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Adomian Polynomial

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Adomian polynomials decompose a function u(x,t) into a sum of components

u(x,t)=sum_(n=0)^inftyu_n(x,t)
(1)

for a nonlinear operator F as

F(u(x,t))=sum_(n=0)^inftyA_n.
(2)

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

One possible set of polynomials is given by

A_0=F(u_0)
(3)
A_1=u_1F^'(u_0)
(4)
A_2=u_2F^'(u_0)+1/(2!)u_1^2F^('')(u_0)
(5)
A_3=u_3F^'(u_0)+u_1u_2F^('')(u_0)+1/(3!)u_1^3F^(''')(u_0).
(6)

These polynomials have the property that A_n depends only on u_0, u_1, ..., u_n, and that the sum of subscripts for the component u_n is equal to n.

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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.

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Adomian Polynomial

Cite this as:

Weisstein, Eric W. "Adomian Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AdomianPolynomial.html

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