Theory and Numerical Analysis of Volterra Functional Equations

The qualitative and quantitative analyses of numerical methods delay differential equations (DDEs) are now quite well understood, as reflected in the recent monograph by Bellen and Zennaro (2003). This is in remarkable contrast to the situation in the numerical analysis of more general Volterra functional equations in which delays occur in connection with memory terms described by Volterra integral operators. The complexity of the convergence and asymptotic stability analysis has its roots in a number of aspects not present in DDEs: the problems have distributed delays; kernels in the Volterra operators may be weakly singular; a second discretisation step (approximation of the memory term by feasible quadrature processes) will in general be necessary before solution approximations can be computed. These notes are intended to provide an introduction to functional integral and integrodifferential equations of Volterra type and their numerical analysis, focusing on collocation methods. They contain background material (and references), and also describe the “state of the art” in the numerical analysis. In addition, they reveal that we still have a long way to go before we reach a level of insight into the numerical analysis of Volterra functional equations comparable to the one that has been achieved for delay differential equations.