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UWO ORCCA TR-08-03 Summary

A Survey on Solution Methods for Integral Equations
Ilias S. Kotsireas

Abstract: Integral Equations arise naturally in applications, in many areas of Mathematics, Science and Technology and have been studied extensively both at the theoretical and practical level. It is noteworthy that a MathSciNet keyword search on Integral Equations returns more than eleven thousand items. In this survey we plan to describe several solution methods for Integral Equations, illustrated with a number of fully worked out examples. In addition, we provide a bibliography, for the reader who would be interested in learning more about various theoretical and computational aspects of Integral Equations.

Our interest in Integral Equations stems from the fact that understanding and implementing solution methods for Integral Equations is of vital importance in designing efficient param- eterization algorithms for algebraic curves, surfaces and hypersurfaces. This is because the implicitization algorithm for algebraic curves, surfaces and hypersurfaces based on a nullvector computation, can be inverted to yield a parameterization algorithm.

Integral Equations are inextricably related with other areas of Mathematics, such as Integral Transforms, Functional Analysis and so forth. In view of this fact, and because we made a conscious effort to limit the length in under 50 pages, it was inevitable that the present work is not self-contained.

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