A general multiple-time-scale method for solving an $n$-th order weakly nonlinear differential equation with damping

M. Abul Kalam Azad; M. Shamsul Alam; M. Saifur Rahman; Bimolendu Shekhar Sarker

doi:10.4134/CKMS.2011.26.4.695

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2011; 26(4): 695-708

Printed December 1, 2011

https://doi.org/10.4134/CKMS.2011.26.4.695

A general multiple-time-scale method for solving an $n$-th order weakly nonlinear differential equation with damping

M. Abul Kalam Azad, M. Shamsul Alam, M. Saifur Rahman, and Bimolendu Shekhar Sarker

Rajshahi University of Engineering and Technology (RUET), Rajshahi University of Engineering and Technology (RUET), Rajshahi University of Engineering and Technology (RUET), Rajshahi University of Engineering and Technology (RUET)

Abstract

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Abstract

Based on the multiple-time-scale (MTS) method, a general formula has been presented for solving an $n$-th, $n= 2,3,\ldots,$ order ordinary differential equation with strong linear damping forces. Like the solution of the unified Krylov-Bogoliubov-Mitropolskii (KBM) method or the general Struble's method, the new solution covers the un-damped, under-damped and over-damped cases. The solutions are identical to those obtained by the unified KBM method and the general Struble's method. The technique is a new form of the classical MTS method. The formulation as well as the determination of the solution from the derived formula is very simple. The method is illustrated by several examples. The general MTS solution reduces to its classical form when the real parts of eigen-values of the unperturbed equation vanish.

Keywords: oscillation, non-oscillation, asymptotic method

MSC numbers: 3.40E+06