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Unformatted text preview: The Method of Averaging II CDS140B Lecturer: Wang Sang Koon Winter, 2003 1 Introduction Remarks: The method leads generally to asymtotic series as opposed to convergent series. It is not restriced to periodic solutions. Averaging Method. Put the equation ¨ x + x = f ( x, ˙ x ) into Lagrange stardard form and do the averaging. Example 11.1 ¨ x + x = (- ˙ x + x 2 ) . 2 The Lagrange standard form Unperturbed Equation is Linear. ˙ x = A ( t ) x + g ( t,x ) , x (0) = x . 3 Avaraging in the Periodic Case Asymptotic Validity of Averaging Method. Consider equation (11.17) ˙ x = f ( t,x ) + 2 g ( t,x, ) , x (0) = x . We assume that f ( t,x ) is T-periodic in t and we introduce the average f ( y ) = 1 T Z T f ( t,y ) dt. Consider now equation (11.18) ˙ y = f ( y ) , y (0) = x . 1 Theorem 11.1 Consider the initial value problem 11.7 and 11.8 with x,y,x ∈ D ⊂ R n ,t ≥ 0. Suppose that 1. f,g and ∂f/∂x are defined, continuous and bounded by a constant M in [0 , ∞ ) × D ; 2. g is Lipschitz-continuous in...
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- Fall '10
- Constant of integration, Lipschitz continuity