This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations (VIEs), ranging from Volterra's fundamental contributions and the resulting classical theory to more recent developments that include Volterra functional integral equations with various kinds of delays, VIEs with highly oscillatory kernels, and VIEs with non-compact operators. It will act as a 'stepping stone' to the literature on the advanced theory of VIEs, bringing the reader to the current state of the art in the theory. Each chapter contains a large number of exercises, extending from routine problems illustrating or complementing the theory to challenging open research problems. The increasingly important role of VIEs in the mathematical modelling of phenomena where memory effects play a key role is illustrated with some 30 concrete examples, and the notes at the end of each chapter feature complementary references as a guide to further reading.

Volterra Integral Equations: An Introduction to Theory and Applications

We survey continuous and discrete regularization methods for first-kind Volterra problems with continuous kernels Classical regularization methods tend to destroy the non-anticipatory (or causal) nature of the original Volterra problem because such methods typically rely on computation of the Volterra adjoint operator, an anticipatory operator In this survey we pay special attention to particular regularization methods, both classical and nontraditional, which tend to retain the Volterra structure of the original problem Our attention will primarily be focused on linear problems, although extensions of methods to nonlinear and integro-operator Volterra equations are mentioned when known

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A Survey of Regularization Methods for First-Kind Volterra Equations

Toughening by partial or full bridging of cracks in ceramics and fiber reinforced composites

A survey of singularly perturbed Volterra equations

The method makes use of the asymptotic matching principle of Van Dyke. Its application is mechanical and it appears to allow, in principle, the construction of asymptotic solutions to any order

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Singular perturbation solution of a class of singular integral equations

A formal methodology is developed to obtain a uniformly valid asymptotic solution to a class of singularly perturbed Volterra integral equations. The methodology systematically determines the appropriate integral equations which govern the inner and outer solutions at each order of the perturbation analysis. Several examples are presented to illustrate the procedure.

J. S. Angell

Papers

Singularly perturbed Volterra integral equations II