which are termed as multiplicative PCH operators. We initially formulate Bullen-type (Bln-type) identity applicable to multiplicative (geometric) convex function via multiplicative PCH operators. Subsequently, leveraging certain characteristics of multiplicative convex functions, we present novel inequalities related to multiplicative convex functions via multiplicative PCH operators. Additionally, we explore applications of these findings to special functions and special means.

This study presents a novel computational neural networks process for solving the prey-predator mathematical model (PPMM). The dynamical form of the PPMM has two species, prey and predator. The importance of this model is signified due to its oscillatory behavior based on the population of two species, as the population of prey enhances, the predator population reduces, and vice versa. A two-layered radial basis deep neural network is performed by using RB in both layers, 22 and 36 neurons in the hidden layer 1 and 2, while the optimization tests are performed through the Bayesian regularization. An implicit Runge-Kutta is used to obtain the reference data, which is divided into training as 82%, while 9% for both authentication and testing. The correctness of the designed radial basis deep neural networks process is approved through the matching of the outputs, best authentication values calculated as 10(-12) to 10(-13), and insignificant absolute error performances found around 10(-7) to 10(-8). The reliability of the radial basis deep neural network process is observed by using different measures based on regression and error histogram.

The goal of this conducted study is to provide the arithmetical performances through the stochastic computing procedure for the anthrax disease in animals (ADiA) model, which splits the populations between vaccinated, infected, susceptible, and recovered. A specific type of neural network, which is the novel radial basis is exploited by the radial basis and twenty-two neurons in the neural network’s hidden layer along with the optimization of Levenberg-Marquardt Backpropagation for solving the ADiA model. An Adam solver is generated to get the dataset and minimize the mean square error by dividing the data into testing as 14

This research’s goal is to demonstrate the numerical computing capabilities of the epidemic computer virus delay differential model (ECV-DDM) by relating the deep neural network along with a scale conjugate gradient scheme (DNNP-SCGS). The deep neural network process is implemented by taking 20 and 35 neurons and log-sigmoid transfer function in hidden layers. The mathematical form of the ECV-DDM is divided into uninfected S(u), latently infected L(u), breaking-out B(u), and the antivirus aptitude R(u). The framework based on stochastic computing is accessible for the ECV-DDM by using the data selection as 12