Gronwall

# Gronwall - Gronwall’s Inequality We suppose that we have...

This preview shows pages 1–2. 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: Gronwall’s Inequality We suppose that we have two solutions, x 1 ( t ) and x 2 ( t ), to the differential equation x = f ( x,t ), and we want an estimate of | x 1 ( t )- x 2 ( t ) | in terms of | x 1 ( t )- x 2 ( t ) | . Think of this as measuring how much the two solutions can move away from each other in terms of how far apart they are when t = t . We look at the equation on a rectangle R = { a < t < b, c < x < d } , and use the number M , M = max ( t,x ) ∈ R | ∂f ∂x ( t,x ) | in the estimate. The estimate is | x 1 ( t )- x 2 ( t ) | ≤ | x 1 ( t )- x 2 ( t ) | e M | t- t | . (0) This holds as long as the points ( t,x 1 ( t )) and ( t,x 2 ( t )) stay in R . Notice that when x 1 ( t ) = x 2 ( t ) the estimate says that x 1 ( t )- x 2 ( t ) ≡ 0. That’s the uniqueness theorem. In the examples where uniqueness fails, like x = x 2 / 3 , you will always find ∂f/∂x is unbounded, and the estimate doesn’t say anything when M = ∞ ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

Gronwall - Gronwall’s Inequality We suppose that we have...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online