The decomposition method (G. Adomian, "Stochastic Systems," Academic Press, New York, 1983) developed to solve nonlinear stochastic differential equations has recently been generalized to nonlinear (and/or) stochastic partial differential equations, systems of equations, and delay equations and applied to diverse applications. As pointed out previously (see reference above) the methodology is an operator method which can be used for nondifferential operators as well. Extension has also been made to algebraic equations involving real or complex coefficients. This paper deals specifically with quadratic, cubic, and general higher-order polynomial equations and negative, or nonintegral powers, and random algebraic equations. Further work on this general subject appears elsewhere (G. Adomian, "Stochastic Systems II," Academic Press, New York, in press). 0 1985 Academic Press. Inc. The decomposition method [ 1, 21 is applied to generic operator equations of the form 7~ =g where ST may be a nonlinear (and/or) stochastic operator and x a stochastic process on a appropriate probability space. The basic equation is considered in the form L/u + Mu = g, or Lu + Nu = g in the deterministic case where L is a linear (deterministic) operator and N a nonlinear (deterministic) operator. (In the case where the operators involve stochasticity, script letters 9, JP" are preferred.) (If we write an ordinary quadratic equation ax2 + bx + c = 0 in the form Lu + Nu = g, identifying Nx=ax',L=b, andg=-c we have Lu=g-Nu or bx=-c-ax2. 141 0022-247x/85 S3.00
Adomian and Rach REF generalized the method to solve nonlinear systems of equations.
On the solution of algebraic equations by the decomposition method
Journal of Mathematical Analysis and Applications
Mathematics