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

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For comparison, note that both the Runge-Kutta method with h = 0.05 and the improved Euler method with h = 0.025 require 160 evaluations of f to reach t = 2. The improved Euler method yields a result at t = 2 that is in error by 1.22%. While this error may be acceptable for some purposes, it is more than 135 times the error yielded by the Runge-Kutta method with comparable computing effort. Note also that the Runge-Kutta method with h = 0.2, or 40 evaluations of f, produces a value at t = 2

438

Chapter 8. Numerical Methods

with an error of 1.40%, which is only slightly greater than the error in the improved Euler method with h = 0.025, or 160 evaluations of f. Thus we see again that a more accurate algorithm is more efficient; it produces better results with similar effort, or similar results with less effort.

TABLE 8.3.1 A Comparison of Results for the Numerical Solution of the Initial Value Problem ó1 = 1 — t + 4y, y(0) = 1

t Improved Runge-Kutta Exact

Euler

h = 0.025 h = 0.2 h = 0.1 h = 0.05

0 1.0000000 1 . 0000000 1.0000000 1 . 0000000 1 . 0000000

0.1 1.6079462 1.6089333 1.6090338 1 . 6090418

0.2 2.5020619 2.5016000 2.5050062 2.5053060 2.5053299

0.3 3.8228282 3.8294145 3.8300854 3.8301388

0.4 5.7796888 5.7776358 5.7927853 5.7941198 5.7942260

0.5 8.6849039 8.7093175 8.7118060 8.7120041

1.0 64.497931 64.441579 64.858107 64.894875 64.897803

1.5 474.83402 478.81928 479.22674 479.25919

2.0 3496.6702 3490.5574 3535.8667 3539.8804 3540.2001

PROBLEMS In each of Problems 1 through 6 find approximate values of the solution of the given initial value

ã problem at t = 0.1, 0.2, 0.3, and 0.4. Compare the results with those obtained by using other

methods and with the exact solution (if available).

(a) Use the Runge-Kutta method with h = 0.1.

(b) Use the Runge-Kutta method with h = 0.05.

> 1. / = 3 + t - y, y(0) = 1 > 2. Ó

> 3. y(0) = 1 > 4. Ó

=

2

y

1

3

,

> 5. Ó = ^ 5. > 6. Ó

2 0.

=

0()

y(

-ty

y(0) = 2 y(0) = 1

Ó(0) = -1

2

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. Compare the results with those given by other methods.

(a) Use the Runge-Kutta method with h = 0.1.

(b) Use the Runge-Kutta method with h = 0.05.

> 7. Ó = 0.5 - t + 2ó, ó(0) = 1 > 8. Ó = 5t - 3v/y, y(0) = 2

> 9. Ó = yt + ó, ó(0) = 3 > 10. 1 y(0) = 1

e

+

2t

=

> 11. / = (4 - ty)/(1 + ó2), ó(0) = -2

> 12. Ó = (ó2 + 2ty)/(3 + t2), ó(0) = 0.5

> 13. Confirm the results in Table 8.3.1 by executing the indicated computations.

> 14. Consider the initial value problem

/ = t2 + y2, y(0) = 1.

(a) Draw a direction field for this equation.

8.4 Multistep Methods

439

(b) Use the Runge-Kutta or other methods to find approximate values of the solution at t = 0.8, 0.9, and 0.95. Choose a small enough step size so that you believe your results are accurate to at least four digits.

(c) Try to extend the calculations in part (b) to obtain an accurate approximation to the solution at t = 1. If you encounter difficulties in doing this, explain why you think this happens. The direction field in part (a) may be helpful.

15. Consider the initial value problem

y> = 3t2/(3y2 — 4), y(0) = 0.

(a) Draw a direction field for this equation.

(b) Estimate how far the solution can extended to the right. Let tM be the right endpoint of the interval of existence of this solution. What happens at tu to prevent the solution from continuing farther?

(c) Use the Runge-Kutta method with various step sizes to determine an approximate value of ‘m .

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