Applied Mathematics for Science and Engineering

9Integro-Differential Equations

Introduction

Integro-differential equations (IDEs) arise in a variety of contexts; there are applications to process control and to diffusion along grain boundaries (Antipov and Gao, 2000), as well as modeling for neural networks (Jackiewicz et al., 2008), option prices (Cont and Voltchkova, 2005), and the spread of infectious diseases (Medlock and Kot, 2003). Only rarely can analytic solutions be found for such problems, so frequently, the analyst must resort to numerical methods. Note that IDEs also figure prominently in the analysis of multiphase processes where countable entities such as bubbles, drops, and particles are borne by a fluid phase. Examples include solvent extraction/emulsification, flocculation, crystallization, sedimentation, and the operation of biochemical reactors. Because so many industrial processes involve countable entities, the importance of IDEs to process engineering and the applied sciences cannot be overstated.

To provide a historical framework for our consideration of IDEs, we will explore their role in the early twentieth-century study of biological systems. Vito Volterra was an eminent Italian mathematician (1860–1940) whose career was effectively ended by his refusal to sign the oath of allegiance to the Fascist government in 1931. It is worth noting that only 12 Italian university professors refused to sign, a small number but one that is understandable given the consequences (essentially exclusion from the Italian ...