Abel Integral Equations: Analysis and Applications

Rudolf Gorenflo, Sergio Vessella354053668X, 9783540536680

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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Table of contents :
front-matter……Page 1
1Introduction……Page 8
2Basic theory and representation formulas……Page 15
3Applications of Abel’s original integral equation Determination of potentials……Page 33
4Applications of a transformed abel integral equation……Page 42
5Smoothing properties of the abel operators……Page 71
6Existence and uniqueness theorems……Page 90
7Relations between abel transform and other integral transforms……Page 102
8Nonlinear abel integral equations of second kind……Page 136
9Illposedness and stabilization of linear abel integral equations of first kind……Page 161
9On numerical treatment of first kind abel integral equations……Page 189
back-matter……Page 202