In this exercise we derive the theoretical bounds

Chapter , Problem 11

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In this exercise we derive the theoretical bounds mentioned in the section. Let R be a rectangle {(t, y) | a t b, c y d} in the t y-plane and suppose that f (t, y) is continuously differentiable on R. Given an initial-value problem dy dt = f (t, y), y(t0) = y0, and an interval t0 t tn such that the point (t0, y0) is in R and tn b, then we can bound the error en involved in the Euler approximation assuming the Euler approximate values y1, y2,..., yn all lie between c and d. To do so, let M1 = max f t + f y f on R and let M2 = max f y on R. (a) Show that e1 M1 (t)2 2 . (b) Show that e2 e1 + M2e1t + M1 (t)2 2 and explain the significance of each of these three terms. Copyright 2011 Cen