## A review of the decomposition method in applied mathematics.(English)Zbl 0671.34053

The paper is a kind of selfreview. Let $$Fu=g$$ be an ordinary nonlinear equation, $$F=L+R+N$$, where L is “easily invertible linear operator”, R is the remainder of the linear part of F, N is the nonlinearity. Then $$u=u_ 0+L^-Ru+L^-Nu,$$ where $$Lu_ 0=0$$. Write $$u=\sum^{\infty}_{0}u_ n$$, $$Nu=\sum^{\infty}_{n=0}A_ n$$, where $$\{A_ n\}$$ are special polynomials, $$A_ n$$ depends only on $$u_ 0,u_ 1,...,u_ n$$. Then $$u_{n+1}=-L^-Ru_ n+L^{-1}A_ n$$ and $$u_ n$$ can be found successively. Polynomials $$A_ n$$ should be constructed for each nonlinearity and the author proposes several formal schemes of such constructions which are the essence of the decomposition method by the author. He discusses the applications of this method to the systems of equations, stochastic equations, partial differential equations, considering for them both initial value problems and boundary problems. These applications are given in the numerous papers by the author and his colleagues (the list of references consists of 58 such papers). However the general or rigorous statements about convergence and error estimates are absent, although when numerical examples are considered, one can observe rather fast convergence, at least for fixed time. My opinion is that this formal method may happen to be a kind of variational method but its mathematical status is still not understood and justified.
Reviewer: L.Pastur

### MSC:

 34F05 Ordinary differential equations and systems with randomness 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34A34 Nonlinear ordinary differential equations and systems 35R60 PDEs with randomness, stochastic partial differential equations
Full Text:

