next up previous

2.2.2 Runge-Kutta Methods     continued...

To obtain an approximate solution of order p = 2 to the IVP (17), let and generate the sequences

where , , .

Euler's method is the special case, , and has order 1; the improved Euler method has and the Euler-Cauchy method has .

Approximate the solution to , at using the Euler-Cauchy method with . The recurrence relation for is

and the resulting approximations are given in Table 4. The IVP has the solution . The approximate and exact solutions are represented graphically in Figure 3 where the approximating values have been joined by straight line segments.

Table 4: Approximate solution using the Euler-Cauchy method.

Figure 3: Analytical solution y(t) = tan t vs. numerical solution given at t = 0.0, 0.1, ..., 1.0, as in Table 5.