In this paper, we extend the block hybrid method with equally spaced intra-step points to solve linear and nonlinear third-order initial value problems. The proposed block hybrid method uses a simple iteration scheme to linearize the equations. Numerical experimentation demonstrates that equally spaced grid points for the block hybrid method enhance its speed of convergence and accuracy compared to other conventional block hybrid methods in the literature. This improvement is attributed to the linearization process, which avoids the use of derivatives. Further, the block hybrid method is consistent, stable, and gives rapid convergence to the solutions. We show that the simple iteration method, when combined with the block hybrid method, exhibits impressive convergence characteristics while preserving computational efficiency. In this study, we also implement the proposed method to solve the nonlinear Jerk equation, producing comparable results with other methods used in the literature.

This article presents a study of a rotating hybrid nanofluid flow in a porous medium past a stretching surface using an overlapping grid multi-domain bivariate spectral simple iteration method (OMD-BSSIM). The objective is to appraise the performance of the OMD-BSSIM on a system of nonlinear partial differential equations that describe the unsteady flow of a hybrid nanofluid composed of copper and aluminum oxide nanoparticles in water. Additionally, the study aims to evaluate the impact of different fluid and surface parameters on the behavior of the hybrid nanofluid. The improved OMD-BSSIM algorithm is implemented in MATLAB and tested for its convergence and accuracy. The results are compared with previously published findings that were obtained using the overlapping spectral local linearization method and the bivariate spectral relaxation method. The study explores the use of R line graphs as a visualization tool to identify patterns and relationships among the skin friction coefficients, local Nusselt number, and local Sherwood number of the hybrid nanofluid and parameters over time. The findings reveal several key trends and patterns, such as the positive correlation between the shrinking ratio parameter and the skin friction coefficient and the positive relationship between increasing the rotation parameter and an inverse relationship between an increase in the alumina volume fraction coefficient and the local Nusselt number of the hybrid nanofluid. This article provides valuable insights into the flow behavior of hybrid nanofluids and highlights the potential of alumina nanoparticles as a tool for enhancing the thermal and mechanical properties of the fluid.

This paper investigates the flow of a second-grade hybrid nanofluid through a Darcy–Forchheimer porous medium under Cattaneo–Christov heat and mass flux models. The hybrid nanofluid, composed of alumina and copper nanoparticles in water, enhances thermal and mass transport, while the second-grade model captures viscoelastic effects, and the Darcy–Forchheimer medium accounts for both linear and nonlinear drag. Using similarity transformations and the spectral quasilinearisation method, the nonlinear governing equations are solved numerically and validated against benchmark results. The results show that hybrid nanoparticles significantly boost heat and mass transfer, while Cattaneo–Christov fluxes delay thermal and concentration responses, reducing the near-wall temperature and concentration. The distributions of velocity, temperature, concentration, and microorganism density are markedly affected by porosity, the Forchheimer number, the bio-convection Peclet number, and relaxation times. The results illustrate that hybrid nanoparticles significantly increase heat and mass transfer, whereas thermal and concentration relaxation factors delay energy and species diffusion, thickening the associated boundary layers. Viscoelasticity, porous medium resistance, Forchheimer drag, and bio-convection all have an influence on flow velocity and transfer rates, highlighting the subtle link between these mechanisms. These breakthroughs may be beneficial in establishing and enhancing bioreactors, microbial fuel cells, geothermal systems, and other applications that need hybrid nanofluids and non-Fourier/non-Fickian transport.

The Fitzhugh–Nagumo equation, a key model for excitable systems in biology and neuroscience, requires efficient numerical methods due to its nonlinear nature. A spectral optimized multiderivative hybrid block method is proposed, constructed using a multistep collocation and interpolation technique with an approximated power series as the basis function. Incorporating two optimal intra‐step points, the method demonstrates improved accuracy, with its consistency, convergence, and absolute stability rigorously analyzed. By combining the optimized multiderivative hybrid block method in time with a spectral collocation method in space, the approach demonstrates potency and flexibility in solving partial differential equations. Prior to using the spectral method, the partial differential equation is linearized using a linear partition technique. Numerical experiments confirm the accuracy and efficiency of the method compared to existing methods, demonstrating the potential of the method as a robust framework for solving partial differential equations requiring both high accuracy and stability.