An algebraic approach to nonlinear functional expansions

A new theory of functional expansion is presented which makes use of formal power series in several noncommutative variables and of iterated integrals. A simple closed-form expression for the solution of a nonlinear differential equation with forcing terms is derived, which enables us to give the corresponding Volterra kernels with utmost precision. The noncommutative variables give birth to a symbolic calculus which generalizes in a nonlinear setting many features of the Laplace and Fourier transforms and which is developed in order to simplify some computations, like the so-called association of variables, related to high-order transfer functions.