Sec 21: Analysis of the Euler Method

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Presentation transcript:

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

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.

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

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

Sec 21: Analysis of the Euler Method