Solutions of Nonlinear Differential Equations

TL;DR

This paper investigates solutions to nonlinear differential equations within a new algebra of generalized functions, establishing conditions for when these solutions relate to ordinary functions and describing all such associated functions.

Contribution

It introduces a framework for solving nonlinear differential equations in a new algebra of generalized functions and characterizes when solutions are associated with ordinary functions.

Findings

Conditions for solutions to be associated with ordinary functions

Complete description of all possible associated functions

Extension of differential equation solutions in a new algebraic setting

Abstract

We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the article we formulate necessary and sufficient conditions when the solution of the given equation in algebra of new generalized functions is associated with an ordinary function. Moreover the class of all possible associated functions was described.