An explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation

TL;DR

This paper introduces an explicit Maclaurin series method to solve non-autonomous abstract evolution equations for bounded linear operators on Banach spaces, extending solutions beyond scalar cases.

Contribution

It provides a novel explicit Maclaurin series solution for non-autonomous evolution equations in operator theory, which was previously unavailable.

Findings

Derived the Maclaurin series solution for operator evolution equations.

Validated the solution's convergence and applicability.

Extended scalar solution concepts to operator equations.

Abstract

It is well known that the non-autonomous scalar differential equation of evolution has a unique solution given by an elementary exponential function. In general there is no such analogous solution to the corresponding non-autonomous evolution equation for square matrices. In this paper we propose and justify an explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation for bounded linear operators on Banach space.