An Operator Approach to the Integration of Linear Differential Equations

TL;DR

This paper introduces an operator-based method for solving linear differential equations using intertwining relations, simplifying the problem to Riccati equations in some cases, and applies it to partial differential equations like the Klein-Gordon equation.

Contribution

It presents a novel operator approach utilizing intertwining relations for integrating linear differential equations, including partial differential equations, with conditions for existence and solution construction.

Findings

Conditions for intertwining operators are established.

Method reduces to Riccati equations in low-order cases.

Successfully applied to the Klein-Gordon equation.

Abstract

We develop an operator approach to the integration of linear differential equations based on intertwining relations between differential operators. Conditions for the existence of intertwining operators are obtained, and it is shown that, in low-order cases, the problem reduces to Riccati-type equations. The method is applied to linear partial differential equations, which makes it possible to construct their solutions. The linear Klein--Gordon equation is presented as an illustrative example.