The method is an extension of the unified Kry lov-Bogoliubov-Mitropolskii method , which was initially developed for un-darn ped , under-clamped and over-clamped cases of the second order ordinary different ia l equation. The methods also cover a special condition of the over-damped case in which the general solution is useless. | V iet na m J ourn al of Mechanics, VAST , Vol. 30, No . 1 (2008) , pp. 11 - 19 A UNIFIED KRYLOV-BOGOLIUBOVMITROPOLSKII METHOD FOR SOLVING HYPERBOLIC-TYPE NONLINEAR PARTIAL DIFFERENTIAL SYSTEMS M. Zahurul Islam Department of A pplied M athematics, R ajshahi Un i1;ersity, Rajshahi-6205, Bangladesh M. Shamsul Alam and M. Bellal Hossain Departmen t of M athematics, Rajshahi University of E ngin eering and Techn ology, R ajshahi-6204, B angladesh, E mail address : msalam l 964 @yahoo .com Abstract . A general asy m ptotic solution is prese nted fo r invest igating t he transient response of non- linear systems modeled by hyperbolic-type part ia l different ia l equat ions wit h sm a ll nonlineari t ies. T he method covers all t he cases when eige n- values of t he correspo nding unp er t urbed systems are real, complex conjugate , or purely imaginary. It is shown t hat by suitable substit utio n for t he eigen-values in t he general res ul t t hat t he solut ion correspo nd ing to each of t he t hree cases can be obtained. T he met hod is an extension of t he unified Kry lov-Bogoliu bov-M itropolskii met hod , whi ch was init iall y developed for un-da rn ped , un der-clam ped a nd over-clamped cases of t he second order ordin ary different ia l eq uat ion . T he m et hods a lso cover a specia l condit ion of t he over-damped case in which t he general solu t ion is useless. Keywords: Unified KB:M method , Osc ill at ion , Non-oscillat ion 1. INTRODUCTION Krylov-Bogoliubov-Mitropolskii (KB ivI) [1, 2] method is one of t he widely used t echniques to obtain analyt ical solutions of weakly nonlinear ordinary differenti al equations. The method was originally developed t o find periodic solutions of second-order nonlinear ordinary d ifferent ial equations . P opov [3] extended t he method to damp ed nonlinear syst ems. Murty, D eekshatulu and K risna [4] investigat ed nonlinear over-damp ed systems by t his met hod . Ivlurty [5] used t heir earli er sol ution [4] as a .
