Higher-Order Averaging, Formal Series and Numerical Integration I: B-series

Higher-Order Averaging, Formal Series and Numerical Integration I: B-series

We show how B-series may be used to derive in a systematic way the analytical expressions of the high-order stroboscopic averaged equations that approximate the slow dynamics of highly oscillatory systems. For first-order systems we give explicitly the form of the averaged systems with O( j ) errors, j = 1,2,3 (2π denotes the period of the fast oscillations). For second-order systems with large...

Higher-Order Averaging, Formal Series and Numerical Integration III: Error Bounds

In earlier papers, it has been shown how formal series like those used nowadays to investigate the properties of numerical integrators may be used to construct highorder averaged systems or formal first integrals of Hamiltonian problems. With the new approach the averaged system (or the formal first integral) may be written down immediately in terms of (i) suitable basis functions and (ii) scal...

Erratum to: Higher-Order Averaging, Formal Series and Numerical Integration II: The Quasi-Periodic Case

The paper considers non-autonomous oscillatory systems of ordinary differential equations with d ≥ 1 nonresonant constant frequencies. Formal series like those used nowadays to analyze the properties of numerical integrators are employed to construct higher-order averaged systems and the required changes of variables. With the new approach, the averaged system and the change of variables consis...

ALGEBRAIC INDEPENDENCE OF CERTAIN FORMAL POWER SERIES (I)

We give a proof of the generalisation of Mendes-France and Van der Poorten's recent result over an arbitrary field of positive characteristic and then by extending a result of Carlitz, we shall introduce a class of algebraically independent series.

A Numerical Integration Method by Using Generalized Series of Functions