# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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method and the backward Euler method in Section 8.1 and with the exact solution (if available).

(a) Use the improved Euler method with h = 0.05.

(b) Use the improved Euler method with h = 0.025.

(c) Use the improved Euler method with h = 0.0125.

? 1. / = 3 + t - y, y(0) = 1 ? 2. y = 5t - 3Jy, y(0) = 2

? 3. / = 2y - 3t, y(0) = 1 ? 4. / = 2t + e-ty, y(0) = 1

, y2 + 2ty

? 5. y = ^+/’ y(0) =0.5

? 6. ? = (t2 - y2) siny, y(0) = -1

In each of Problems 7 through 12 find approximate values of the solution of the given initial value problem at t = 0.5, 1.0, 1.5, and 2.0.

(a) Use the improved Euler method with h = 0.025.

(b) Use the improved Euler method with h = 0.0125.

? 7. / = 0.5 - t + 2y, y(0) = 1 ? 8. / = 5t - 3Jy, y(0) = 2

? 9. y = Jt+y, y(0) = 3 ? 10. y = 2t + e-ty, y(0) = 1

? 11. / = (4 - ty)/(1 + y2), y(0) = -2

? 12. y = (y2 + 2ty)/(3 + t2), y(0) = 0.5

? 13. Complete the calculations leading to the entries in columns four and five of Table 8.2.1.

14. In this problem we establish that the local truncation error for the improved Euler formula is

proportional to h3. If we assume that the solution ? of the initial value problem y = f (t, y), y(t0) = y0 has derivatives that are continuous through the third order ( f has continuous second partial derivatives), it follows that

, ?" (t ) , ?"'( t ) ,

0(t„ + h) = ?(tn) + ? (tn)h +—^h2 + 3! n h3,

where t„ < t„ < t„ + h. Assume that y„ = ?(t„).

(a) Show that for y„+1 as given by Eq. (5)

e„+1 = ?(t„+1) - y„+1

4>"(t„)h - {f [t„ + h, y„ + hf(tn, yn)] - f (t„, y„)}

2!

3

h

+ . (i)

3!

(b) Making use of the facts that 4>"(t) = ft [t,^(t)] + f [t,$(t)]$'(t), and thatthe Taylor approximation with a remainder for a function F(t, y) of two variables is

F(a + h, b + k) = F(a, b) + Ft(a, b)h + Fy(a, b)k

+ — (h2 F + 2hkF + k2 F ) I

2! ( ‘‘ ‘y yy \x=yy=n

where f lies between a and a + h and n lies between b and b + k, show that the first term on the result.

on the right side of Eq. (i) is proportional to h3 plus higher order terms. This is the desired

8.3 The Runge-Kutta Method

435

(c) Showthatif f (t, y) islinearin tandy,thenen+1 = $>'"( tn)h3/6, where tn < ~tn < t r itfontHint: What are ftt, ft and fyy?

15. Consider the improved Euler method for solving the illustrative initial value problem y = 1 — t + 4y, y(0) = 1. Using the result of Problem 14(c) and the exact solution of the initial value problem, determine en+1 and a bound for the error at any step on 0 < t < 1. Compare this error with the one obtained in Eq. (26) of Section 8.1 using the Euler method. Also obtain a bound for e1 for h = 0.1 and compare it with Eq. (27) of Section 8.1.

In each of Problems 16 and 17 use the actual solution $(t) to determine en+1 and a bound for en+1 at any step on 0 < t < 1 for the improved Euler method for the given initial value problem. Also obtain a bound for e1 for h = 0. 1 and compare it with the similar estimate for the Euler method and with the actual error using the improved Euler method.

16. / = 2y — 1, y(0) = 1 17. / = 0.5 — t + 2y, y(0) = 1

In each of Problems 18 through 21 carry out one step of the Euler method and of the improved Euler method using the step size h = 0.1. Suppose that a local truncation error no greater than 0.0025 is required. Estimate the step size that is needed for the Euler method to satisfy this requirement at the first step.

18. y = 0.5 — t + 2y, y(0) = 1 19. / = 5t — 3yy, y(0) = 2

20. y = yt + y, y(0) = 3

21. / = (y2 + 2ty)/(3 + t2), y(0) = 0.5

22. The modified Euler formula for the initial value problem y = f (t, y), y(t0) = y0 is

given by

Yn+1 = Yn + hf[tn + 2h yn + 1 hf(tn, yn)].

Following the procedure outlined in Problem 14, show that the local truncation error in the modified Euler formula is proportional to h3.

In each of Problems 23 through 26 use the modified Euler formula of Problem 22 with h = 0.05 to compute approximate values of the solution of the given initial value problem at t = 0. 1, 0.2, 0.3, and 0.4. Compare the results with those obtained in Problems 1 through 4.

? 23. y = 3 + t — y, y(0) = 1 ? 24. y = 5t — 3yy, y(0) = 2

? 25. y = 2y — 3t, y(0) = 1 ? 26. y = 2t + e-‘y, y(0) = 1

27. Show that the modified Euler formula of Problem 22 is identical to the improved Euler

formula of Eq. (5) for y = f (t, y) if f is linear in both t and y.

8.3 The Runge-Kutta Method

In preceding sections we have introduced the Euler formula, the backward Euler formula, and the improved Euler formula as ways to solve the initial value problem

y f , y), y(t0) = y0

(1)

436

Chapter 8. Numerical Methods

numerically. The local truncation errors for these methods are proportional to h2, h2, and h3, respectively. The Euler and improved Euler methods belong to what is now called the Runge-Kutta1 class of methods.

In this section we discuss the method originally developed by Runge and Kutta. This method is now called the classic fourth order four-stage Runge-Kutta method, but it is often referred to simply as the Runge-Kutta method, and we will follow this practice

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