The numerical treatment of two-point singular boundary value problems has always been a difficult and challenging task due to the singularity behaviour that occurs at a point. Various efficient numerical methods have been proposed to deal with such boundary value problems. We present a new efficient modification of the Adomian decomposition method for solving singular boundary value problems, both linear and nonlinear. Numerical examples illustrate the efficiency and accuracy of the proposed method.

Let ?(x) : ?x/2, ?2/3, ?3/4, ?4/5, ... be a sequence with a real variable x > 0 and let W?(x) be the associated weighted shift with weight sequence ?(x). In [17], Exner-Jung-Park provided an algorithm to distinguish weak k-hyponormality and k-hyponormality of weighted shift W?(x), and obtained sn > 0 for some low numbers n = 4,..., 10, such that W?(sn) is weakly n-hyponormal but not n-hyponormal. In this paper, we obtain a formula of sn (for all positive integer n) such that W?(sn) is weakly n-hyponormal but not n-hyponormal, which improves Exner-Jung-Park?s result above.