The Chronological Operator Algebra and Formal Solutions of Differential Equations

The aim of this paper is twofold. First, we obtain the explicit exact formal solutions of differential equations of different types in the form with Dyson chronological operator exponents. This allows us to deal directly with the solutions to the equations rather than the equations themselves. Second, we consider in detail the algebraic properties of chronological operators, yielding an extensive family of operator identities. The main advantage of the approach is to handle the formal solutions at least as well as ordinary functions. We examine from a general standpoint linear and non-linear ODEs of any order, systems of ODEs, linear operator ODEs, linear PDEs and systems of linear PDEs for one unknown function. The methods and techniques involved are demonstrated on examples from important differential equations of mathematical physics.