A Survey of Regularization Methods for First-Kind Volterra Equations

Abstract

We survey continuous and discrete regularization methods for first-kind Volterra problems with continuous kernels. Classical regu-larization 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 regular-ization 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.