3 - Volterra integro-differential equations with smooth kernels

Summary

In 1909 Volterra wrote (following the study of the modelling of hysteresis problems) that one is led ‘… ad equazioni che hanno tipo misto, cioé in parte quello delle equazioni differenziali a derivate parziale ed in parte quello delle equazioni integrali. Mi permetto perciò di chiamarle equazioni integro-differenziali.’ He then used such ‘equations of mixed type’, namely linear integro-differential equations involving Volterra integral operators, as models describing heredity effects (see Volterra (1913, pp. 138–162)). Related, but more general (nonlinear) versions became famous in Volterra's work, starting around 1926, on the growth of single-species or interacting populations. At the end of his 1909 paper (p. 174) he added, however, a cautionary note when he observed that ‘… il problema della risoluzione delle equazioni integro-differenziali costituisce in generale un problema essenzialmente distinto dai problemi delle equazioni differenziali e da quelli ordinarii delle equazioni integrali’ [his italics].

Although such functional equations may be viewed formally as ODEs perturbed by a ‘memory’ term given by a Volterra integral operator, the analysis of collocation methods will be more complex (perhaps not ‘essentially distinct’ – except when it comes to the analysis of qualitative properties) than simply a synthesis of the techniques employed in Chapters 1 and 2. The convergence results we establish in this chapter will of course yield those of Chapter 1 as special cases.