Convergence of Euler’s 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 = φ(t) = (y0 − t0)et−t0 + t. (b) Using the Euler formula, show that k = (1 + h)yk−1 + h − htk−1, k = 1, 2, . . . (c) Noting that y1 = (1 + h)(y0 − t0) + t1, show by induction that n = (1 + h)n(y0 − t0) + tn 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 → φ(t) as n→∞. Hint: lim n→∞ (1 + a/n)n = ea.
Convergence of Euler’s 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 = φ(t) = (y0 − t0)et−t0 + t.
(b) Using the Euler formula, show that k = (1 + h)yk−1 + h − htk−1, k = 1, 2, . . .
(c) Noting that y1 = (1 + h)(y0 − t0) + t1, show by induction that n = (1 + h)n(y0 − t0) + tn 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 → φ(t) as n→∞.
Hint: lim n→∞ (1 + a/n)n = ea.
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