Development and investigation of the nonlinear models of the mathematical physics is one of the most important problems of the modern science. This project will shed light into modeling and control of an important class of nonlinear processes with phase transition, so called free boundary problems, arising in thermophysics and mechanics of continuous media, bioengineering and materials science. Free boundary problems involve the solution of the partial differential equations in domains that are unknown apriori: not only the solutions of the equations but also the domain of definition of the equations must be determined.
The main goal of this project is to gain insight into inverse free boundary problems. These problems arise when not only the solution of the PDEs and the free boundaries are unknown, but also some other characteristics of the media are not available. For example, temperature or heat flux on the fixed boundaries, intensity of the source term or phase transition temperature may not be available. Another important motivation for the analysis of the inverse Stefan problem arises in optimal control of processes with phase transitions.
In summer REU session of 2014 students will pursue research on the optimal control of the free boundary problem for the heat equation motivated by the new variational formulation suggested in a recent paper:
U. G. Abdulla, On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations.
I.Well-posedness and Convergence of the Method of Lines,
Inverse Problems and Imaging, Volume 7, Number 2(2013), 307-340.