Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 52

We take a simple autonomous non-linear scalar ODE to show how to calculate Picard iterates:

(3.6)

whose solution is given by:

We now compute the Picard iterates (3.4) for this ODE in order to determine the values of t for which the ODE has a solution. For convenience, let us take . Some simple integration shows that:

(3.7)

We can see that the series is beginning to look like . We know that this series is convergent for . A nice exercise is to compute the Picard iterates in the most general case (that is, ) and to determine under which circumstances the ODE (3.6) has a solution. In this case we have represented the solution of an ODE as a series, and we then analysed this series for which there are many convergence results, such as the root test and the ratio test.