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.
· · · For E = 0.2 and for initial values [u(x , 0) = 0.90667 sin(l.1 44465x), u 1(x , 0) = ], u(x, t, E) has been evaluated and the co rres ponding numerical solution of (19) computed The results for x = respecyively and x = are presented respectively in Fig 2(a) and Fig (b) From the figures, it is clear that solution (32) compares well with the numerical solution For the under damp ed case, we consider p = 1, k = 0.2 , l = 2, K = l The solutions of (24) are 1.144465 , 2.543493, 4.048082 ·· · or eigen-values are - 0.2 ± 1.126854 , -0 ± 2.535618 , -0.2 ± 4.043138 · · · For E = 0.5 and for initial values [u( x, 0) = 0.90667 sin(l.144465x), Ut(x, 0) =OJ, u( x, t, E) evaluated and the corresponding numerical solution of (19) has been computed The results for x = and x = are presented in Fig (a) and Fig.3 (b) respectively From the figures, it is clear that solution (32 ) co mpares well with the numerical solution When k = 2w1, then solution Eq (52) is useful for an over-damped solution of equation (19) We are interested to compare it with numerical solution (generated by finite difference method) Let us considerp = 1, k = 1.3215 , l = 2, K = The solutions of (24 ) are l.144465, 2.543493, 4.048082 · · · or eigen- values are - 1.3215 ± 0.660728 , - 1.3215 ± l 73245i , - 1.3215 ± 3.826304i · · · and one set of eigen-value is real For E = 0.5 and initial values [u(x, O) = 0.90667sin(l.144465.r,), 1J,t(x, 0) = OJ, 11(x , t , c) has been evaluated and the corresponding numerical solution of (19) has been computed The results for x = respectively and x = are presented respectively in Fig 4( a) and Fig 4(b) From the fi gures, it is clear that solution Eq (52) compares well with the numerical solution CONCLUSION A general formula is presented for obtaining the transient response of nonlinear systems governed by a hyp erbolic-type partial differential equation with small nonlinearities According to.the unified theory [4, 5] there exists a general solution , used in three cases, i.e o,ver~ damp ed , under-damped and un-damped In previous p apers [5, 7] only ordinary differential equations are considered In the present paper, we observe a similar result for partial d ifferential equations ACKNOWLEDGEMENT The authors are grateful to two potential reviewers for their helpful comments /suggestions to prepare the revised manuscript A unified Krylo v-Bogoliubov-Mitropolskii method 19 REFERENCES N N Krylov a nd N N Bogoliubov, Introduction to Nonlinear M echanics , Princeton Univer- sity Press , New J esey, 1943 N N Bigoliubov and Yu A Mit ropolskii , As ympt otic methods in the ·t h eory of nonlinear oncillations, Gordan and Breach , New York , 1961 I P P opov , A generali zation of the Bogoliubov asymptotic method in the theory of 11onlinear oscillations (in Rusia n), Dok! Akad Nauk SSSR, (1956), 308-310 I S N Murty, B L Deekshatulu and G Krisna, General asymptotic method of KrylovBogo liubov for over-damped nonlinear systems, J Frank Inst 288 (1969 ) 49-64 I S N Murty, A univi ed Krylov-Bogoliubov method for solving second order nonlinear systems, Int J Nonlinear M ech (1971 ) 45-53 G N Boj adz iev and J Edwards , On some asymptotic method for non-oscillatory and oscillatory processes, J Nonlinear Vibrat ion problems 20 (1981) 69-79 M Shamsul Alam, A unified Krylov-Bogoliubov-Mi tropolskii method for solving n - th order nonlinear system , J Frar;k Inst 339 (2 002 ) 239-248 M Shamsul Alam , A unifi ed Krylov-Bogoliubov-Mitropolskii method for solving n - th order nonli near systems wit h varying coeffici ents , ] Sound and Vibration 265 (2003 ) 987-1002 Yu A Mi tropo lskii and R I Mosencov , Lectures on t he appli cation of asymp totic methods of the solut ion of equations with partial derivatives, (in Russian), Ac of Sci Ukr SSR, Kiev, 1968 10 G N Boj adziev and R , W Lardner , Monofrequent osci llations in mecha ni cal syste governed by seco nd- order hyperbolic different ia l equation with small nonlinearities, Int ] Nonlinear M ech (1973) 289-302 11 G N Boj adziev and R W Lardner , Asymptot ic solu tions of parti al differential equations with damping and delay, J Quart Appl Math 33 (1975) 205-21 12 M Shamsul Alam , M Zahurul Islam and M Ali Akber, A general form of Krylov- BogoliobovMitropolskii m et hod for solving nonlinear part ial differential equations, ] Sound and Vi bration 285 (2005 ) 173-1 85 13 M Shamsul Alam , Asymptotic methods for second order over-damped and crit ically damped nonlinear systems, Soochow J Math 27 (2001 ) 187-200 R eceive.d Ma rch 10, 2008 GIA.I HE PHL'd NG TRINH DAO HAM RIENG PHI T UYEN DANG HYPERBOLIC BANG PHVdNG PHAP KROLOV-BOGOLIUBOV-MITROPOLSKII Mot nghiem tiem c~n t6ng quat dl1Q C bi§u dien d§ khao sat di;ic tr ung cua he phi tuyE\n cl\19~ mo hlnh bing cac phuong trlnh d