averaging_2

# averaging_2 - The Method of Averaging II CDS140B Lecturer...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

averaging_2 - The Method of Averaging II CDS140B Lecturer...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online