Inversion of First Kind Volterra Equations: Back to Direct Methods?

Abstract

Abstract Numerical inversion of the first kind Volterra equation (the Abel inversion included) has been extensively studied. Direct methods were probably the first methods used to attempt the inversion. Together with the computer hardware evolution, new methods were devised in order to deal with the inherent problem of this kind of equation, that is, error magnification. Using a large number of data points (several thousands) most methods are difficult to use, specially when the inversion and its error are required on line, that is, while performing the experiments. Further, error propagation (coming from the input data and from the parameters of the problem) is, usually, a difficult task and has not been extensively studied. On the other hand, direct methods together with an adequate filter give good resolution, are fast, and error propagation is easily performed. In this work we used the so called Matrix Method for inverting three different equations, showing how to build the resolvent nucleus and how errors propagate through the solution.