# Computational Analysis of Parallel Techniques for Nonlinear Biomedical Engineering Problems
## 1 Introduction  
Nonlinear equations have been used for practical problem-solving from the time of ancient civilizations such as Babylon and Egypt [1]. In ancient mathematics, they were fundamental to astronomy, land measuring, building, and the computation of areas, volumes, and geometric difficulties [2; 3]. Methods for dealing with increasingly complex nonlinear equations arose with contributions from Greek and Arab mathematicians [4], resulting in advances in algebra [5; 6]. In many branches of science and engineering today, nonlinear equations are still essential. They are employed in mechanical and biomedical engineering to simulate biological systems [7], including pharmacokinetics and enzyme kinetics [8], which forecast medication concentration over time [9]. Nonlinear models also help with medical imaging, specifically reconstruction methods used in CT and MRI scans [10]. For effective manufacturing, polynomials are used in chemical engineering to model material attributes, improve process conditions, and define reaction rates. In order to ensure structural integrity and performance efficiency, nonlinear equations in mechanical engineering help define stress-strain relationships, characterize aerodynamic features, and support mechanical design optimization [11; 12]. Thus, nonlinear equations nevertheless serve as a link between cutting-edge scientific and engineering applications and old mathematical truths. Nonlinear equations are important because they may be used to describe and predict complex systems precisely, which can lead to important advancements in a variety of engineering and scientific fields [13]. Recent advances in computing technologies, as well as the increasing complexity of current engineering issues, have highlighted the importance of efficient and accurate solutions to nonlinear equations. Designing cutting-edge technology in the fields of renewable energy, aerospace, and telecommunications requires these solutions. Nonlinear equations are significant in biomedical engineering [14] because they represent complex systems and biological processes that are intrinsically nonlinear. They are crucial in modeling physiological dynamics such as blood flow in arteries [15], oxygen diffusion in tissues [16], and electrical activity in the heart and brain [17]. These equations are fundamental to the design of medical devices such as pacemakers, artificial organs, and prosthetics, guaranteeing that their functionalities are compatible with natural systems. Nonlinear models aid in the simulation of drug distribution and pharmacokinetics, allowing therapies to be optimized. They also simplify the study of biomechanics subjects including joint movement and tissue deformation [18]. Nonlinearities in biomedical engineering also replicate sickness progression, resulting in improvements in specific treatment and early detection. They are utilized in systems biology, which investigates the interactions between biological networks [19] and bioinformatics [20] using nonlinear equations to better data interpretation and simulations. Despite its importance, solving nonlinear equations remains difficult due to their intrinsic complexity and extensive processing resources required for advanced numerical iterative methods to obtain approximate solutions; see, e.g., [21; 22; 23]. The underlying non-locality of these models, in which the derivative at a location is determined by the complete history of the function, makes them infamously difficult to solve both analytically and numerically. The main contribution of this research work is the development and analysis of parallel schemes designed specifically for solving nonlinear equations in diverse biomedical engineering applications. Indeed, these strategies have been carefully tested against conventional methods for computational efficiency, convergence rate, and even CPU time.