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.

In this chapter, we present a weakly nonlinear stability analysis of the flow of a nanofluid in a porous medium with stress-free boundary conditions. Some previous studies have investigated cross-diffusion in a nanofluid layer although in most cases these studies mostly deal with linear stability analysis. It is important to study the nonlinear stability in flows subject to cross-diffusion due to the wide range of applications where such flows arise such as in hydrothermal growth, compact heat exchanges, the solidification of binary mixtures, geophysical systems, solar pond, etc. Here we consider flow between parallel plates with an applied magnetic field and zero nanoparticle flux at the boundaries. A truncated Fourier series is introduced reducing the flow equations to a Lorenz-type system of nonlinear evolution equations. The multidomain spectral method is used to solve the equations that describe the growth of the convection amplitudes. The solutions are obtained as sets of trajectories in the phase space. Some interesting spiral trajectories and their sensitivity to the Rayleigh number are given.

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.