I am involved in research with Prof. J. R. Whiteman focusing on:
(i) the elliptic Volterra problem, find u such that
(ii) the parabolic Volterra problem, find u such that
and, (iii) the hyperbolic Volterra problem, find u such that
Such equations arise when modelling physical problems associated with viscoelastic materials such as polymers. For example: (H) arises by combining Newton's second law with a "Hooke's law with memory", and is used to model dynamic response; (V) models the quasistatic response resulting from neglecting inertial effects; and, (P) models the non-Fickian diffusion effect that results from the memory in the constitutive law.
When mathematics is properly supported I hope to make available a detailed synopsis of our research which will include links to relevant reports, publications etc.