Abel Integral Equations

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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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.

• Abelsche Funktion
• Analysis
• Integral equation
• Integral equations
• Linear and nonlinear Abel integral equations
• Potential
• ill-posed problems
• numerical analysis
• numerical methods
• operator
• smoothing properties

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