Recursive analysis of singular ordinary differential equations

We investigate systems of ordinary differential equations with a parameter. We show that under suitable assumptions on the systems the solutions are computable in the sense of recursive analysis. As an application we give a complete characterization of the recursively enumerable sets using Fourier coefficients of recursive analytic functions that are generated by differential equations and elementary operations. (C) 2010 Elsevier B.V. All rights reserved

General theory for integer-type algorithm for higher order differential equations

Based on functional analysis, we propose an algorithm for finite-norm solutions of higher-order linear Fuchsian-type ordinary differential equations (ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only the four arithmetical operations on integers. This algorithm is based on a band-diagonal matrix representation of the differential operator P(x,d/dx), though it is quite different from the usual Galerkin methods. This representation is made for the respective CONSs of the input Hilbert space H and the output Hilbert space H' of P(x,d/dx). This band-diagonal matrix enables the construction of a recursive algorithm for solving the ODE. However, a solution of the simultaneous linear equations represented by this matrix does not necessarily correspond to the true solution of ODE. We show that when this solution is an l^2 sequence, it corresponds to the true solution of ODE. We invent a method based on an integer-type algorithm for extracting only l^2 components. Further, the concrete choice of Hilbert spaces H and H' is also given for our algorithm when p_m is a polynomial or a rational function with rational coefficients. We check how our algorithm works based on several numerical demonstrations related to special functions, where the results show that the accuracy of our method is extremely high.Comment: Errors concerning numbering of figures are fixe

Solution of reduced equations derived with singular perturbation methods

For singular perturbation problems in dynamical systems, various appropriate singular perturbation methods have been proposed to eliminate secular terms appearing in the naive expansion. For example, the method of multiple time scales, the normal form method, center manifold theory, the renormalization group method are well known. In this paper, it is shown that all of the solutions of the reduced equations constructed with those methods are exactly equal to sum of the most divergent secular terms appearing in the naive expansion. For the proof, a method to construct a perturbation solution which differs from the conventional one is presented, where we make use of the theory of Lie symmetry group.Comment: To be published in Phys. Rev.

NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions

This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast high-precision numerical evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our descriptions contain improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun

N=1 SUSY Conformal Block Recursive Relations

We present explicit recursive relations for the four-point superconformal block functions that are essentially particular contributions of the given conformal class to the four-point correlation function. The approach is based on the analytic properties of the superconformal blocks as functions of the conformal dimensions and the central charge of the superconformal algebra. The results are compared with the explicit analytic expressions obtained for special parameter values corresponding to the truncated operator product expansion. These recursive relations are an efficient tool for numerically studying the four-point correlation function in Super Conformal Field Theory in the framework of the bootstrap approach, similar to that in the case of the purely conformal symmetry.Comment: 12 pages, typos corrected, reference adde