Algebraic computation of the solution of some nonlinear differential equations

Lamnabhi‐Lagarrigue, Françoise; Lamnabhi, M.

doi:10.1007/3-540-11607-9_23

Cited by 8 publications

(1 citation statement)

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“…Like the Laplace or Fourier transforms, the generating series offered a duality between time responses mediated by differential equations and a purely algebraic approach based on polynomials in the transform variables. Once solutions were established in the transform domain, they could be taken back into the time domain using the Laplace-Borel transform [10,11]. Although the theory of the generating series is extremely elegant, it sadly did not gain a great deal of traction in the arXiv:2209.05169v1 [math.DS] 12 Sep 2022 structural dynamics community.…”

Section: Introductionmentioning

confidence: 99%

On the Application of the Generating Series for Nonlinear Systems with Polynomial Stiffness

Gowdridge¹,

Dervilis²,

Worden³

2021

Nonlinear Structures &Amp; Systems, Volume 1

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Analytical solutions to nonlinear differential equations -where they exist at all -can often be very difficult to find. For example, Duffing's equation for a system with cubic stiffness requires the use of elliptic functions in the exact solution. A system with general polynomial stiffness would be even more difficult to solve analytically, if such a solution was even to exist. Perturbation and series solutions are possible, but become increasingly demanding as the order of solution increases. This paper aims to revisit, present and discuss a geometric/algebraic method of determining system response which lends itself to automation. The method, originally due to Fliess and co-workers, makes use of the generating series and shuffle product, mathematical ideas founded in differential geometry and abstract algebra. A family of nonlinear differential equations with polynomial stiffness is considered; the process of manipulating a series expansion into the generating series follows and is shown to provide a recursive schematic, which is amenable to computer algebra. The inverse Laplace-Borel transform is then applied to derive a time-domain response. New solutions are presented for systems with general polynomial stiffness, both for deterministic and Gaussian white-noise excitation.

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Section: Introductionmentioning

confidence: 99%