A modified Adomian's decomposition method (Q2723267)

In some domain \(\Omega\) the authors consider a nonlinear boundary value problem for the unknown functions \(\big\{u_i\big\}_{i=1}^n\) NEWLINE\[NEWLINE \begin{gathered} L_iu_i + R_i(u_1,\dots,u_n) + N_i(u_1,\dots,u_n) = g_i,\quad i=1,\dots,n,\\ L_i = \frac{\partial^{k_i}}{\partial\xi^{k_i}},\quad g_i = \sum_{j=0}^\infty g_{ij}\xi^j, \end{gathered}\tag{1} NEWLINE\]NEWLINE with the boundary conditions NEWLINE\[NEWLINE G_j(u_1,\dots,u_n) = \mu_j,\quad j=1,\dots,k,\quad k= k_1+\dots+k_n,\quad \mu_j = \text{const},\tag{2} NEWLINE\]NEWLINE on the boundary \(\partial\Omega\). NEWLINENEWLINENEWLINEIt is shown that the one-dimensional problems (1) and (2) may be solved efficiently by the application of Padé approximants or the method of perturbations. Numerical experiments are discussed.