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 paper presents a one‐step multiderivative hybrid block method of Order 12 that incorporates an overlapping strategy, in which intrastep points from the previous block are reused in the current step to enhance accuracy and stability when solving linear and nonlinear initial value problems. The derivation incorporates a multistep collocation and interpolation technique, using power series as the basis function for the approximate solution. Within a one‐step block, three intrastep points are considered. As a foundational step, a non‐overlapping one‐step multiderivative scheme is first developed and expressed in matrix form. The overlapping aspect of the method is then introduced by incorporating the second‐to‐last intrastep point of the previous step into each integrating block. The accuracy, consistency, and stability properties of the method are analyzed. The features of the method are determined through an error analysis of the numerical solutions of linear and nonlinear second‐order initial value problems. The nonlinear initial value problems are converted into linear ones using a modified Picard iteration technique. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. The results are evaluated against other methods from the literature.

A Multi-Domain Spectral Collocation Approach for Solving Lane-Emden Type Equations | InTechOpen, Published on: 2016-08-24. Authors: Motsa Sandile Sydney, Magagula Vusi Mpendulo, Goqo Sicelo Praisegod, et

In this study, we present a numerical method for solving two--dimensional space--time fractional partial differential equations (FPDEs), where the solutions of the FPDEs are expanded in terms of the shifted Chebyshev polynomials. The numerical approximations are evaluated at the Chebyshev-Gauss-Lobatto points. In the case when the FPDE is nonlinear, we employ a Newton-Raphson approach to linearize the equation. Both the linear and nonlinear cases lead to a consistent system of linear algebraic equations. The scheme is tested on selected FPDEs and the numerical results show that the proposed numerical scheme is accurate and computationally efficient in terms of CPU times. To establish the accuracy of the method, we also present an error analysis which shows the convergence of the numerical method. These positive attributes make the proposed method a good approach for solving two--dimensional fractional partial differential equations with both space and time fractional orders.

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.