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Abel Integral Equations

Analysis and Applications

  • Book
  • © 1991

Overview

Authors:
  1. Rudolf Gorenflo
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    You can also search for this author in PubMed   Google Scholar

  2. Sergio Vessella
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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Lecture Notes in Mathematics (LNM, volume 1461)

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Table of contents (10 chapters)

  1. Front Matter

    Pages I-VII
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  2. Introduction

    • Rudolf Gorenflo, Sergio Vessella
    Pages 1-7

  3. Basic theory and representation formulas

    • Rudolf Gorenflo, Sergio Vessella
    Pages 8-25

  4. Applications of Abel's original integral equation: Determination of potentials

    • Rudolf Gorenflo, Sergio Vessella
    Pages 26-34

  5. Applications of a transformed abel integral equation

    • Rudolf Gorenflo, Sergio Vessella
    Pages 35-63

  6. Smoothing properties of the abel operators

    • Rudolf Gorenflo, Sergio Vessella
    Pages 64-82

  7. Existence and uniqueness theorems

    • Rudolf Gorenflo, Sergio Vessella
    Pages 83-94

  8. Relations between abel transform and other integral transforms

    • Rudolf Gorenflo, Sergio Vessella
    Pages 95-128

  9. Nonlinear abel integral equations of second kind

    • Rudolf Gorenflo, Sergio Vessella
    Pages 129-153

  10. Illposedness and stabilization of linear abel integral equations of first kind

    • Rudolf Gorenflo, Sergio Vessella
    Pages 154-181

  11. On numerical treatment of first kind abel integral equations

    • Rudolf Gorenflo, Sergio Vessella
    Pages 182-194

  12. Back Matter

    Pages 195-217
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Keywords

About this book

In many fields of application of mathematics, progress is crucially dependent on the good flow of information between (i) theoretical mathematicians looking for applications, (ii) mathematicians working in applications in need of theory, and (iii) scientists and engineers applying mathematical models and methods. The intention of this book is to stimulate this flow of information. In the first three chapters (accessible to third year students of mathematics and physics and to mathematically interested engineers) applications of Abel integral equations are surveyed broadly including determination of potentials, stereology, seismic travel times, spectroscopy, optical fibres. In subsequent chapters (requiring some background in functional analysis) mapping properties of Abel integral operators and their relation to other integral transforms in various function spaces are investi- gated, questions of existence and uniqueness of solutions of linear and nonlinear Abel integral equations are treated, and for equations of the first kind problems of ill-posedness are discussed. Finally, some numerical methods are described. In the theoretical parts, emphasis is put on the aspects relevant to applications.

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