Adaptive collocation methods of volterra functional equations

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Many mathematical modelling problems in the physical and biological sciences that involve memory effects lead to (systems of) ordinary or partial Volterra integro-differential equations whose solutions possess a low degree of regularity at certain points in the interval on which the equations have to be solved.  This loss of regularity occurs if the Volterra integral operators in these functional equations possess weakly singular (unbounded) kernels or/and delay arguments (Vito Volterra's delay integro-differential model (1927) for two competing species is a celebrated example of the latter case).  Since numerical schemes that are based on a priori selected (graded or constrained) temporal meshes - for which the mathematical theory is now quite well understood - are in general inefficient, so-called adaptive methods based on `dynamically' selected meshes resulting from a posteriori error estimation techniques have become the methods of choice. The present research proposal is concerned with the mathematical and computational aspects of a posteriori error estimation and adaptive time-stepping schemes for functional equations involving Volterra integral operators of the types described above.  The work will be carried out in collaboration with researchers in China, Italy, Estonia, Scotland and Canada; it will also expose Ph.D. students and postdoctoral fellows to many important developments in modern numerical analysis and scientific computation.