Nonlinear Analysis as a Calculus

Abstract

In the last 60 years, there was formed a universal nonlinear analysis, whose unified algorithms allow to find asymptotic forms and asymptotic expansions of solutions to nonlinear equations and systems of different types: algebraic, ordinary differential (ODE), partial differential (PDE) and systems of mixed-type equations. This calculus contains two main algorithms: (a) Reducing equations to the normal form and (b) Separating truncated equations, and two kinds of transformations of coordinate can be used to simplify the obtained equations: (A) Power and (B) Logarithmic. Here we show that for algebraic equation, single ODE, autonomous system of ODE’s, Hamiltonian system, single PDE. Some applications are mentioned as well.