相关论文: On the convergence of formal power-log series solu…
We propose a sufficient condition of the convergence of a generalized power series formally satisfying an algebraic (polynomial) ordinary differential equation. The proof is based on the majorant method.
A sufficient condition of the convergence of an exotic formal series (a kind of power series with complex exponents) solution to an ODE of a general form is proposed.
We propose a sufficient condition of the convergence of a Dulac series formally satisfying an algebraic ordinary differential equation (ODE). Such formal solutions of algebraic ODEs appear rather often, in particular, the third, fifth, and…
We propose a sufficient condition of the convergence of a complex power type formal series of the form $\varphi=\sum_{k=1}^{\infty}\alpha_k(x^{{\rm i}\gamma})\,x^k$, where $\alpha_k$ are functions meromorphic at the origin and…
Given an algebraic ordinary differential equation (AODE), we propose a computational method to determine when a truncated power series can be extended to a formal power series solution. If a certain regularity condition on the given AODE or…
We study the existence of formal power series solutions to q-algebraic equations. When a solution exists, we give a sufficient condition on the equation for this solution to have a positive radius of convergence. We emphasize on the case…
A sufficient condition for the convergence of a generalized formal power series solution to an algebraic $q$-difference equation is provided. The main result leans on a geometric property related to the semi-group of (complex) power…
We propose an analytic proof of the Malgrange-Sibuya theorem concerning a sufficient condition of the convergence of a formal power series satisfying an ordinary differential equation. The proof is based on the majorant method and allows to…
Given a first-order autonomous algebraic ordinary differential equation, we present a method for computing formal power series solutions by means of places. We provide an algorithm for computing a full characterization of possible initial…
A Maillet-Malgrange type theorem is proved for a Dulac series (in the general case, with complex exponents), which formally satisfies an analytical ordinary differential equation (ODE). This theorem allows to estimate the growth of the…
Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions…
We generalize results of Rosenlicht to give a necessary and sufficient condition for when order one differential equations of the form $D(x) = f(x)$ where $f$ is a rational function is orthogonal to the constants. Following the main results…
Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expensive to compute. In this paper, we derive analytic…
A family of formal power series, such that its coefficients satisfy a recursion formula, is characterized in terms of the summability, in the sense of J. P. Ramis, of its elements along certain well chosen directions. We describe a set of…
The question of the convergence of generalized formal power series (with complex power exponents) solutions of $q$-difference equations is studied in the situation where the small divisors phenomenon arises; a sufficient condition of…
It is a classical fact that domains of convergence of power series of several complex variables are characterized as logarithmically convex complete Reinhardt domains; let $D \subsetneq \mathbb{C}^N$ be such a domain. We show that a…
In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of…
We discuss various compatibility criteria for overdetermined systems of PDEs generalizing the approach to formal integrability via brackets of differential operators. Then we give sufficient conditions that guarantee that a PDE possessing a…
We show a general method allowing the solution calculation, in the form of a power series, for a very large class of nonlinear Ordinary Differential Equations (ODEs), namely the real analytic $\sigma\pi$-ODEs (and, more in general, the real…
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…