A generalization of d'Alembert formula

TL;DR

This paper derives a closed-form solution for a generalized inhomogeneous linear differential equation involving commuting operators, extending the classical d'Alembert formula to a broader class of equations.

Contribution

It provides a new explicit solution formula for factored linear equations with commuting operators, without requiring the operators to be distinct.

Findings

Derived a closed-form solution for the generalized equation

Applicable to partial differential equations with commuting operators

Offers a computational method for explicit solutions

Abstract

In this paper we find a closed form of the solution for the factored inhomogeneous linear equation \begin{equation*} \prod_{j=1}^{n}(\frac{\hbox{d}}{\hbox{d}t}-A_{j}) u(t) =f(t). \end{equation*} Under the hypothesis $A_{1},A_{2}, ..., A_{n}$ are infinitesimal generators of mutually commuting strongly continuous semigroups of bounded linear operators on a Banach space $X$. Here we do not assume that $A_{j}$s are distinct and we offer the computational method to get explicit solutions of certain partial differential equations.