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(a) Use the Euler method with h = 0.025.
(b) Use the Euler method with h = 0.0125.
(c) Use the backward Euler method with h = 0.025.
(d) Use the backward Euler method with h = 0.0125.
? 7. / = 0.5 t + 2y, y(0) = 1 ? 8. y = 5t 3^y, y(0) = 2
? 9. y = yTTy, y(0) = 3 ? 10. y = 2t + e-y, y(0) = 1
? 11. y = (4 ty)/(\ + y2), y(0) = -2
? 12. y = (y2 + 2ty)/(3 + t2), y(0) = 0.5
? 13. Complete the calculations leading to the entries in columns three and four of Table 8.1.1.
? 14. Complete the calculations leading to the entries in columns three and four of Table 8.1.2.
15. Using three terms in the Taylor series given in Eq. (12) and taking h = 0.1, determine approximate values of the solution of the illustrative example y = 1 t + 4y, y(0) = 1 at t = 0.1 and 0.2. Compare the results with those using the Euler method and with the exact values.
Hint: If y = f (t, y), what is /'?
In each of Problems 16 and 17 estimate the local truncation error for the Euler method in terms of the solution y = 0(t). Obtain a bound for en+1 in terms of t and 0(t) that is valid on the interval 0 < t < 1. By using a formula for the solution obtain a more accurate error bound for e 1. For h = 0.1 compute a bound for e1 and compare it with the actual error at t = 0.1. Also compute a bound for the error e4 in the fourth step.
16. y = 2y 1, y(0) = 1 17. y = 1 t + 2y, y(0) = 1
In each of Problems 18 through 21 obtain a formula for the local truncation error for the Euler method in terms of t and the solution 0.
18. y = t2 + y2, y(0) = 1 19. y = 5t 3yy, y(0) = 2
20. y = JtTy, y(1) = 3 21. y = 2t + e-ty, y(0) = 1
? 22. Consider the initial value problem
y = cos 5^ t, y (0) = 1.
(a) Determine the solution y = 0(t) and draw a graph of y = 0(t) for 0 < t < 1.
(b) Determine approximate values of 0 (t) at t = 0.2, 0.4, and 0.6 using the Euler method
with h = 0.2. Draw a broken-line graph for the approximate solution and compare it with the graph of the exact solution.
(c) Repeat the computation of part (b) for 0 < t < 0. 4, but take h = 0. 1.
(d) Show by computing the local truncation error that neither of these step sizes is suffi-
ciently small. Determine a value of h to ensure that the local truncation error is less than 0.05 throughout the interval 0 < t < 1. That such a small value of h is required results from the fact that max 10" (t) | is large.
23. In this problem we discuss the global truncation error associated with the Euler method for the initial value problem y = f (t, y), y(t0) = y0. Assuming that the functions f and fy are continuous in a region R of the ty-plane that includes the point (t0, y0), it can be shown that there exists a constant L such that | f (t, y) f (t, y| < L |y y|, where (t, y) and (t, y) are any two points in R with the same t coordinate (see Problem 15 of Section 2.8). Further, we assume that ft is continuous, so the solution 0 has a continuous second derivative.
8.1 The Euler or Tangent Line Method
(a) Using Eq. (20) show that
\En+11 < \EB| + h| f[tB,$(tn)] f(tn, yn)\ + 1 h2\0"(ln)\<a\En\ + ph2, (i)
where a = 1 + hL and p = max \4>"(t)\/2 on t0 < t < tn.
(b) Accepting without proof that if E0 = 0, and if \ En \ satisfies Eq. (i), then \ En \< ph2(an 1)/(a 1) for a = 1, showthat
(1 + hL)n 1
\En \< -------^--------P h? (ii)
Equation (ii) gives a bound for \ En \ in terms of h, L, n, and p. Notice that for a fixed h, this error bound increases with increasing n; that is, the error bound increases with distance from the starting point t0.
(c) Showthat (1 + hL)n < enhL; hence
enhL - 1 e(tn t0)L- 1
\ En \ < L Ph = l------------------Ph-
For a fixed point t = t0 + nh [that is, nh is constant and h = (t t0)/n] this error bound is of the form of a constant times h and approaches zero as h ^ 0. Also note that for nhL = (t t0)L small the right side of the preceding equation is approximately nh2p = (t t0)Ph, which was obtained in Eq. (24) by an intuitive argument.
24. Derive an expression analogous to Eq. (21) for the local truncation error for the backward Euler formula.
Hint: Construct a suitable Taylor approximation to 0(t) about t = tn+1.
? 25. Using a step size h = 0.05 and the Euler method, but retaining only three digits throughout the computations, determine approximate values of the solution at t = 0.1, 0.2, 0.3, and 0.4 for each of the following initial value problems.
(a) y = 1 t + 4 y, y(0) = 1
(b) y = 3 + t y, y(0) = 1
(c) y = 2y 3t, y(0) = 1
Compare the results with those obtained in Example 1 and in Problems 1 and 3. The small differences between some of those results rounded to three digits and the present results are due to round-off error. The round-off error would become important if the computation required many steps.
26. The following problem illustrates a danger that occurs because of round-off error when nearly equal numbers are subtracted, and the difference then multiplied by a large number. Evaluate the quantity