The objective of this research is to evaluate the heat and mass transfer in a water-based Darcy–Forchheimer hybrid nanofluid (HNF) flow across an expanding cylinder. The fluid flow has been studied under the influence of a magnetic field, viscous dissipation, heat source, thermal radiation, concentration stratification, and chemical reaction. Carbon nanotubes (CNTs) and iron ferrite (Fe₃O₄) nanoparticles (NPs) are added to the water, for the purpose of synthesizing the HNF. The fluid flow has been induced in the presence of gyrotactic microorganisms and the non-Fick’s model. Microorganisms are used to stabilize scattered nanoparticles through the hybrid nanofluid. The phenomena have been modeled in the form of a nonlinear system of partial differential equations (PDEs). The modeled equations are reduced to a dimensionless system of ODEs by using similarity substitution. The numerical solution of the derived sets of nonlinear differential equations is obtained by using the parametric continuation method. The impact of physical constraints on temperature, velocity, concentration, and microorganism profiles is presented through figures and tables. It has been observed that the heat and mass transport rates increase with the rising effect of the curvature parameter, while declining with the effect of the thermal stratification parameter.

Recently, direct methods that involve higher derivatives to numerically approximate higher order initial value problems (IVPs) have been explored, which aim to construct numerical methods with higher order and very high precision of the solutions. This article aims to construct a fourth and fifth derivative, three-point implicit block method to tackle general third-order ordinary differential equations directly. As a consequence of the increase in order acquired via the implicit block method of higher derivatives, a significant improvement in efficiency has been observed. The new method is derived in a block mode to simultaneously evaluate the approximations at three points. The derivation of the new method can be easily implemented. We established the proposed method’s characteristics, including order, zero-stability, and convergence. Numerical experiments are used to confirm the superiority of the method. Applications to problems in physics and engineering are given to assess the significance of the method.

Fractional calculus generalizes well-known differentiation and integration to noninteger orders, allowing a more accurate framework for modeling complex dynamical behaviors. The application of fractional-order systems is quite wide in engineering, biology, and physics because they inherently capture the memory effects and long-range dependencies. Out of these, fractional jerk chaotic systems have gained attention regarding their applications in secure communication, signal processing, and control systems. This work develops a comparative analysis of a fractional jerk system that includes constant- and variable-order derivatives to contribute to chaos–stability analysis. Additionally, this study uncovers novel chaotic behaviors, further expanding our understanding of complex dynamical systems. The results yield new insights into using variable-order dynamics to enable chaotic systems to better adapt to real applications.

This paper proposes an implicit block method with two-point to directly solve the fourth-order Initial Value Problems (IVPs). The implicit block method is derived by adopting Hermite interpolating polynomial as the basis function, incorporating the first derivative offt,y,y′,y′′,y′′′to enhance the solution’s accuracy. A block formulation is presented to acquire the numerical approximation at two points simultaneously. The introduced method’s basic properties, including order, zero stability, and convergence, are presented. Numerical experiments are carried out to verify the accuracy and efficiency of the proposed method compared with those of the several existing methods. Application in ship dynamics is also presented which yield impressive results for the proposed two-point block method.