Practical Applied Mathematics for Scientific Research: Application of ACP Mathematical Methodology in Analyzing Algebraic Functions and Physical Experimental Data (Applications 11 and 12)

We introduce the ACP (asymptotic curve-based and proportionality-based) mathematical methodology to analyze the face, shape, and proportionality of several algebraic forms, including power and inverse functions as first-order nonlinear phenomena, and sigmoidal curves in physical experiments as second-order nonlinear phenomena. The goal is to express both types of nonlinear behavior using simple, straight-line-oriented proportionality graphs supported by appropriate nonlinear equations. In Part I, we examine first-order nonlinear phenomena using algebraic power and inverse functions and demonstrate the need for a combined linear and nonlinear logarithmic graph to obtain a complete representation. In Part II, we model fluidized-bed experimental data exhibiting various sigmoidal profiles and show that a second-order nonlinear equation can represent the full range of S- and C-shaped curves. The resulting formulation yields a concise straight-line graph and a unified nonlinear rate equation that describes the two-variable relationship.