A technique that decomposes nonlinear operators and the unknown solution into series, representing nonlinear terms by Adomian polynomials computed from the solution components, to obtain rapidly convergent analytical approximations without linearization.

Principle

Assume the solution can be written as a sum of components; expand the nonlinear operator into Adomian polynomials built from those components, then derive recursive relations that produce successive terms of the solution series.

Demonstration

For the logistic equation u' = a u − b u^2, write u = sum_{n=0}^3 u_n, compute Adomian polynomials A_n for the term u^2 from {u_n}, then derive u_{n+1} by integrating linear parts and adding A_n contributions to build a convergent series approximation.

Misapplication

Incorrect computation of Adomian polynomials, inappropriate truncation, or applying formal series outside their radius of convergence can yield inaccurate or nonconvergent approximations; naive use for stiff or chaotic dynamics is problematic.

Consequence

Properly applied ADM yields explicit recursive formulae and semi-analytic series that often converge quickly for a variety of nonlinear problems and avoid explicit linearization or small-parameter assumptions.

Reversal

A direct numerical discretization or linearization of the nonlinear term that forgoes series decomposition; such reversal can be simpler to implement but loses the semi-analytical insight and may require fine meshes for accuracy.

Boundary

Best suited to problems where nonlinear operators can be systematically expanded into Adomian polynomials and where series convergence is plausible; less suitable for high-dimensional nonlinear operators where polynomial computation becomes expensive or ill-conditioned.

Semantic Tension

Competes with VIM and HAM since all build series approximations: ADM emphasizes algebraic decomposition of nonlinearities into polynomials, whereas VIM uses variational correction functionals and HAM uses homotopy with convergence control.

Synthesis

ADM constructs the solution as a series whose nonlinear contributions are encoded in Adomian polynomials, producing recursive, often rapidly convergent approximations that preserve the original operator's nonlinearity without external small parameters.