The theoretical foundation of the Adomian method

Adomian’s decomposition method and homotopy perturbation method in solving nonlinear equations

“…Hence the convergence of ∞ n=2x n in the method HPM is just the convergence of ∞ n=2 A n in the method ADM, and the results already obtained in [1][2][3][10][11][12]15] are suitable for the HPM method. Secondly, the approximate solution of (1) obtained by applying ADM can also be obtained by applying HPM, and vice versa, although each method has its own merits.…”

Blow-up solutions obtained using the decomposition method

“…They obtained some results about the speed of convergence of this method for solving nonlinear functional equations. However, at present there is no general proof for the convergence of the series (7) [28], although particular attempts have been made, [10,[29][30][31].…”

“…Straightforward computer codes in a symbolic programming language such as MAPLE or MATHEMATICA can be written reducing considerably the computing memory and time. The method has common features and has been compared with other methods such as successive approximations (Picard iterations) [9,10], Taylor series [11], homotopy [12], finite differences [13], and sinc-Galerkin [14], but it is distinctly different for nonlinear equations on close examination, and one should not be mislead by its apparent simplicity into superficial conclusions [15,16].…”

An efficient algorithm for the multivariable Adomian polynomials

“…The convergence of the method was previously studied in [18][19][20][21][22]. Different algorithms for the Adomian polynomials have been considered for the sake of improving computational efficiency in [1][2][3][4][5][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].…”