Convergence of Eulers Method. It can be shown that, under suitable conditions on f, the

Chapter 2, Problem 20

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Convergence of Eulers Method. It can be shown that, under suitable conditions on f, the numerical approximation generated by the Euler method for the initial value problem y = f(t, y), y(t0) = y0 converges to the exact solution as the step size h decreases. This is illustrated by the following example. Consider the initial value problem y = 1 t + y, y(t0) = y0. (a) Show that the exact solution is y = (t) = (y0 t0)ett0 + t. (b) Using the Euler formula, show that yk = (1 + h)yk1 + h htk1, k = 1, 2, .... (c) Noting that y1 = (1 + h)(y0 t0) + t1, show by induction that yn = (1 + h) n(y0 t0) + tn (i) for each positive integer n. (d) Consider a fixed point t > t0 and for a given n choose h = (t t0)/n. Then tn = t for every n. Note also that h 0 as n . By substituting for h in Eq. (i) and letting n , show that yn (t) as n . Hint: limn(1 + a/n)n = ea.