### References:

 [1] Adomian, G, Nonlinear stochastic operator equations, (1986), Academic Press Orlando, FL · Zbl 0614.35013 [2] Adomian, G, Applications of nonlinear stochastic systems theory to physics, (1988), Reidel Dordrecht · Zbl 0666.60061 [3] Adomian, G, Stochastic systems, (1983), Academic Press New York · Zbl 0504.60066 [4] Adomian, G, Vibration in offshore structures, I, Math. comp. simulation, 29, 199-222, (1987) [5] Adomian, G, Vibration in offshore structures, II, Math. comp. simulation, 29, 1-6, (1987) [6] Adomian, G, A new approach to the efinger model for a nonlinear quantum theory for gravitating particles, Found phys., 17, 4, 419-424, (1987) [7] Adomian, G, Decomposition solution for Duffing and van der Pol oscillators, Internat. J. math. sci., 9, 4, 732-732, (1986) · Zbl 0605.34036 [8] Adomian, G, Convergent series solution of nonlinear equations, J. comput. appl. math., 11, 2, (1984) · Zbl 0549.65034 [9] Adomian, G, On the convergence region for decomposition solutions, J. comput. appl. math., 11, (1984) · Zbl 0547.65053 [10] Adomian, G, Nonlinear stochastic dynamical systems in physical problems, J. math. anal. appl., 111, 1, (1985) · Zbl 0582.60067 [11] Adomian, G, Random eigenvalues equations, J. math. anal. appl., 111, 1, (1985) · Zbl 0579.60061 [12] Adomian, G, On composite nonlinearities and the decomposition method, J. math. anal. appl., 114, 1, (1986) · Zbl 0617.65046 [13] Adomian, G, Linear stochastic operators, () · Zbl 0114.08503 [14] Adomian, G, Stochastic Green’s functions, () · Zbl 0139.34205 [15] Adomian, G, Theory of random systems, () · Zbl 0556.93005 [16] Adomian, G, Stochastic operators and dynamical systems, () · Zbl 0582.60067 [17] Adomian, G, New results in stochastic equations: the nonlinear case, () · Zbl 0453.60062 [18] Adomian, G, The solution of general linear and nonlinear stochastic systems, (), Springer-Verlag, Berlin/New York · Zbl 0426.93048 [19] Adomian, G, Solution of nonlinear stochastic physical problems, () · Zbl 0491.60066 [20] Adomian, G, On the Green’s function in higher-order stochastic differential equations, J. math. anal. appl., 88, 2, (1982) · Zbl 0493.60064 [21] Adomian, G, Stochastic model for colored noise, J. math. anal. appl., 88, 2, (1982) · Zbl 0493.60065 [22] Adomian, G, Stochastic systems analysis, (), 1-18 [23] Adomian, G; Adomian, G.E, Solution of the marchuk model of infectious disease and immune response, () · Zbl 0604.92006 [24] Adomian, G; Bellman, R.E, The stochastic Riccati equation, J. nonlinear anal., theory, methods, appl., 4, 6, (1980) · Zbl 0447.60044 [25] Adomian, G; Bellomo, N, On the Tricomi problems, () · Zbl 0597.35086 [26] Adomian, G; Bellomo, N; Riganti, R, Semilinear stochastic systems: analysis with the method of stochastic Green’s function and application to mechanics, J. math. anal. appl., 96, 2, (1983) · Zbl 0523.60057 [27] Adomian, G; Bigi, D; Riganti, R, On the solutions of stochastic initial-value problems in continuum mechanics, J. math. anal. appl., 110, 2, (1985) · Zbl 0582.60066 [28] Adomian, G; Elrod, M, Generation of a stochastic process with desired first- and second-order statistics, Kyberbetes, 10, 1, (1981) · Zbl 0444.60048 [29] Adomian, G; Rach, R, Coupled differential equations and coupled boundary conditions, J. math. anal. appl., 112, 1, 129-135, (1985) · Zbl 0579.60057 [30] {\scG. Adomian and R. Rach}, A new computational approach for inversion of very large matrices, Internat. J. Math. Modelling. · Zbl 0613.65023 [31] {\scG. Adomian and R. Rach}, Solving nonlinear differential equations with decimal power nonlinearities, J. Math. Anal. Appl. · Zbl 0591.60052 [32] Adomian, G; Rach, R, Algebraic computation and the decomposition method, Kybernetes, 15, 1, (1986) · Zbl 0604.60064 [33] Adomian, G; Rach, R, Algebraic equations with exponential terms, J. math. anal. appl., 112, 1, (1985) · Zbl 0579.60058 [34] Adomian, G; Rach, R, Nonlinear plasma response, J. math. anal. appl., 111, 1, (1985) · Zbl 0575.60063 [35] Adomian, G; Rach, R, Nonlinear differential equations with negative power non-linearities, J. math. anal. appl., 112, 2, (1985) · Zbl 0579.60059 [36] Adomian, G; Rach, R, Applications of decomposition method to inversion of matrices, J. math. anal. appl., 108, 2, (1985) · Zbl 0598.65011 [37] Adomian, G; Rach, R, Polynomial nonlinearities in differential equations, J. math. anal. appl., 109, 1, (1985) · Zbl 0606.34009 [38] Adomian, G; Rach, R, Nonlinear stochastic differential-delay equations, J. math. anal. appl., 91, 1, (1983) · Zbl 0504.60067 [39] Adomian, G; Rach, R; Sarafyan, D, On the solution of equations containing radicals by the decomposition method, J. math. anal. appl., 111, 2, (1985) · Zbl 0579.60060 [40] Adomian, G; Sibul, L.H, On the control of stochastic systems, J. math. anal. appl., 83, 2, (1981) · Zbl 0476.93077 [41] Adomian, G; Sibul, L.H; Rach, R, Coupled nonlinear stochastic differential equations, J. math. anal. appl., 92, 2, (1983) · Zbl 0517.60064 [42] Bellman, R.E; Adomian, G, Partial differential equations: new methods for their treatment and application, (1985), Reidel Dordrecht · Zbl 0557.35003 [43] Bellomo, N; Monaco, R, A comparison between Adomian’s decomposition methods and perturbation techniques for nonlinear random differential equations, J. math. anal. appl., 110, 495-502, (1985) · Zbl 0575.60064 [44] Bellomo, N; Riganti, R, Nonlinear stochastic systems in physics and mechanics, (1987), World Scientific Publ., Singapore · Zbl 0623.60084 [45] Bellomo, N; Riganti, R; Vacca, M.T, On the nonlinear boundary value problem for ordinary differential equations in the statics of long tlethered satellites, (), 347-352 [46] {\scN. Bellomo and D. Sarafyan}, On Adomian’s decomposition method and some comparisons with Picard’s iterative scheme, J. Math. Anal. Appl. · Zbl 0624.60079 [47] Bellomo, N; Sarayan, D, On a comparison between Adomian’s decomposition method and Picard iteration, J. math. anal. appl., 123, (1987) [48] Bigi, D; Riganti, R, Stochastic response of structures with small geometric imperfections, Meccaneca, 22, 27-34, (1987) · Zbl 0654.73033 [49] Bigi, D; Riganti, R, Solution of nonlinear boundary value problems by the decomposition method, Appl. math. modelling, 10, 48-52, (1986) · Zbl 0592.60048 [50] {\scI. Bonzani}, On a class of nonlinear stochastic dynamical systems: Analysis of the transient behavior, J. Math. Anal. Appl., to appear. · Zbl 0626.60061 [51] Bonzani, I, Analysis of stochastic van der Pol oscillators using the decomposition method, (), 163-168 · Zbl 1185.93124 [52] Adomian, S, Application of decomposition to convection-diffusion equations, Appl. math. lett., 1, 7-10, (1988) · Zbl 0631.65119 [53] Bonzani, I; Riganti, R, Soluzioni periodiche di sistemi dinamici nonlineari applicando il metodo di decomposizione, (), 525-530 [54] Bonzani, I; Zavattaro, M.G; Bellomo, N, On the continuous approximation of probability density and of the entropy functions for nonlinear stochastic dynamical systems, Math. comp. simulations, 29, 233-241, (1987) · Zbl 0625.60074 [55] {\scM. Pandolfi and R. Rach}, An application of the Adomian decomposition method to the matrix Riccati equation in neutron transport process, J. Math. Anal. Appl. to appear. · Zbl 0673.34007 [56] Rach, R, A convenient computational form for the Adomian polynomials, J. math. anal. appl., 102, 2, 415-419, (1984) · Zbl 0552.60061 [57] Riganti, R, Transient behavior of semilinear stochastic systems, J. math. anal. appl., 98, 314-327, (1984) · Zbl 0532.93050 [58] Riganti, R, On a class of nonlinear dynamical systems: the structure of a differential operator in the application of the decomposition method, J. math. anal. appl., 123, (1987) · Zbl 0624.34036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.