i) The Adams-Bashforth method:

Here we have

(7.3)

where the constants

(7.4)

are independent of and will be calculated numerically below. If the values are known, the corresponding values can be calculated and the differences are easily formed. The expression on the right of (7.3) is thus known, and and hence can be calculated. The index is then increased by 1, and the same formula is used to calculate , etc. The method breaks down if some of the values are not known. Such is the case at the beginning of the computation, where the initial condition furnishes only one of the required values. In such cases the missing values have to be secured by an independent method.