Adams, Method
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
23-11-2021
2449
Adams' Method
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
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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