The focus of this work is the solution of a fundamental problem that arises in non-dissipative nonlinear oscillators and related applications, namely the rare possibility of explicitly inverting the associated time-integral. Here, the inversion issue is treated by near-minimax approximation of the restoring force via fifth-order Cebishev polynomials on a normalised integration interval: this gives rise to a Duffing-type quintic oscillator, whose solutions effectively represent those of the original problem. Indeed, when an odd function describes the restoring force, the elliptic time-integral associated with the quinticate oscillator can be inverted in closed form. This is obtained here, by observing that the integrand involves a quadratic polynomial, built on the quinticate oscillator coefficients, and by studying its discriminant. Based on these findings, we provide a novel solution procedure, implemented within the Mathematica scientific environment, that exploits elliptic integrals of the first kind and whose effectiveness is tested on three well-known conservative nonlinear oscillator models.

Exact time–integral inversion via Čebyšëv quintic approximations for nonlinear oscillators / Martina Boschi, Daniele Ritelli, Giulia Spaletta. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - ELETTRONICO. - 533:1 (1 May 2024)(2024), pp. 128015.1-128015.16. [10.1016/j.jmaa.2023.128015]

Exact time–integral inversion via Čebyšëv quintic approximations for nonlinear oscillators

Daniele Ritelli;Giulia Spaletta
2024

Abstract

The focus of this work is the solution of a fundamental problem that arises in non-dissipative nonlinear oscillators and related applications, namely the rare possibility of explicitly inverting the associated time-integral. Here, the inversion issue is treated by near-minimax approximation of the restoring force via fifth-order Cebishev polynomials on a normalised integration interval: this gives rise to a Duffing-type quintic oscillator, whose solutions effectively represent those of the original problem. Indeed, when an odd function describes the restoring force, the elliptic time-integral associated with the quinticate oscillator can be inverted in closed form. This is obtained here, by observing that the integrand involves a quadratic polynomial, built on the quinticate oscillator coefficients, and by studying its discriminant. Based on these findings, we provide a novel solution procedure, implemented within the Mathematica scientific environment, that exploits elliptic integrals of the first kind and whose effectiveness is tested on three well-known conservative nonlinear oscillator models.
2024
Exact time–integral inversion via Čebyšëv quintic approximations for nonlinear oscillators / Martina Boschi, Daniele Ritelli, Giulia Spaletta. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - ELETTRONICO. - 533:1 (1 May 2024)(2024), pp. 128015.1-128015.16. [10.1016/j.jmaa.2023.128015]
Martina Boschi, Daniele Ritelli, Giulia Spaletta
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/964646
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