ON THE RECENT DEVELOPMENT OF SOME ASYMPTOTIC METHODS FOR NONLINEAR VIBRATION EQUATIONS

Authors

  • MAHMOUD BAYAT "DEPARTMENT OF CIVIL ENGINEERING, ISLAMIC AZAD UNIVERSITY, SHIRVAN BRANCH,IRAN" AND "DEPARTMENT OF CIVIL ENGINEERING, SHOMAL UNIVERSITY, AMOL, IRAN, P.O. BOX 731" (I JUST PUT ONE OF THEM IN MY PAPERS),

Keywords:

NONLINEAR VIBRATION, VARIATIONAL ITERATION METHOD(VIM), HOMOTOPY PERTURBATION METHOD (HPM), ENERGY BALANCE METHOD (EBM), ITERATION PERTURBATION METHOD, PARAMETER-EXPANSION METHOD(PEM), ANCIENT CHINESE MATHEMATICS METHOD, PARAMETERIZED PERTURBATION METHOD(PPM)

Abstract

THIS PAPER FEATURES A SURVEY OF SOME RECENT DEVELOPMENTS IN ASYMPTOTIC TECHNIQUES, WHICH ARE VALID NOT ONLY FOR WEAKLY NONLINEAR EQUATIONS, BUT ALSO FOR STRONGLY ONES. FURTHER, THE ACHIEVED APPROXIMATE ANALYTICAL SOLUTIONS ARE VALID FOR THE WHOLE SOLUTION DOMAIN. THE LIMITATIONS OF TRADITIONAL PERTURBATION METHODS ARE ILLUSTRATED, VARIOUS MODIFIED PERTURBATION TECHNIQUES ARE PROPOSED, AND SOME MATHEMATICAL TOOLS SUCH AS VARIATIONAL THEORY, HOMOTOPY TECHNOLOGY, AND ITERATION TECHNIQUE ARE INTRODUCED TO OVER-COME THE SHORTCOMINGS.

IN THIS PAPER THE FOLLOWING CATEGORIES OF ASYMPTOTIC METHODS ARE EMPHASIZED: (1) VARIATIONAL ITERATION METHOD (2) HOMOTOPY PERTURBATION METHOD (3) HE'S VARIATIONAL ITERATION METHOD FOR NONLINEAR OSCILLATORS (4) ENERGY BALANCE METHOD (5) PARAMETER-EXPANSION METHOD (6) PARAMETERIZED PERTURBATION METHOD (7) AMPLITUDE-FREQUENCY FORMULATION FOR NONLINEAR OSCILLATORS BASED ON ANCIENT CHINESE MATHEMATICS (8) ITERATION PERTURBATION METHOD.

KEYWORDS: NONLINEAR VIBRATION ; VARIATIONAL ITERATION METHOD(VIM); HOMOTOPY PERTURBATION METHOD (HPM); ENERGY BALANCE METHOD (EBM); ITERATION PERTURBATION METHOD; PARAMETER-EXPANSION METHOD(PEM); ANCIENT CHINESE MATHEMATICS METHOD; PARAMETERIZED PERTURBATION METHOD(PPM).

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Published

2012-07-10

Issue

Vol. 9 No. 2 (2012)

Section

Articles

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