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Equations of Mathematical Physics

Generalized Functions and Historical Notes

  • Book
  • © 2023

Overview

Authors:
  1. A. S. Demidov

    1. Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

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  • Theory of pseudodifferential operators & Sobolev spaces to the inverse magneto/electroencephalography problem
  • Presents derivation of generalized Sobolev functions & L. Schwartz distributions
  • Suitable for both novice & advanced readers

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About this book

This concise volume presents an overview of equations of mathematical physics and generalized functions. While intended for advanced readers, the accessible introduction and text structure allows beginners to study at their own pace as the material gradually increases in difficulty. The text introduces the concept of generalized Sobolev functions and L. Schwartz distributions briefly in the opening section, gradually approaching a more in-depth study of the “generalized” differential equation (also known as integral equality). In contrast to the traditional presentation of generalized Sobolev functions and L. Schwartz distributions, this volume derives the topology from two natural requirements (which are equivalent to it). The text applies the same approach to the theory of the canonical Maslov operator. It also features illustrative drawings and helpful supplementary reading in the footnotes concerning historical and bibliographic information related to the subject of the book. Additionally, the book devotes a special chapter to the application of the theory of pseudodifferential operators and Sobolev spaces to the inverse magneto/electroencephalography problem.

Explicit numerically realizable formulas related to the Cauchy problem for elliptic equations (including quasilinear ones) and also to the Poincaré--Steklov operators are presented. The book is completed by three additions, which were written by famous mathematicians Yu. V. Egorov, A. B. Antonevich, and S. N. Samborski.

Keywords

Table of contents (3 chapters)

  1. Front Matter

    Pages i-xv
    Download chapter PDF

  2. Chapter 1:Introduction to Problems of Mathematical Physics

    • A. S. Demidov
    Pages 1-69

  3. Chapter 2: The Spaces \(\mathcal D^\flat \), \(\mathcal D^\sharp \) and \(\mathcal D'\): Elements of the Distribution Theory (Generalized Functions in the Sense of Schwartz)

    • A. S. Demidov
    Pages 71-90

  4. Chapter 3: Pseudo-Differential Operators and Fourier Operators

    • A. S. Demidov
    Pages 91-192

  5. Back Matter

    Pages 193-248
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Authors and Affiliations

  • Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    A. S. Demidov

Bibliographic Information

  • Book Title: Equations of Mathematical Physics

  • Book Subtitle: Generalized Functions and Historical Notes

  • Authors: A. S. Demidov

  • DOI: https://doi.org/10.1007/978-3-031-30358-6

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Hardcover ISBN: 978-3-031-30357-9Published: 28 June 2023

  • Softcover ISBN: 978-3-031-30360-9Published: 29 June 2024

  • eBook ISBN: 978-3-031-30358-6Published: 27 June 2023

  • Edition Number: 1

  • Number of Pages: XV, 248

  • Number of Illustrations: 1 b/w illustrations, 2 illustrations in colour

  • Topics: Functional Analysis, Mathematical Physics

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