Highly Stable Multistage Numerical Methods for Functional Equations: Theory and Implementation Issues

Functional equations provide the best and most natural way to describe evolution in time and space, also in presence of memory. In fact, the spread of diseases, the growth of biological populations, the brain dynamics, elasticity and plasticity, heat conduction, fluid dynamics, scattering theory, seismology, biomechanics, game theory, control, queuing theory, design of electronic filters and many other problems from physics, chemistry, pharmacology, medicine, economics can be modelled through systems of functional equations, such as Ordinary Differential Equations (ODEs) and Volterra Integral Equations (VIEs). For instance, ODEs based models can be found in the context of evolution of biological populations, mathematical models in physiology and medicine, such as oncogenesis and spread of infections and diseases, economical sciences, analysis of signals. Concerning VIEs based models, the following books and review papers contain sections devoted to this subject in the physical and biological sciences: Brunner, Agarwal and O’Regan, Corduneanu and Sandberg, Zhao. Most of these also include extensive lists of references. Regarding some specific applications of VIEs, they are for example models of population dynamics and spread of epidemics, wave problems, fluido-dynamics, contact problems,electromagnetic signals.