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Date: 27-5-2018
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Date: 27-5-2018
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Date: 27-5-2018
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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
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