# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

**Download**(direct link)

**:**

**232**> 233 234 235 236 237 238 .. 609 >> Next

(1 + hL)B - 1

|En |< ----------y-----â h. (ii)

Equation (ii) gives a bound for | En | in terms of h, L, n, and â. 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)B < enhL; hence

eBhL - 1 e(tn-t0)L - 1

I En | < L âh = ----------l------âh.

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 nh2â = (t - ^)âh, 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 ô(¥) 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) Ó = 1 - t + 4 y, y(0) = 1

(b) Ó = 3 + t - y, y(0) = 1

(c) Ó = 2Ó - 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

1000-

6.010 18.04

2.004 6.000

as follows.

(a) First round each entry in the determinant to two digits.

(b) First round each entry in the determinant to three digits.

(c) Retain all four digits. Compare this value with the results in parts (a) and (b).

27. The distributive law a(b - c) = ab - ac does not hold, in general, if the products are

rounded off to a smaller number of digits. To show this in a specific case take a = 0.22,

b = 3.19, and c = 2.17. After each multiplication round off the last digit.

430

Chapter 8. Numerical Methods

8.2 Improvements on the Euler Method

Since for many problems the Euler method requires a very small step size to produce sufficiently accurate results, much effort has been devoted to the development of more efficient methods. In the next three sections we will discuss some of these methods. Consider the initial value problem

Ó = f (t, y), y(t0) = Óî (1)

and let y = ô(´) denote its solution. Recall from Eq. (10) of Section 8.1 that by

integrating the given differential equation from tn to tn+1 we obtain

f n+i

ô(ï+1) = Ô(ï) + / f[t,ô(t)] dt. (2)

Jtn

The Euler formula

Óï+1 = Óï + hf(tn, Óï) (3)

is obtained by replacing f [t, ô(³)] in Eq. (2) by its approximate value f (tn, yn) at the

left endpoint of the interval of integration.

Improved Euler Formula. A better approximate formula can be obtained if the integrand in Eq. (2) is approximated more accurately. One way to do this is to replace the integrand by the average of its values at the two endpoints, namely, { f [tn, Ô(¿ï)] + f [tn+1, Ô(¿ï+1)]}/2. This is equivalent to approximating the area under the curve in Figure 8.2.1 between t = tn and t = tn+1 by the area of the shaded trape-zoid. Further, we replace ô(´ï) and Ô(^ï+1) by their respective approximate values yn and yn 1 . In this way we obtain from Eq. (2)

, f (tn, Óï) + f(tn+1’ Óï+1) u Óï+1 = Óï + ----------------2-------------- (4)

Since the unknown yn+1 appears as one of the arguments of f on the right side of Eq. (4), this equation defines yn+1 implicitly rather than explicitly. Depending on the nature of the function f, it may be fairly difficult to solve Eq. (4) for yn+1. This

**232**> 233 234 235 236 237 238 .. 609 >> Next