Lower bounds and approximations of the locations of movable singularities of some nonlinear differential equations using parameterized bounded operators Steven G. From Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182-0243, United States Abstract In this paper, we consider methods for easily computing lower bounds for the loca- tions of movable singularities of certain nonlinear differential equations. The types of singularities include poles, movable branch points, other types of vertical asymptotes, and derivative blow-ups. Most methods use the idea of parametrized bounded opera- tors. These lower bounds can then be used within a numerical procedure, such as Runge–Kutta (4, 4) algorithm, to approximate the locations of these movable singular- ities. This paper extends the work of Eliason [S.B. Eliason, Vertical asymptotes and bounds for certain solutions of a class of second order differential equations, SIAM J. Math. Anal. 3 (3) (1972) 474–484], Bobisud [L.E. Bobisud, The distance to vertical asymptotes for solutions of second order differential equations, Mich. Math. J. 19 (1972) 277–283] and From [S.G. From, Bounds for asymptote singularities of certain nonlinear differential equations, submitted for publication] to more general and higher order nonlinear differential equations. The importance of methods for locating singular- ities is discussed by Tourigny and Grinfeld [Y. Tourigny, M. Grinfeld, Deciphering 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.07.040 E-mail address: sfrom@mail.unomaha.edu Applied Mathematics and Computation 175 (2006) 16–37 www.elsevier.com/locate/amc