tailieunhanh - Effect of higher approximation of krylov-bogoliubov-mitropolskii's solution and matched asymptotic solution of a differential system with slowly varying coefficients and damping near to a turning point

The second or higher order approximate solution is able to give better results than first approximate solution when the reduced frequency is many times larger than the small parameter. On the contrary, higher order solution diverges faster than the lower order solution when the reduced frequency becomes small (i .e., near to a turning point). In these situations matched asymptotic solution is important. An example is made to illustrate the matter. | Vietnam Journal of Mechanics, VAST , Vol. 26, 2004, No. 3 (182 - 192) EFFECT OF HIGHER APPROXIMATION OF KRYLOV-BOGOLIUBOV-MITROPOLSKII'S SOLUTION AND MATCHED ASYMPTOTIC SOLUTION OF A DIFFERENTIAL SYSTEM WITH SLOWLY VARYING COEFFICIENTS AND DAMPING NEAR TO A TUR NING POINT ROY K . C. AN D SHAMSU L ALAM M . Dept. of M ath, R ajshahi University of E ngineering and Technology, R aj shahi 6204, Bangladesh ABSTRACT. Second approximate solut ion of a second order differential equation with slowly varying coefficients and damping is obtained by Krylov-Bogoliubov-Mitropolskii method. The method is illustrated by an example. T he second or higher order approximate solution is able to give better results t han first approximate solution when the reduced frequency is many times larger than the small parameter. On the contrary, higher order solution diverges faster than the lower order solution when the reduced frequency becomes small (i .e., near to a t urning point). In t hese situations matched asymptotic solution is important . An example is made to illustrate the matter. 1 Introduction There are some well known perturbation methods ( ., Poincare method [1], WKB method [2-4], t wo-scale method [5-6] or Krylov-Bogoliubov-Mitropolskii (KBM) method [7-9] for handling linear and nonlinear different ial systems involving slowly varying coefficients. Among t he above pro cedures, KBM method is convenient and is widely used. The method has been extended to damped oscillatory and purely non-oscillatory systems with slowly varying coeffi cients by Boj adziev and Edwards [10]. Recent ly, Shamsul [11] has presented a brief way t o determine KBM solut ion (first order) of an n -t h, n = 2, 3, · · · order different ial system. In an another recent pap er , Shamsul et al [12] have presented an asymptotic solut ion of a second order d ifferent ial system in presence of strong linear damping for ce based on [13] . Sometimes fir st approximat e solut ion obtained in [10-11] gives .