Abel Integral Equations Analysis and Applications
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1461
Rudolf Gorenflo Sergio Vessella
Abel Integral Equations Analysis and Applications
Springer-Verlag Berlin Heidelberg New York London ParisTokyo Hong Kong Barcelona
Lecture Notes in Mathematics Edited by A. Oold, B. Eckmann and F. Takens
1461
Rudolf Gorenflo Sergio Vessella
Abel Integral Equations Analysis and Applications
Springer-Verlag Berlin Heidelberg New York London ParisTokyo Hong Kong Barcelona
Authors
Rudolf Gorenflo Fachbereich Mathematik Freie Universitat Berlin Arnimallee 2-6 1000 Berlin 33, Federal Republic of Germany Sergio Vessella Facolta di Ingegneria Universita di Salerno 84100 Salerno, Italy
Mathematics Subject Classification (1980): 45E10, 45005, 44A15, 65R20 ISBN 3-540-53668-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-53668-X Springer-Verlag New York Berlin Heidelberg
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Preface Abel's integral equation, one of the very first integral equations seriously studied, and the corresponding integral operator (investigated by Niels Henrik Abel in 1823 and by Liouville in 1832 as a fractional power of the operator of anti-derivation) have never ceased to inspire mathematicians to investigate and to generalize them. Abel was led to his equation by a problem of mechanics, the tautochrone problem. However, his equation and slight or not so slight variants of it have in the meantime found applications in such diverse fields (let us mention a few from outside of mathematics, arisen in our century) as inversion of seismic travel times, stereology of spherical particles, spectroscopy of gas discharges (more generally: " tomography" of cylindrically or spherically symmetric objects like e.g. globular clusters of stars), and determination of the refractive index of optical fibres. More pertinent to mathematics think of particular (inverse) problems in partial differential equations (e.g. heat conduction, Tricomi's equation, potential theory, theory of elasticity - we recommend here the books of Bitsadze and of Sneddon) and of special problems in the theory of Brownian motion. Of course, these variants of Abel's original equation comprise linear and nonlinear equations, equations of first and of second kind, systems of equations, and the widest generalizations consist in simply retaining in the kernel of the integral equation its integrability in the sense that this ker
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