Recurrence formula for some higher order evolution equations

TL;DR

This paper introduces a recurrence formula that generates higher order evolution equations from first order equations using a logarithmic operator transform, expanding the understanding of differential equations in Banach spaces.

Contribution

It presents a novel recurrence formula linking first and higher order evolution equations via a logarithmic operator transform, generalizing Riccati's equation.

Findings

Derived a recurrence formula for higher order evolution equations.

Established a transform between first and higher order evolution equations.

Provided examples of classes of evolution equations with different orders.

Abstract

Riccati's differential equation is formulated as abstract equation in finite or infinite dimensional Banach spaces. Since the Riccati's differential equation with the Cole-Hopf transform shows a relation between the first order evolution equations and the second order evolution equations, its generalization suggests the existence of recurrence formula leading to a sequence of differential equations with different order. %%% In conclusion, by means of the logarithmic representation of operators, a transform between the first order evolution equations and the higher order evolution equation is presented. Several classes of evolution equations with different orders are given, and some of them are shown as examples.