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Asymptotic solution of the high order partial differential equation

In the present paper, the authors have constructed an asymptotic solution of the high order equation with partial derivatives by means of the asymptotic method for the high order systems. The improved first approximation of the solution of the given boundary value problem is determined. | Vietnam Journal of Mechanics, VAST , Vol.31, No. 2 (2009), pp . 65 - 74 ASYMPTOTIC SOLUTION OF THE HIGH ORDER PARTIAL DIFFERENTIAL EQUATION 1 Hoang Van Da, 1 Tran Dinh Son, 2 Nguyen Due Tinh 1 Ha Noi University of Mining and Geology 2 Quang Ninh University of Industry Abstract. In the present paper , the authors have constructed an asymptotic solution of the high order equation with partial derivatives by means of the asymptotic method for the high order systems. The improved first approximation of the solution of the given boundary value problem is determined. INTRODUCTION 1. The problem of the oscillation of the creepy elastic beam with linear boundary conditions in the autonomous case has been studied [3]. In this work, the authors inverstigate the oscillation of the creepy elastic beam, in the non-autonomous case, described by the third order equation as follows: o3y f)t3 a2y + ~ f)t2 + w 2 o5y 8tf)x4 + ~w 2 o4y 8x4 . = cF(x, y, Y, e, . ), w, n are real constants, Eis a small parameter, e = e(t), y homogeneous boundary conditons are where~' a2y I ;:i 2 = 0, Ylx=O = 0, ux x=O It is supposed that the resonance relation takes the from. p ni = -1 + c.6., q Ylx=O = 0, de dt = ~2; I X (1) = y(x, t). The relevant x=O = 0. 1' (2) (3) (4) p, q are integers. 2. CONTRUCTION OF THE NON-AUTONOMOUS SYSTEM With the boundary conditions (2) we get the fundamental functions and the eigenvalues in the form (5) 66 Hoang Van Da, Tran Dinh Son, Nguyen Due Tinh In this case, the partial solution of the equation ( 1) is found in form of the following series [6] y(x, t) =a cos r.pZ1 + EU1 (x, a, r.p, ()) + E2U2(x, a, r.p, ()) + . + ~) r.p = ( ;e where a, ~ (6) (7) are the functions satisfy the following differential equations da . 2 dt = EA1(a, 'if!)+ E A2(a, ~) ~~ + . (8) = ( !11 - :')') + EB1 (a,~) + E2B2 (a,~) + . (9) Now different iating the function y(x, t) in the form (6) with respect to argument t, after calculation we .