Probabilistic Evolution for the Most General First Order Single Unknown Explicit ODEs: Autonomization, Triangularization, and, Certain Important Aspects in the Analysis

摘要

Now it is well known that a basis set expansion depending on the unknowns serves us to get an infinite set of linear ODEs with an infinite coefficient matrix we called "Evolution Matrix", the infinite solution vector of which elementwisely gives the solution of the original ODE. The initial vector of this set of equation is basically power vector even though it does not have to be. Nonpower type initial vectors involves the probabilistic natures hence we give the name "Probabilistic Evolution Equation" to this infinite set of ODEs. Until know we assumed the autonomy in the descriptive functions to facilitate the analysis. This work focuses on the one unknown first order explicit ODEs in their most general, therefore nonautonomous, forms. Autonomization and the triangularization of the Evolution matrix are given. The essential issues like the generators which are responsible for the entire dynamics are also emphasized.