Computational Solution of Nonlinear Integro-differential Equations

The mathematical modeling of physical or chemical processes where memory effects play an important role (for example heat conduction in materials with memory, rheology, turbulence, chemical combustion theory) leads to nonstandard ordinary or partial integro-differential equations in which the underlying nonlinear integral operators contain the unknown solution with different time arguments, or where the spatial domain is unbounded. While the theory of these nonlinear problems is quite well understood, there is still a distinct lack of efficient computational methods.

This research proposal aims to fill this gap and thus make a significant contribution to the design, analysis, and computational implementation of efficient numerical methods for such nonstandard integro-differential equations, including singularly perturbed problems. The theoretical part of the proposal will be complemented by extensive numerical simulations for various model equations arising in the above-mentioned applications.