41 research outputs found
Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables
We investigate Gevrey order and 1-summability properties of the formal
solution of a general heat equation in two variables. In particular, we give
necessary and sufficient conditions for the 1-summability of the solution in a
given direction. When restricted to the case of constants coefficients, these
conditions coincide with those given by D.A. Lutz, M. Miyake, R. Schaefke in a
1999 article, and we thus provide a new proof of their result.Comment: 16 page
Divergent series and differential equations
We develop the various known approaches to the summability of a class of series that contains all divergent series solutions of ordinary differential equations in the complex field. We first study the case when the divergence depends only on one parameter (the level k or critical time) called k-summability. We study then generalizations to the case when the divergence depends on several levels called multi-summability. We prove the coherence of the definitions and their equivalences and we provide some applications. We also provide the necessary basics on Gevrey asymptotics and a survey of sheaf theory, cohomology and linear ordinary differential equations. Various examples are worked on, including the example of tangent-to-identity germs of diffeomorphisms in the complex plane
Second-order linear differential equations with two irregular singular points of rank three: the characteristic exponent
For a second-order linear differential equation with two irregular singular
points of rank three, multiple Laplace-type contour integral solutions are
considered. An explicit formula in terms of the Stokes multipliers is derived
for the characteristic exponent of the multiplicative solutions. The Stokes
multipliers are represented by converging series with terms for which limit
formulas as well as more detailed asymptotic expansions are available. Here
certain new, recursively known coefficients enter, which are closely related to
but different from the coefficients of the formal solutions at one of the
irregular singular points of the differential equation. The coefficients of the
formal solutions then appear as finite sums over subsets of the new
coefficients. As a by-product, the leading exponential terms of the asymptotic
behaviour of the late coefficients of the formal solutions are given, and this
is a concrete example of the structural results obtained by Immink in a more
general setting. The formulas displayed in this paper are not of merely
theoretical interest, but they also are complete in the sense that they could
be (and have been) implemented for computing accurate numerical values of the
characteristic exponent, although the computational load is not small and
increases with the rank of the singular point under consideration.Comment: 33 page
Divergent series, summability and resurgence II: simple and multiple summability
Addressing the question how to “sum” a power series in one variable when it diverges, that is, how to attach to it analytic functions, the volume gives answers by presenting and comparing the various theories of k-summability and multisummability. These theories apply in particular to all solutions of ordinary differential equations. The volume includes applications, examples and revisits, from a cohomological point of view, the group of tangent-to-identity germs of diffeomorphisms of C studied in volume 1. With a view to applying the theories to solutions of differential equations, a detailed survey of linear ordinary differential equations is provided which includes Gevrey asymptotic expansions, Newton polygons, index theorems and Sibuya’s proof of the meromorphic classification theorem that characterizes the Stokes phenomenon for linear differential equations. This volume is the second of a series of three entitled Divergent Series, Summability and Resurgence. It is aimed at graduate students and researchers in mathematics and theoretical physics who are interested in divergent series, Although closely related to the other two volumes it can be read independently