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To determine the error for Euler's method, we need look no further than the Taylor series:
We know, from the IVP, that y(1)(t) = f(t, y(t)), and therefore the error is the other term: ½ y(2)(τ)h2 where τ ∈ [t, t + h].
Thus, the error is O(h2). As with Newton's method, the knowledge of the derivative gives us this formula.
To demonstrate, consider y(1)(t) = t y(t) − y(t) + t − 1 with the initial value y(0) = 1. We may approximate y(0.1) by setting y0 = y(0) and calculating y1 = y0 + 0.1 f(0, 1) = 1 − 2 ⋅ 0.1 = 0.8.
The actual answer is 0.8187458691, and to check this, we can use implicit differentiation:
We can substitute y(1)(τ) into this formula to get (after simplification):
We know that τ ∈ [0, 0.1] and if we assume that y(τ) ∈ [0.8, 1], then we get that the second derivative is bounded by [3.258, 4]. Thus, the error should be ½ y(2)(τ) 0.12 ∈ [0.01629, 0.02]. We found above that the error was 0.01875 which is approximately in the middle of this interval.
Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.
