Asymptotic expansions and summability : application to partial differential equations / Pascal Remy.

Format:

Book

Author/Creator:

Remy, Pascal, author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 2351.

Lecture notes in mathematics ; 2351

Language:

English

Subjects (All):

Mathematical analysis.

Mathematical physics.

Physical Description:

xiii, 244 pages ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer, [2024]

Summary:

"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."-- Provided by publisher.

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

Part IV. Appendices. Some related equations

Naguno norms

Generalized binomial and multinomial coefficients

Mittag-Leffler's function.

Notes:

Includes bibliographical references (pages 237-242) and index.

ISBN:

3031590937

9783031590931

OCLC:

1427940482