R Rach

On composite nonlinearities and the decomposition method

Abstract 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.

On linear and nonlinear integro-differential equations

Abstract 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.

A new approach to boundary value equations and application to a generalization of Airy's equation

Abstract 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.

Coupled differential equations and coupled boundary conditions

Abstract 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.

On the solution of partial differential equations with specified boundary conditions

Abstract The decomposition method is applied to solution of partial differential equations in two and three dimensions with specified boundary conditions.

Nonlinear differential equations with negative power nonlinearities

Abstract Differential equations involving the term y−m, where m is a positive integer, are solved by the decomposition method 1. , 3. , 441–452).

Solving nonlinear differential equations with decimal power nonlinearities

Abstract 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.

Algebraic equations with exponential terms

Abstract The decomposition method is applied to algebraic equations containing exponential terms. The n term approximation φ n is rapidly damped as n increases, yielding an oscillating convergence of superior accuracy ( n = 5 and n = 9 for k = 2).

ON THE SOLUTION OF NONLINEAR DIFFERENTIAL EQUATIONS WITH CONVOLUTION PRODUCT NONLINEARITIES

Abstract Transform techniques can be used with differential equations containing convolution product nonlinearities to yield an algebraic equation for which the Adomian polynomials are more easily obtained for solution by the decomposition method.

Nonlinear plasma response

Abstract 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.