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

This article reviews convolution quadrature and its uses, extends the known approximation results for the case of sectorial Laplace transforms to finite-part convolutions with non-integrable kernel, and gives new, unified proofs of the optimal error bounds for both locally integrable and non-integrable convolution kernels.

Convolution Quadrature Revisited

A perspective on the numerical treatment of Volterra equations

New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets

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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J.-P. Kauthen

Papers

A survey of singularly perturbed Volterra equations

Implicit Runge-Kutta methods for singularly perturbed integro-differential systems