Adomian, G. A review of the decomposition method in applied mathematics. (English) Zbl 0671.34053 J. Math. Anal. Appl. 135, No. 2, 501-544 (1988). 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 Cited in 9 ReviewsCited in 568 Documents 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 Keywords:decomposition method; stochastic equations; applications; error estimates; numerical examples × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Adomian, G., Nonlinear Stochastic Operator Equations (1986), Academic Press: Academic Press Orlando, FL · Zbl 0614.35013 [2] Adomian, G., Applications of Nonlinear Stochastic Systems Theory to Physics (1988), Reidel: Reidel Dordrecht · Zbl 0666.60061 [3] Adomian, G., Stochastic Systems (1983), Academic Press: 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, (Ph. D. dissertation (1963), University of California: University of California Los Angeles) · Zbl 0114.08503 [14] Adomian, G., Stochastic Green’s functions, (Bellman, R. E., Stochastic Processes in Mathematical Physics and Engineering (1964), Amer. Math. Soc.,: Amer. Math. Soc., Providence, RI) · Zbl 0139.34205 [15] Adomian, G., Theory of random systems, (Trans. of Fourth Prague Conf. on Information Theory, Statistical Decision, and Random Processes (1967), Prague Publ. House) · Zbl 0556.93005 [16] Adomian, G., Stochastic operators and dynamical systems, (Wang, P. C.C, Information Linkage Between Applied Mathematics and Industry (1979), Academic Press: Academic Press New York) · Zbl 0582.60067 [17] Adomian, G., New results in stochastic equations: The nonlinear case, (Lakshmikanthum, V., Nonlinear Equations in Abstract Spaces (1978), Academic Press: Academic Press New York) · Zbl 0453.60062 [18] Adomian, G., The solution of general linear and nonlinear stochastic systems, (Rose, J., Modern Trends in Cybernetics and Systems (1976), Editura Technica: Editura Technica Romania), Springer-Verlag, Berlin/New York · Zbl 0426.93048 [19] Adomian, G., Solution of nonlinear stochastic physical problems, (Rendiconti del Seminario Matematico, Stochastic Problems in Mechanics. Rendiconti del Seminario Matematico, Stochastic Problems in Mechanics, Torino, Italy (1982)) · 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, (Adomian, G., Applied Stochastic Processes (1980), Academic Press: Academic Press New York), 1-18 · Zbl 0474.60050 [23] Adomian, G.; Adomian, G. E., Solution of the Marchuk model of infectious disease and immune response, (Witten, M., Mathematical Models in Medicine Diseases and Epidemics (1987), Pergamon: Pergamon Elmsford, NJ) · 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, (Witten, M., Hyperbolic Partial Differential Equations, Vol. 3 (1986), Pergamon: Pergamon Elmsford, NJ) · 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] G. Adomian and R. RachInternat. J. Math. Modelling; G. Adomian and R. RachInternat. J. Math. Modelling · Zbl 0613.65023 [31] G. Adomian and R. RachJ. Math. Anal. Appl.; G. Adomian and R. RachJ. 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: 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.,: 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, (Tzafestas, S. G.; Borne, P., Complex and Distributed Systems: Analysis, Simulation, and Control (1986), Elseven Publ. (North-Holland): Elseven Publ. (North-Holland) Amsterdam), 347-352 [46] N. Bellomo and D. SarafyanJ. Math. Anal. Appl.; N. Bellomo and D. SarafyanJ. 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) · Zbl 0624.60079 [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] I. BonzaniJ. Math. Anal. Appl.; I. BonzaniJ. Math. Anal. Appl. · Zbl 0626.60061 [51] Bonzani, I., Analysis of stochastic Van der Pol oscillators using the decomposition method, (Tzafestas, S.; Borne, P., Complex and Distributed Systems: Analysis, Simulation, and Control (1986), North-Holland: North-Holland Amsterdam), 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, (Atti. VIII Congresso Naz. AIMETA, 2 (1986)), 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] M. Pandolfi and R. RachJ. Math. Anal. Appl.; M. Pandolfi and R. RachJ. Math. Anal. Appl. · 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. 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