Mathematics – Classical Analysis and ODEs
Scientific paper
2006-12-30
Int.J.Geom.Meth.Mod.Phys.4:577-646,2007
Mathematics
Classical Analysis and ODEs
International Journal of Geometric Methods in Modern Physics, in press
Scientific paper
10.1142/S0219887807002211
The theory of exact and of approximate solutions for non-autonomous linear differential equations forms a wide field with strong ties to physics and applied problems. This paper is meant as a stepping stone for an exploration of this long-established theme, through the tinted glasses of a (Hopf and Rota-Baxter) algebraic point of view. By reviewing, reformulating and strengthening known results, we give evidence for the claim that the use of Hopf algebra allows for a refined analysis of differential equations. We revisit the renowned Campbell-Baker-Hausdorff-Dynkin formula by the modern approach involving Lie idempotents. Approximate solutions to differential equations involve, on the one hand, series of iterated integrals solving the corresponding integral equations; on the other hand, exponential solutions. Equating those solutions yields identities among products of iterated Riemann integrals. Now, the Riemann integral satisfies the integration-by-parts rule with the Leibniz rule for derivations as its partner; and skewderivations generalize derivations. Thus we seek an algebraic theory of integration, with the Rota-Baxter relation replacing the classical rule. The methods to deal with noncommutativity are especially highlighted. We find new identities, allowing for an extensive embedding of Dyson-Chen series of time- or path-ordered products (of generalized integration operators); of the corresponding Magnus expansion; and of their relations, into the unified algebraic setting of Rota-Baxter maps and their inverse skewderivations. This picture clarifies the approximate solutions to generalized integral equations corresponding to non-autonomous linear (skew)differential equations.
Cariñena José F.
Ebrahimi-Fard Kurusch
Figueroa Hector
Gracia-Bondia Jose M.
No associations
LandOfFree
Hopf algebras in dynamical systems theory does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Hopf algebras in dynamical systems theory, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hopf algebras in dynamical systems theory will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-486270