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

ISBN 0-471-31999-6

**Download**(direct link)

**:**

**189**> 190 191 192 193 194 195 .. 486 >> Next

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

s 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.

? A 1 + m II y(0) = 1 ? 2. /

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

? 5. / = ^ + f, 2 y(0) = 0.5 ? 6. /

-ty

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

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

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

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

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

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

? 14. Consider the initial value problem

y = 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

/ = 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 tM to prevent the solution from continuing farther?

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

(d) If you continue the computation beyond tM, you can continue to generate values of y. What significance, if any, do these values have?

(e) Suppose that the initial condition is changed to y(0) = 1. Repeat parts (b) and (c) for this problem.

8.4 Multistep Methods

In previous sections we have discussed numerical procedures for solving the initial value problem

y = f (t, y), y(t0) = y0> (1)

in which data at the point t = tn are used to calculate an approximate value of the solution 0(tn+j) at the next mesh point t = tn+r In other words, the calculated value of 0 at any mesh point depends only on the data at the preceding mesh point. Such methods are called one-step methods. However, once approximate values of the solution y = 0(t) have been obtained at a few points beyond t0, it is natural to ask whether we can make use of some of this information, rather than just the value at the last point, to calculate the value of 0(t) at the next point. Specifically, if y1 at tj, y2 at t2,, yn at tn are known, how can we use this information to determine yn 1 at tn 1? Methods that use information at more than the last mesh point are referred to as multistep methods. In this section we will describe two types of multistep methods, Adams3 methods and backward differentiation formulas. Within each type one can achieve various levels of accuracy, depending on the number of preceding data points that are used. For simplicity we will assume throughout our discussion that the step size h is constant.

3John Couch Adams (1819-1892), English astronomer, is most famous as codiscoverer with Joseph Leverrier of the planet Neptune in 1846. Adams was also extremely skilled at computation; his procedure for numerical integration of differential equations appeared in 1883 in a book with Francis Bashforth on capillary action.

440

Chapter 8. Numerical Methods

Adams Methods. Recall that

(2)

where $( t) is the solution of the initial value problem (1). The basic idea of an Adams

method is to approximate $(t) by a polynomial Pk(t) of degree k — 1 and to use the polynomial to evaluate the integral on the right side of Eq. (2). The coefficients in Pk (t) are determined by using k previously calculated data points. For example, suppose that we wish to use a first degree polynomial P2(t) = At + B. Then we need only the two data points (tn, yn) and (tn_p yn_ 1). Since P2 is to be an approximation to $, we require that P2(tn) = f (tn, yn) and that P2(tn—1) = f (tn—1, yn—1). Recall that we denote f (tj, y-j) by fj for an integer j. Then A and B must satisfy the equations

**189**> 190 191 192 193 194 195 .. 486 >> Next