Nested Computation Framework of Generalized Mapping and Its Applications

This paper innovatively proposes a nested computation framework for generalized mapping, aiming to provide a unified and efficient descriptive approach for ordinary differential equations, partial differential equations, stochastic differential equations, and stochastic integral equations. By nesting the two forms of generalized mapping (ordinary form and probabilistic form) in multiple layers, this framework constructs a powerful mathematical model. First, it reviews the basic forms, tensor extension, and iterative extension of generalized mapping, then introduces the nested form and elaborates on its construction logic in detail. Through classic examples of differential equations and stochastic equations, the application of this framework in different types of equations is demonstrated. The research shows that the nested computation framework of generalized mapping not only simplifies the expression of complex equations but also provides a new perspective and method for solving related mathematical and physical problems. The theory of generalized mapping is still in its infancy, and there is much in-depth research work to be carried out in the future. This study is expected to inspire new research directions and application explorations in various fields such as mathematics, physics, and engineering.