Application of the decomposition method to identify the distributed parameters of an elliptical equation
\textit{G. Adomian} [Math. Comput. Modelling 13, No. 7, 17-43 (1990; Zbl 0713.65051)] has proposed to solve a nonlinear equation \(u= G(u)+ f\) (\(G\) operator from Hilbert space \(H\) to \(H\)) through an infinite series \(u= \sum u_ n\) with \(u_ 0= f\), \(u_{n+ 1}= A(u_ 0, u_ 1, \dots, u_ n)\), \(n\geq 0\), \(A\) polynomial. The authors show how the method yields the solution (approximated by truncation of the series \(u\)) of a special second-order elliptic boundary value problem in the unit square and a related inverse problem (determination of a coefficient in the equation). The equation is special, indeed, however, the efficiency is surprising. The authors hope to apply this technique to identification problems in biology and medicine, see e.g. their work in Math. Comput. Modelling 18, No. 8, 77-82 (1993; Zbl 0792.92009).
second-order elliptic boundary value problem
scientific article; zbMATH DE number 751285