Adomian Polynomial

Adomian polynomials decompose a function  into a sum of components

(1)

for a nonlinear operator  as

(2)

There appears to be no well-defined method for constructing a definitive set of polynomials for arbitrary , but rather slightly different approaches are used for different specific functions.

One possible set of polynomials is given by

			(3)
			(4)
			(5)
			(6)

These polynomials have the property that  depends only on , , ..., , and that the sum of subscripts for the component  is equal to .

REFERENCES:

Adomian, G. "Linear Stochastic Operators." Ph.D. Dissertation. Los Angeles, CA: University of California, Los Angeles, 1963.

Adomian, G. Stochastic Systems. New York: Academic Press, 1983.

Adomian, G. "A New Approach to Nonlinear Partial Differential Equations." J. Math. Anal. Appl. 102, 420-434, 1984.

Adomian, G. Nonlinear Stochastic Operator Equations. Orlando, FL: Academic Press, 1986.

Adomian, G. "A Review of the Decomposition Method in Applied Mathematics." J. Math. Anal. Appl. 135, 501-544, 1988.

Adomian, G. Nonlinear Stochastic Systems Theory and Applications to Physics. Dordrecht, Netherlands: Kluwer, 1989.

Adomian, G. Solving Frontier Problems of Physics: The Decomposition Method. Boston, MA: Kluwer, 1994.

Bellman, R. and Adomian, G. Partial Differential Equations: New Methods for their Treatment and Solution. Dordrecht, Netherlands: Reidel, 1985.

Cherruault, Y. Modèles et mŽthodes mathŽmatiques pour les sciences du vivant. Paris, France: Presses Universitaires de France, 1998.

Rach, R. "A Convenient Computational Form for the Adomian Polynomials." J. Math. Anal. Appl. 102, 415-419, 1984.

Rach, R. C. "A New Definition of the Adomian Polynomials." Kybernetes 37, 910-955, 2008.

Wazwaz, A. M. "A New Algorithm for Calculating Adomian Polynomials for Nonlinear Operators." Appl. Math. Comput. 111, 53-69, 2000.

Wazwaz, A.-M. Partial Differential Equations: Methods and Applications. Lisse, Netherlands: Balkema Publishers, 2002.