A reliable modification of the Adomian decomposition method for higher‐order nonlinear differential equations

The purpose of this paper is to propose a new modification of the Adomian decomposition method for resolution of higher‐order inhomogeneous nonlinear initial value problems.

First the authors review the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. Next, the advantages of Duan's new algorithms and subroutines for fast generation of the Adomian polynomials to high orders are discussed. Then algorithms are considered for the solution of a sequence of first‐, second‐, third‐ and fourth‐order inhomogeneous nonlinear initial value problems with constant system coefficients by the new modified recursion scheme in order to derive a systematic algorithm for the general case of higher‐order inhomogeneous nonlinear initial value problems.

The authors investigate seven expository examples of inhomogeneous nonlinear initial value problems: the exact solution was known in advance, in order to demonstrate the rapid convergence of the new approach, including first‐ through sixth‐order derivatives and quadratic, cubic, quartic and exponential nonlinear terms in the solution and a sextic nonlinearity in the first‐order derivative. The key difference between the various modified recursion schemes is the choice of the initial solution component, using different choices to partition and delay the subsequent parts through the recursion steps. The authors' new approach extends this concept.

The new modified decomposition method provides a significant advantage for computing the solution's Taylor expansion series, both systematically and rapidly, as demonstrated in the various expository examples.