Anwendung der Methode von Šapkarev zum Auflosen Einige Randwertaufgaben für die Lineare Komplexe Differentialgleichung II Ordnung
Auszug aus einem Schreiben des Herrn L. Fuchs an C. W. Borchardt.
Convergence of formal solutions of first order singular partial differential equations of nilpotent type
Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: , where . For this equation, our aim is to characterize the solvability on by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.
Elementary Solution of Vecua Equation With Analytic Coefficients on Z,
Euler's integral transformation for systems of linear differential equations with irregular singularities
Dettweiler and Reiter formulated Euler's integral transformation for Fuchsian systems of differential equations and applied it to a definition of the middle convolution. In this paper, we formulate Euler's integral transformation for systems of linear differential equations with irregular singularities. We show by an example that the confluence of singularities is compatible with Euler's integral transformation.
Intégration, sous forme finie, de trois espèces d'équations différentielles linéaires, à coefficients variables.
Multisummability and ordinary meromorphic differential equations
In this expository paper we consider various approaches to multisummability. We apply it to nonlinear ODE's and give a somewhat modified proof of multisummability of formal solutions of ODE's with levels 1 and 2 via Écalle's method involving convolution equations.
On q-asymptotics for q-difference-differential equations with Fuchsian and irregular singularities
This work is devoted to the study of a Cauchy problem for a certain family of q-difference-differential equations having Fuchsian and irregular singularities. For given formal initial conditions, we first prove the existence of a unique formal power series X̂(t,z) solving the problem. Under appropriate conditions, q-Borel and q-Laplace techniques (firstly developed by J.-P. Ramis and C. Zhang) help us in order to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion...
On the exact WKB analysis of microdifferential operators of WKB type
We first introduce the notion of microdifferential operators of WKB type and then develop their exact WKB analysis using microlocal analysis; a recursive way of constructing a WKB solution for such an operator is given through the symbol calculus of microdifferential operators, and their local structure near their turning points is discussed by a Weierstrass-type division theorem for such operators. A detailed study of the Berk-Book equation is given in Appendix.
Polynomial solutions to the Hele--Shaw problem.
Resurgence in a Hamilton-Jacobi equation
We study the resurgent structure associated with a Hamilton-Jacobi equation. This equation is obtained as the inner equation when studying the separatrix splitting problem for a perturbed pendulum via complex matching. We derive the Bridge equation, which encompasses infinitely many resurgent relations satisfied by the formal solution and the other components of the formal integral.
Resurgence relations for classes of differential and difference equations
Résurgence-sommabilité de séries formelles ramifiées dépendant d’un paramètre et solutions d’équations différentielles linéaires
Dans cet article, nous établissons le caractère résurgent-sommable de séries formelles ramifiées solutions d’une classe d’équations différentielles linéaires. Nous analysons d’une part le problème de la dépendance analytique des sommes de Borel de telles séries par rapport aux paramètres de cette classe d’équations différentielles linéaires d’ordre deux, et d’autre part, nous analysons la structure résurgente complète associée à ces séries formelles via l’outil des singularités générales (ou microfonctions)....
Riemann-Hilbert problem and solvability of differential equations.
Semi-formal theory and Stokes' phenomenon of non-linear meromorphic systems of ordinary differential equations
This article continues earlier work of the author on non-linear systems of ordinary differential equations, published in Asymptotic Analysis 15 (1997), MR no. 98g:34015b. There, a completely formal theory was presented, while here we are concerned with a semi-formal approach: Solutions of non-linear systems of ordinary meromorphic differential equations are represented as, in general divergent, power series in several free parameters. The coefficients, aside from an exponential polynomial, a general...
Some addition to the generalized Riemann-Hilbert problem
We consider the generalized Riemann-Hilbert problem for linear differential equations with irregular singularities. If one weakens the conditions by allowing one of the Poincaré ranks to be non-minimal, the problem is known to have a solution. In this article we give a bound for the possibly non-minimal Poincaré rank. We also give a bound for the number of apparent singularities of a scalar equation with prescribed generalized monodromy data.
Ueber die Funktionen, welche durch Umkehrung der Integrale von Lösungen der linearen Differentialgleichungen entstehen
Ueber eine Klasse von Funktionen mehrerer Variablen, welche durch Umkehrung der Integrale von Lösungen der linearen Differenzialgleichungen mit rationalen Coeffizienten entstehen