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

Abstract

We compare and discuss the respective efficiency of three methods (with two variants for each of them), based respectively on Taylor (Maclaurin) series, Padé approximants and conformal mappings, for solving quasi-analytically a two-point boundary value problem of a nonlinear ordinary differential equation (ODE). Six configurations of ODE and boundary conditions are successively considered according to the increasing difficulties that they present. After having indicated that the Taylor series method almost always requires the recourse to analytical continuation procedures to be efficient, we use the complementarity of the two remaining methods (Padé and conformal mapping) to illustrate their respective advantages and limitations. We emphasize the importance of the existence of solutions with movable singularities for the efficiency of the methods, particularly for the so-called Padé-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 sought and show how this distribution controls the efficiency of the two methods. In general the method based on Padé approximants is easy to use and robust but may be awkward in some circumstances whereas the conformal mapping method is a very fine method which should be used when high accuracy is required.

Final version

References in corpus (7)

• Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation
• Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings
• Rational Approximation for Two-Point Boundary value problems
• An analytical approximation scheme to two point boundary value problems of ordinary differential equations
• Solving the Anharmonic Oscillator: Tuning the Boundary Condition
• A Modified Borel Summation Technique
• Rational approximation to the Thomas--Fermi equation

Cited by in corpus (13)

• A proper fixed functional for four-dimensional Quantum Einstein Gravity
• Status of the differential transformation method
• Global Wilson-Fisher fixed points
• Critical $O(N)$ models in the complex field plane
• On the Kármán momentum-integral approach and the Pohlhausen paradox
• Revisiting the local potential approximation of the exact renormalization group equation
• A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions
• Highly accurate potential energy curves for the hydrogen molecule ion
• On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs
• Using Hermite Function for Solving Thomas-Fermi Equation
• A comparison between pre-Newton and post-Newton approaches for solving a physical singular second-order boundary problem in the semi-infinite interval
• A stability preserved time-integration method for nonlinear advection-diffusion models
• Accurate calculation of the solutions to the Thomas-Fermi equations