In this paper, we investigate a (discretization) Caputo fractional glucose-insulin model qualitatively with incommensurate orders that appear in Bergman's minimal model. After intravenous tolerance testing, the model is used to characterize the blood insulin and glucose metabolism. We also prove that the presented model possesses existence, uniqueness, non-negative, and boundedness solution. We also proceed a systematical studies on the stability of the (discretization) Caputo fractional. Comparisons between the results of the fractional-order, the integer order and the measured real data obtained from patients are presented. These comparisons is shown that the presented Caputo fractional order model is better representative of the system than its integer order form. Numerical solutions of the Caputo fractional model are obtained by using the method of Adams-Bashforth-Moulton type to handle the fractional derivatives. Also, numerical simulations of the discretization fractional derivative order model are used to support the analytical results.

The purpose of this article is to formulate a simplified nonlinear fractional mathematical model to illustrate the dynamics of the new coronavirus (COVID-19). Based on the infectious characteristics of COVID-19, the population is divided into five compartments: susceptible S(t), asymptomatic infection I(t), unreported symptomatic infection U(t), reported symptomatic infections W(T) and recovered R(t), collectively referred to as (SIUWR). The existence, uniqueness, boundedness, and non-negativeness of the proposed model solution are established. In addition, the basic reproduction number R ₀ is calculated. All possible equilibrium points of the model are examined and their local and global stability under specific conditions is discussed. The disease-free equilibrium point is locally asymptotically stable for R ₀ leq1 and unstable for R ₀ > 1. In addition, the endemic equilibrium point is locally asymptotically stable with respect to R ₀ > 1. Perform numerical simulations using the Adams–Bashforth–Moulton-type fractional predictor–corrector PECE method to validate the analysis results and understand the effect of parameter variation on the spread of COVID-19. For numerical simulations, the behavior of the approximate solution is displayed in the form of graphs of various fractional orders. Finally, a brief conclusion about simulation on how to model transmission dynamics in social work.

This study investigates the complex dynamics and control mechanisms of fractional-order glucose–insulin regulatory systems, incorporating memory-dependent properties through fractional derivatives. Employing the Laplace–Adomian Decomposition Method (LADM) and the Generalized Euler Method (GEM), the research models glucose–insulin interactions with time-varying fractional orders to simulate long-term physiological processes. Key aspects include the derivation of Lyapunov exponents, bifurcation diagrams, and phase diagrams to explore system stability and chaotic behavior. A novel control strategy using simple linear controllers is introduced to stabilize chaotic oscillations. The effectiveness of this approach is validated through numerical simulations, where Lyapunov exponents are reduced from positive values (${\lambda }_1=0.123$) in the uncontrolled system to negative values (${\lambda }_1=-0.045$) post-control application, indicating successful stabilization. Additionally, bifurcation analysis demonstrates a shift from chaotic to periodic behavior when control is applied, and time-series plots confirm a significant reduction in glucose–insulin fluctuations. These findings underscore the importance of fractional calculus in accurately modeling nonlinear and memory-dependent glucose–insulin dynamics, paving the way for improved predictive models and therapeutic strategies. The proposed framework provides a foundation for personalized diabetes management, real-time glucose monitoring, and intelligent insulin delivery systems.

All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The Newton–Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software.

In this study, we construct a family of single root finding method of optimal order four and then generalize this family for estimating of all roots of non-linear equation simultaneously. Convergence analysis proves that the local order of convergence is four in case of single root finding iterative method and six for simultaneous determination of all roots of non-linear equation. Some non-linear equations are taken from physics, chemistry and engineering to present the performance and efficiency of the newly constructed method. Some real world applications are taken from fluid mechanics, i.e., fluid permeability in biogels and biomedical engineering which includes blood Rheology-Model which as an intermediate result give some nonlinear equations. These non-linear equations are then solved using newly developed simultaneous iterative schemes. Newly developed simultaneous iterative schemes reach to exact values on initial guessed values within given tolerance, using very less number of function evaluations in each step. Local convergence order of single root finding method is computed using CAS-Maple. Local computational order of convergence, CPU-time, absolute residuals errors are calculated to elaborate the efficiency, robustness and authentication of the iterative simultaneous method in its domain.

Fractional differential equations have recently gained popularity due to their ability to simulate a wide range of complex processes in various fields, including engineering, physics, biology, and finance. These equations provide a powerful framework for describing phenomena with memory effects and hereditary features that standard integer-order models cannot account for. In this study, we present fractional versions of numerical algorithms specifically designed for solving fractional-order differential equations. We thoroughly investigate the proposed approaches for stability under various fractional parameter values and compare their stability performance with existing methods. The schemes’ consistency and local truncation error are calculated to ensure their accuracy. In terms of stability surface, our methods have a larger stability zone than existing fractional schemes. Two engineering applications are addressed utilizing both fixed and adaptive step-length algorithms to assess efficiency. In both cases, our methods outperform existing approaches, as evidenced by less local and global errors, reduced CPU time, and fewer function and derivative evaluations. Our newly developed fractional order technique outperforms modern high-performance algorithms in solving fractional differential equations, demonstrating superior computational efficiency and stability. These findings demonstrate the robust and efficient capabilities of the proposed methods to solve fractional-order problems.