This article introduces an approximation method for solving singularly perturbed differential equations with a large positive shift parameter, a problem that has received limited attention in existing research. A key challenge in this context is to estimate the solution in [ 0 , 2 ] by using a function in the range [ 2 , 3 ] ) . The proposed numerical scheme is implemented using the Gauss elimination method in MATLAB R2022a and exhibits a linear convergence rate on a uniform grid. Both theoretical and numerical analyses confirm the scheme's accuracy and reliability. To demonstrate its effectiveness, several examples are presented in tables and charts, validating the theoretical results. Compared to existing approaches, our method delivers improved accuracy with significantly lower error rates for the tested examples.

A new class of Bazilevič functions of the type κ + iμ is defined involving differential characterizations inspired by the study of multiplicative calculus. The defined function class unifies the studies of well-known subclasses like convex, starlike and alpha-convex functions. To further make our study more versatile, we have studied the new function class involving a differential operator defined using the Mittag-Leffler function. Estimates of Taylor–Maclaurin coefficients a 2 and a 3 of the functions which belong to the defined function class are obtained. Further, the Fekete–Szeg˝o inequality of this new class which was computationally cumbersome is part of our main results in this paper.

In recent years, both degenerate versions and probabilistic extensions of many special numbers and polynomials have been explored. For instance, degenerate Bernstein polynomials and probabilistic Bernstein polynomials were investigated earlier. Assume that Y is a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to study probabilistic degenerate Bernstein polynomials associated with Y which are both probabilistic extension of the degenerate Bernstein polynomials and degenerate version of the probabilistic Bernstein polynomials associated with Y. We derive several explicit expressions and certain related identities for those polynomials. In addition, we treat the special cases of the Poisson random variable, the Bernoulli random variable and of the binomial random variable.

ABSTRACT The objective of the presented study is to develop a neuro-evaluation-based algorithm for the mathematical solution of the SEIRC model that describes the dynamics of campylobacteriosis transmission (CBT) using the artificial neural network along with log-sigmoid as an activation function (ANN-LS) and the properties of both genetic algorithms and sequential quadratic programming (GASQP). Five groups compose the mathematical SEIRC framework for the CBT model: susceptible (S), exposed (E), infected (I), recovered (R), and the bacterial population (C). The objective function is expressed in the mean squared error sense in the proposed ANN-LS-GASQP approach employing the approximate differential mapping of ANN for the SEIRC model. To evaluate the reliability, accuracy, and consistency of the proposed methodology, the designed intelligent solver ANN-LS-GASQP is compared with the state of the art Adam numerical solver. The solver has successfully achieved the absolute error in excellent ranges, which confirms the effectiveness and reliability of the solver ANN-LS-GASQP. The effectiveness of the suggested stochastic approach is examined using 100 trials with 5 neurons each, along with extra statistical indicators.

This study focuses on impulsive Volterra-Fredholm integro-differential equations (VFIDEs) under specific order conditions, conducting a rigorous analysis involving fractional Caputo derivatives. By applying the Schauder fixed-point theorem, we prove the existence of solutions and examine the interaction between fractional Caputo derivatives and the equation's integro-differential structure. Furthermore, we explore the application of artificial neural networks (ANNs) to predict system states based on input features, demonstrating how these techniques enhance the understanding of control inputs and system responses. These findings improve the theoretical understanding of impulsive VFIDEs and demonstrate their applications in science and engineering, particularly regarding existence conditions and controllability.

Obesity is a risk factor for several chronic diseases, including type 2 diabetes, heart disease, and high blood pressure. Developing models of obesity and its complications allows for close monitoring of these epidemics, informing the creation of effective control strategies. This research focuses on presenting an endemic model of obesity and its health complications, taking into account, social factors and relationships among people in society. The term epidemic is related to the influence of the health lifestyle of obese people on healthy individuals. The transmission of obesity and its complications is through peer pressure to consume fast food and acquire bad eating habits from relatives and friends. So, in our model, the effect of these factors is expressed with parameters β i . We demonstrate theoretically the asymptotic local stability of disease-free equilibrium and the asymptotic local stability of endemic equilibrium, proving the existence and uniqueness of the solution to the obesity model and its complications. To confirm the model's theoretical results, we conduct numerical simulations to study the impact of variation in social factor parameters on the diseases studied.

ABSTRACT Recently, the monomiality principle has been extended to q-polynomials, namely, the q-monomiality principle of q-Appell polynomials has been considered. Also, certain results including the monomiality properties of the q-Gould-Hopper polynomials are derived and applications of monomiality are explored for a few members of the q-Appell polynomial families. In this paper, the primary purpose of this paper is to define 2-variable q-Laguerre–Appell polynomials by applying the q-monomiality principle techniques and to study their quasi-monomial properties and applications. We provide some operational identities and quasi-monomial features. Also, we derive some q-differential equations of these polynomials. As applications, using the operational identity of 2-variable q-Laguerre–Appell polynomials we draw specific conclusions regarding several q-Laguerre–Appell polynomial families. Furthermore, we define the family of q-Laguerre-Sheffer polynomials by an operational approach and give some of its fundamental properties.

In this paper, we analyse the unique characteristics of hyperparasites and discharge incidents in the initial stage of parasite populations. If hyperparasites are effectively developed in a population, they can play a crucial role in controlling host–parasite interactions. First, we expanded the integer-order host–parasite–hyperparasite model by using different fractional-order operators. We then examined the uniqueness and existence of the solutions of the non-integer order host–parasite–hyperparasite model. We inspected the stability of the model using the Ulam–Hyers stability theorem and employed a numerical technique to find a solution to the host–parasite–hyperparasite model using the fractional operator. Finally, we discussed the graphical results with the help of the Toufik–Atangana (T–A) numerical scheme. To examine the host–parasite–hyperparasite models and approximate results, we employed the Caputo fractional operator and the Atangana–Baleanu (A–B) fractional operator. The numerical results demonstrate that every important aspect of the model is preserved by the anticipated numerical method, which also provides an understandable solution for the fractional problem. The findings of this study bear significant implications for the theory and methodology of biological control of plant diseases associated with hyperparasites. It sheds light on the ecology of biological invasions, which is a subject of great importance in today's world. This research provides valuable insights that can design effective strategies to combat plant diseases and prevent the spread of invasive species.

The subdivision scheme is a valuable tool for designing shapes and representing geometry in computer-aided geometric design. It has excellent geometric properties, such as fractals and adjustable shape. In this research paper, we explore the generation of fractal curves using a novel binary 8-point interpolatory subdivision scheme with two parameters. We analyse different properties of the proposed scheme, including convergence, special cases, and fractals. Additionally, we demonstrate through various examples the relationship between the shape parameters and the fractal behaviour of the resulting curve. Our research also identifies a specific range of shape parameters that can effectively produce fractal curves. The findings of this study provide a fast and efficient method for generating fractals, as demonstrated by numerous examples. Modelling examples show that the 8-point interpolatory scheme can enhance the efficiency of computer design for complex models.