On Solving the Cauchy Problem with Propagators

TL;DR

This paper introduces a method to solve the first order Cauchy problem using Taylor series and propagators, extending the approach to higher order problems by expressing derivatives in terms of lower order ones.

Contribution

It presents a novel operator-based approach to solving higher order Cauchy problems through Taylor series and propagators, simplifying the process.

Findings

Solution of first order Cauchy problem via propagators

Extension of method to higher order problems

Simplification by expressing derivatives in terms of lower order derivatives

Abstract

The abstract first order Cauchy problem is solved in terms of Taylor's series leading to a series of operators which is a propagator. It is found that higher order Cauchy problems can be solved in the same way. Since derivatives of order lower than the Cauchy problem are built-in to the problem, it suffices to solve the higher order derivatives in terms of the lower order ones, in a simple way.