Adams' method is a numerical method for solving linear first-order ordinary differential equations of the form

(1)

Let

(2)

be the step interval, and consider the Maclaurin series of  about ,

(3)

(4)

Here, the derivatives of  are given by the backward differences

			(5)
			(6)
			(7)

etc. Note that by (◇),  is just the value of .

For first-order interpolation, the method proceeds by iterating the expression

(8)

where . The method can then be extended to arbitrary order using the finite difference integration formula from Beyer (1987)

(9)

to obtain

(10)

Note that von Kármán and Biot (1940) confusingly use the symbol normally used for forward differences  to denote backward differences .

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 896, 1972.

Bashforth, F. and Adams, J. C. Theories of Capillary Action. London: Cambridge University Press, 1883.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 455, 1987.

Jeffreys, H. and Jeffreys, B. S. "The Adams-Bashforth Method." §9.11 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 292-293, 1988.

Kármán, T. von and Biot, M. A. Mathematical Methods in Engineering: An Introduction to the Mathematical Treatment of Engineering Problems. New York: McGraw-Hill, pp. 14-20, 1940.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 741, 1992.

Whittaker, E. T. and Robinson, G. "The Numerical Solution of Differential Equations." Ch. 14 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 363-367, 1967