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The operator-theoretic method (Adomian and Malakian, J. Math. Anal. Appl. 76(1), (1980), 183–201) recently extended Adomian's solutions of nonlinear stochastic differential equations (G. Adomian, Stochastic Systems Analysis, in “Applied Stochastic Processes,” Nonlinear Stochastic Differential Equations, J. Math. Anal. Appl. 55(1) (1976), 441–452; On the modeling and analysis of nonlinear stochastic systems, in “Proceeding, International Conf. on Mathematical Modeling.” Vol. 1, pp. 29–40) to provide an efficient computational procedure for differential equations containing polynomial, exponential, and trigonometric nonlinear terms N(y). The procedure depends on the calculation of certain quantities An and Bn. This paper generalizes the calculation of the An and Bn to much wider classes of nonlinearities of the form N(y, y′,…). Essentially, the method provides a systematic computational procedure for differential equations containing any nonlinear terms of physical significance. This procedure depends on a recurrence rule from which explicit general formulae are obtained for the quantities An and Bn for any order n in a convenient form. This paper also demonstrates the significance of the iterative series decomposition proposed by Adomian for linear stochastic operators in 1964 and developed since 1976 for nonlinear stochastic operators. Since both the nonlinear and stochastic behavior is quite general, the results are extremely significant for applications. Processes need not, for example, be limited to either Gaussian processes, white noise, or small fluctuations.

The decomposition method ( G. Adomian, “Stochastic Systems,” Academic Press, New York/London, 1983 ; G. Adomian, “Stochastic Systems II,” in press) (and earlier work of Adomian referenced therein) allows solution without linearization of problems normally requiring the self-consistent field technique. The latter technique can provide a sufficiently accurate approximation in weakly coupled plasma, but the decomposition method of Adomian provides a superior and physically preferable methodology for the general case.

Differential equations involving the term y−m, where m is a positive integer, are solved by the decomposition method (Adomian, “Stochastic Systems,” Academic Press, New York, 1983; “ Nonlinear Stochastic Operator Equations,” Academic Press, New York, in press; J. Math. Anal. Appl. 55, No. 1 (1976), 441–452).

Equations containing radicals are solved using Adomian's decomposition method.

It is demonstrated that the high accuracy for approximations requiring only a few terms which is typical of the decomposition method for nonlinear stochastic operator equations, or special cases (linear or deterministic), holds for coupled equations and coupled boundary conditions as well.

A procedure reported elsewhere for solution of linear and nonlinear, deterministic or stochastic, delay differential equations developed by the authors as an extension of the first author's methods for nonlinear stochastic differential equations is now applied to a nonlinear delay-differential equation arising in population problems and studied by Kakutani and Markus. Examples involving time-dependent constants and even stochastic coefficients and delays can also be done.

Systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co-workers. Examples include two coupled Riccati equations, coupled linear equations, stochastic coupled equations with product terms, and n coupled stochastic differential equations.

The anharmonic oscillator is solved quickly, easily, and elegantly by Adomian's methods for solution of nonlinear stochastic differential equations emphasizing its applicability to nonlinear deterministic equations as well as stochastic equations. No difficulty is encountered in treating the case of the forced anharmonic oscillator or the stochastic case or of any nonlinear oscillating system such as the Duffing or Van der Pol oscillators, for example, with coefficients, as well as forcing functions, which are stochastic processes, since statistical separability is inherent in the Adomian method.

The decomposition method is demonstrated to provide a convenient and computationally simple solution of equations, such as ▽ 2 u = ƒ + ku for given boundary conditions, and can easily be adapted to solution of nonlinear versions and for complex (linear, nonlinear, or even random or coupled) boundary conditions.

Accurate, convergent, computable solutions using the decomposition method have been demonstrated in and papers for wide classes of nonlinear and/or stochastic differential, partial differential, or algebraic equations. It is shown specifically in this paper that composite nonlinearities of the form Nx = N0(N1(N2(···(x)···) appearing in such equations where the Ni are nonlinear operators can also be handled with the Adomian An polynomials.

