Asymptotic Expansions and Summability: Application to Partial Differential Equations

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Pascal Remy

Asymptotic Expansions and Summability

Application to Partial Differential Equations

Pascal Remy

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English

Springer International Publishing AG
02 July 2024

Differential calculus & equations; Mathematical physics

Series: Lecture Notes in Mathematics

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.

By: Pascal Remy
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 2024 ed.
Volume: 2351
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9783031590931
ISBN 10: 3031590937
Series: Lecture Notes in Mathematics
Pages: 246
Publication Date: 02 July 2024
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

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.