A generalization of d'Alembert formula

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.