Lower bounds and approximations of the locations of movable singularities of some nonlinear differential equations using parameterized bounded operators

摘要

In this paper, we consider methods for easily computing lower bounds for the locations 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 operators. 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 singularities. 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 singularities is discussed by Tourigny and Grinfeld [Y. Tourigny, M. Grinfeld, Deciphering singularities by discrete methods, Math. Comput. 62 (205) (1994) 155–169], who used methods based upon Taylor series coefficients.