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2.3.1 The Adams-Bashforth and Adams-Moulton Formulas

On reaching a mesh point with approximate solution , there are (usually) available approximate solutions for . From the differential equation itself, approximations to the derivatives can be obtained from

This information can be exploited for solution values prior to the current point by using the integrated form of the differential equation:

The Adams Bashforth formula of order p is obtained by integrating the polynomial that interpolates at for , in place of f:

Formula (42) involves only one evaluation of f per step. An attractive feature of the approach is the underlying polynomial approximation to because it can be used to approximate between mesh points: