An explicit Maclaurin series solution to non-autonomous and non-homogeneous evolution equation, Omega Calculus, and associated applications

We give a new Omega Calculus (a.k.a MacMahon’s Partition Analysis) based integral-free representation for the solution of a non-autonomous and non-homogeneous evolution equation. Our new representation generalizes some of the main results of the recent work of Francisco Neto (2024, A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus. Ann. Inst. Henri Poincaré D, 11, 383) and Bassom et al. (2023, An explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation. Appl. Math. Lett., 139, 108537) and show that we can indeed compute the coefficients of the Maclaurin series solution associated with the evolution equation starting with the Peano-Baker series. Furthermore, we discuss in the context of our framework the inverse problem for homogeneous evolution equations in a Hilbert space answering a question left open by Bassom et al. in this case; that is, assuming the solution of the homogeneous evolution equation is a known analytic function the problem concerns the determination of the associated generator of the dynamics. Finally, in order to illustrate the versatility of our approach we explicitly determine the Maclaurin series solution related to the power series method in the context of the vibration problems for the non-uniform (tapered) Euler-Bernoulli beam and thus we explicitly solve the recursion relations considered by Adair and Jaeger (2018, A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams. J. Vib. Control, 24, 3855-3864).