1. Sec 21: Analysis of the Euler Method
Initial Value Problem
Lipschitz condition
Consider a differential equation
Where f is continuous and satisfies a Lipschitz condition
Euler Method
We divide the interval [a, b]
given

2. Sec 21: Analysis of the Euler Method
Initial Value Problem
Euler Method
Consider a differential equation
We divide the interval [a, b]
given
Where f is continuous and satisfies a Lipschitz condition
Define a function, by the formula
We are interested in the quality of this function as an approximation to y.

3. Sec 21: Analysis of the Euler Method
Local truncation error
Euler Method
1)Taylor expansion 2)
The local truncation error is the error in one interval assuming we have started at a point on the solution curve
Lemma:
Let
be positive constants and let
Global Error if
then
Proof: start by recursive

4. Sec 21: Analysis of the Euler Method
Theorem:
Let f be continuous and satisfies a Lipschitz condition
and be the solution of
Then there is a constant K such that
Proof:
Euler Method is a first order method

5. Sec 21: Analysis of the Euler Method