The study aims to clarify and understand the importance of the algebraic properties of transformation symbols in developing more efficient methods for analyzing differential systems, and to analyze these properties mathematically. The study also aims to clarify future studies on the application of these concepts to multi-input, multi-output, and complex nonlinear systems. This is achieved through an applied methodology based on developing a mathematical model that links the algebraic structure of transformation symbols with the solution of differential systems. In addition, a descriptive methodology is used to describe data, transformations, and symbols, an analytical methodology is used to analyze the algebraic properties of the proposed model, and a comparative methodology is used to compare the results. The algebraic results indicate that the transformation process facilitates the symbols and reduces computational complexity, with a focus on integrating abstract algebra and mathematical analysis to provide more accurate and efficient solutions. Furthermore, analyzing the algebraic properties of these symbols mathematically provides more accurate and efficient solutions and opens new horizons for developing algorithms that can rely on these properties to solve complex problems in abstract algebra.
Abdulsalalam et al. (Mon,) studied this question.