Computational Integrated-Block-Based Simulation Developed to Approximate Solutions of Multi-Degree Regular Differential Equations

This paper introduces a novel unified numerical approach to obtain explicit approximations of initial value formulations (IVFs) encompassing ODEs ranging from first- to third-order. The proposed technique leverages Chebyshev polynomials as basic functions and is developed using continuous schemes formulated through both collocation and interpolation strategies. It operates on a block-by-block basis, providing an efficient framework for numerically solving ODEs of multiple orders. The convergence properties of this method are thoroughly examined through the lens of zero-stability and consistency. In-depth discussions unfold, shedding light on the efficacy of this approach in addressing first, second, and third-order ODEs. Through comparative analyses against existing methods, it is distinctly evident that the proposed model surpasses its counterparts in terms of accuracy, marking a significant advancement in the numerical treatment of IVPs. This model not only introduces a unified approach for diverse ODE orders but also stands as a testament to its superior performance, establishing itself as a noteworthy contribution to the realm of numerical integration methodologies.