Generalized solutions of algebraic differential equations
Two related notions are introduced of a function or distribution u being a ''solution'' of certain non-linear ordinary or partial differential equations. They are the notion of N-solution and IN-solution. The driving idea of the paper is that a certain u ''should be'' a solution except only that it is not differentiable enough to plug into the equation. But by using Ritt's differential Nullstellensatz in a new way, the lack of sufficient differentiability is circumvented. There are two main advantages of this procedure. First, it gives an intriguing new context in which a Schwartz distribution may be a solution of a non-linear equation, despite the impossibility of a completely satisfactory multiplication of distributions. Next, it gives a new context in which certain classical theorems of differential algebra (like the Ritt-\(Raudenbush\) basis theorem) are valid, despite the absence of higher derivatives. Several examples are worked out, and open problems are described.
scientific article; zbMATH DE number 3924934