Analytic continuations of Taylor series and the two-point boundary value problem of some nonlinear ordinary differential equations

S. Abbasbandy , C. Bervillier (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

We compare and discuss the respective efficiency of two methods, based respectively on Pad\'{e} approximants and on conformal mappings, for solving quasi-analytically a two-point boundary value problem of a nonlinear ordinary differential equation (ODE) with the help of Taylor (Maclaurin) series. Six configurations of ODE and boundary conditions are successively considered according to the increasing difficulties that they offer. We use the complementarity of the two methods to illustrate their respective advantages and limits. We emphasize the importance of the existence of solutions with movable singularities in the efficiency of the methods particularly for the so-called Pad\'{e}-Hankel method. (We show that this latter method is equivalent to pushing a movable pole to infinity.) For each configuration, we determine the singularity distribution (in the complex plane of the independent variable) of the solution looked for and show how this distribution controls the efficiency of the two methods. In general the method based on Pad\'{e} approximants is easy to use and robust but heavier than the conformal mapping method which is a very refined method and should be used when high accuracy is required.

Domaines: Physique des Hautes Energies - Théorie [hep-th]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

S. Abbasbandy, C. Bervillier. Analytic continuations of Taylor series and the two-point boundary value problem of some nonlinear ordinary differential equations. Applied Mathematics and Computation, 2011, 218, pp.2178. ⟨10.1016/j.amc.2011.07.035⟩. ⟨hal-00589924⟩