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.

Synopsis

In many fields of mathematical application, progress is dependent upon the flow of information between theoretical mathematicians looking for applications, mathematicians in need of theory, and scientists and engineers applying mathematical models and methods. The intention of this book is to stimulate sources of information regarding integral equations. In the first three chapters, applications of Abel integral equations are surveyed, including determination of potentials, stereology, seismic travel times, spectroscopy and optical fibres. Subsequent chapters, which require some background in functional analysis, map properties of Abel integral operators and their relation to other integral transforms in various function spaces. Numerical methods are described, and issues concerning the existence and uniqueness of solutions of linear and nonlinear Abel integral equations are explored.

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