averaging_05

averaging_05 - The Method of Averaging CDS140B Lecturer:...

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The Method of Averaging CDS140B Lecturer: Wang Sang Koon Winter, 2005 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 ) . 1
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2 The Lagrange standard form Unperturbed Equation is Linear. ˙ x = A ( t ) x + ±g ( t, x ) , x (0) = x 0 . Remarks: (1) The procedure is called “introducing co-moving coordinates”. (2) If the unper- turbed equation is nonlinear, the variation of paramters techniques still applies. In practice, how- ever, there are usually many technical obstructions while carrying out the procedure 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 0 . We assume that
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averaging_05 - The Method of Averaging CDS140B Lecturer:...

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