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A Toolbox of Averaging Theorems

Ordinary and Partial Differential Equations

  • Book
  • © 2023

Overview

Authors:
  1. Ferdinand Verhulst

    1. Mathematical Institute, University of Utrecht, Utrecht, The Netherlands

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  • Includes numerous hands-on applications of the stated theorems in each chapter
  • The van der Pol, Duffing and Mathieu equations are discussed in detail
  • Provides scientists and engineers with a concise survey of averaging theorems for ODE's and PDE's

Part of the book series: Surveys and Tutorials in the Applied Mathematical Sciences (STAMS, volume 12)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Introduction

    • Ferdinand Verhulst
    Pages 1-17

  3. First Order Periodic Averaging

    • Ferdinand Verhulst
    Pages 19-34

  4. Periodic Solutions

    • Ferdinand Verhulst
    Pages 35-53

  5. Second Order Periodic Averaging

    • Ferdinand Verhulst
    Pages 55-64

  6. First Order General Averaging

    • Ferdinand Verhulst
    Pages 65-78

  7. Approximations on Timescales Longer than \(1/ \varepsilon \)

    • Ferdinand Verhulst
    Pages 79-88

  8. Averaging over Spatial Variables

    • Ferdinand Verhulst
    Pages 89-105

  9. Hamiltonian Resonances

    • Ferdinand Verhulst
    Pages 107-136

  10. Quasi-Periodic Solutions and Tori

    • Ferdinand Verhulst
    Pages 137-161

  11. Averaging for Partial Differential Equations

    • Ferdinand Verhulst
    Pages 163-185

  12. Back Matter

    Pages 187-193
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Keywords

About this book

This primer on averaging theorems provides a practical toolbox for applied mathematicians, physicists, and engineers seeking to apply the well-known mathematical theory to real-world problems. With a focus on practical applications, the book introduces new approaches to dissipative and Hamiltonian resonances and approximations on timescales longer than 1/ε.

Accessible and clearly written, the book includes numerous examples ranging from elementary to complex, making it an excellent basic reference for anyone interested in the subject. The prerequisites have been kept to a minimum, requiring only a working knowledge of calculus and ordinary and partial differential equations (ODEs and PDEs).


In addition to serving as a valuable reference for practitioners, the book could also be used as a reading guide for a mathematics seminar on averaging methods. Whether you're an engineer, scientist, or mathematician, this book offers a wealth of practicaltools and theoretical insights to help you tackle a range of mathematical problems.

Authors and Affiliations

  • Mathematical Institute, University of Utrecht, Utrecht, The Netherlands

    Ferdinand Verhulst

About the author

Ferdinand Verhulst was born in Amsterdam and studied astrophysicist with a specialisation in numerical analysis. He spent five years at the Technical University of Delft, which stimulated his interest in theoretical engineering. Verhulst obtained his PhD from the University of Utrecht and became a Professor of Dynamical Systems there in 1990. He has been a visiting professor in prestigious universities worldwide and has completed five trips to Indonesia to teach mathematics, leading to three successful PhDs. Verhulst has authored numerous research papers and books, including five in English and two in Dutch, as well as edited two Springer Lecture Notes publications. Additionally, he founded Epsilon Uitgaven, a publishing house aimed at supporting teachers and students with Dutch language mathematics books, eventually contributing himself to a couple of their books, one on mathematical proofs and the other on chaos, both aimed at secondary school students.

Bibliographic Information

  • Book Title: A Toolbox of Averaging Theorems

  • Book Subtitle: Ordinary and Partial Differential Equations

  • Authors: Ferdinand Verhulst

  • Series Title: Surveys and Tutorials in the Applied Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-3-031-34515-9

  • 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

  • Softcover ISBN: 978-3-031-34514-2Published: 23 July 2023

  • eBook ISBN: 978-3-031-34515-9Published: 22 July 2023

  • Series ISSN: 2199-4765

  • Series E-ISSN: 2199-4773

  • Edition Number: 1

  • Number of Pages: X, 193

  • Number of Illustrations: 9 b/w illustrations, 30 illustrations in colour

  • Topics: Analysis, Engineering Mathematics

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