The Maclaurin series is quite limited in comparison to the (Adomian) series obtained in the decomposition method. By adding procedures from the decomposition method and the expansion of nonlinearities using the Adomian polynomials, as well as a recent result of Adomian and Rach on transformation of series using the above polynomials, the Maclaurin series can be made much more useful in its applicability. However, the convergence is still slower than for Adomian's results using decomposition.

The decomposition method is applied to algebraic equations containing exponential terms. The n term approximation φn is rapidly damped as n increases, yielding an oscillating convergence of superior accuracy (

This paper shows that the Adomian decomposition method (“Stochastic Systems,” Academic Press, New York,1983) is not to be identified with any of the earlier methods since it solves a wide class of nonlinear and stochastic equations without the often unphysical assumptions which have become customary. Similarities between the Picard method and the decomposition method are purely superficial.

A new algorithm is presented for the well-known harmonic oscillator which can easily be extended to general nonlinear oscillator equations [1].

A damped oscillating convergence to a correct solution sometimes observed in decomposition method solutions of differential and partial differential equation is investigated.

The decomposition method is viewed as a unifying concept connecting solution methods.

The decomposition method (Adomian, “Nonlinear Stochastic Operator Equations,” Academic Press, New York, in press; “Stochastic Systems,” Academic Press, New York 1983) is shown to be applicable to integro-differential operator equations.

The decomposition method (“Stochastic systems,” Academic Press, New York 1983; “Nonlinear Stochastic Operator Equations,” Academic Press, New York, in press) is applied to the solution of nonlinear differential equations of the form Ly + Ny = x ( t ), where L is a linear differential operator and Ny is a nonlinear term of the form y γ with γ a decimal number.

The paper provides solution of the nonlinear partial differential equation ∇2p=k2 sinh p previously obtained by Plint and Breig [J. Appl. Phys. 35, 2745 (1964)] with the objective of confirmation of expected temperature dependence of scattering of light by certain crystals. The solution is of interest both for the experimental problem and as a solution of a nonlinear partial differential equation without customary linearizing assumptions. This is done using Adomian’s decomposition method [G. Adomian, Stochastic Systems (Academic, New York, 1983)]. This has been shown elsewhere in Adomian’s work and numerous following papers to provide a rapidly convergent series solution which is accurate and computable even if strong nonlinearities or stochastic behavior is involved.

It is shown that in addition to its advantages for nonlinear and/or stochastic differential equations [1,2], the decomposition method may be preferable even for equations, such as linear deterministic ordinary differential equations which are easily solvable by well-known methods in integral form because the evaluations of the integrals is easier. It is also shown that since solutions of differential equations are easily obtained by decomposition, it can be convenient to change a difficult integration problem to an easily solved differential equation and consequently evaluate the integral in an easily computed convergent series.

The decomposition method has been applied to nonlinear differential equations which have a linear differential operator term. Here we adapt the method to study purely nonlinear equations as well.

Carbon nano-tube (CNT) is applied to fabricate nano-probes, nano-switches, nano-sensors and nano- actuators. In this paper, a continuum model is employed to obtain the nonlinear constitutive equation and pull-in instability of CNT-based probe/actuators, which includes the effect of electrostatic interaction and intermolecular van der Waals (vdW) forces. The modified Adomian decomposition (MAD) method is applied to solve the nonlinear governing equation of the CNT-based actuator. Furthermore a simple and useful lumped parameter model was developed to investigate trends for various pull-in parameters. The influence of the vdW force and the geometrical dimensionless parameter on the pull-in deflection and voltage of the system is investigated. The obtained results are compared with those available in the literature as well as numerical solutions. The results demonstrate that our developed continuum based model is in good agreement with experimental results.

The effect of nonzero initial conditions in Adomian's solution of linear, or nonlinear, stochastic differential equations is demonstrated for an N th-order equation in which the decomposition of the linear part into L + R is made so as to simplify the Green's function.