Asymptotic Expansions and Summability: Application to Partial Differential Equations

by Pascal Remy

Overview

This book provides a comprehensive exploration of the theory of summability of formal power series with analytic coefficients at the origin of Cn, aiming to apply it to formal solutions of partial differential equations (PDEs). It offers three characterizations of summability and discusses their applications to PDEs, which play a pivotal role in understanding physical, chemical, biological, and ecological phenomena.

Determining exact solutions and analyzing properties such as dynamic and asymptotic behavior are major challenges in this field. The book compares various summability approaches and presents simple applications to PDEs, introducing theoretical tools such as Nagumo norms, Newton polygon, and combinatorial methods. Additionally, it presents moment PDEs, offering a broad class of functional equations including classical, fractional, and q-difference equations. With detailed examples and references, the book caters to readers familiar with the topics seeking proofs or deeper understanding, as well as newcomers looking for comprehensive tools to grasp the subject matter. Whether readers are seeking precise references or aiming to deepen their knowledge, this book provides the necessary tools to understand the complexities of summability theory and its applications to PDEs.

Product Details

ISBN-13:	9783031590931
Publisher:	Springer Nature Switzerland
Publication date:	07/02/2024
Series:	Lecture Notes in Mathematics , #2351
Edition description:	2024
Pages:	246
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Pascal Remy is a research associate at the Laboratoire de Mathématiques de Versailles, at the University of Versailles Saint-Quentin (France). His main interest is the theory of summation of divergent formal power series (including Gevrey estimates, summability, multi-summability, and Stokes phenomenon). His research extends to applications such as formal solutions of meromorphic linear differential equations, partial differential equations and integro-differential equations, both linear and nonlinear.

Table of Contents

- Part I Asymptotic expansions.- Taylor expansions.- Gevrey formal power series.- Gevrey asymptotics.- Part II Summability.- k-summability: definition and first algebraic properties.- First characterization of the k-summability: the successive derivatives.- Second characterization of the k-summability: the Borel-Laplace method.- Part III Moment summability.- Moment functions and moment operators.- Moment-Borel-Laplace method and summability.- Linear moment partial differential equations.

Editorial Reviews

This work is a valuable source for both researchers in the field and those interested in understanding the transition from formal power series solutions to sectorial, analytic ones for some wide families of PDEs and more general equations.” (Javier Sanz, Mathematical Reviews, June, 2025)

“The text are illustrated with many applications to concrete partial differential equations such as the heat equation, the Burgers equation, etc. and also to applications in physics. Moreover, the author provides references of the main results and applications for a deepened reading.” (Alberto Lastra, zbMATH 1555.35003, 2025)

From the Publisher