Executive Summary : | The diffusion/heat equation with free or moving boundary problems involve the parabolic partial differential equation in a time-dependent domain, with an unknown boundary. These problems have practical applications in various fields such as science, engineering, and industries, such as melting, solidification, thawing, food conservation, tumor treatment, fluid flow in porous media, sediment transport, oxygen diffusion problem, and crystal growth. Various models and approximate/numerical solutions exist in the literature, but several important and complex physical processes associated with these problems have not been effectively studied. To improve understanding, detailed studies are needed to present more realistic mathematical models and establish a simple, efficient, and accurate numerical method. This project focuses on complex problems of non-classical diffusion/heat equation with free boundary problems that occur in phase-change processes, tumor treatment, food conservation, and diffusion processes of solute release from polymer matrix. The project will involve rigorous study of these problems, modifications to existing models, and the development of algorithms based on neural networks to solve free boundary problems. The project aims to document all established results for publication in international journals, and hopes that the outcomes will be beneficial to researchers working in the field of partial differential equations with free boundaries in the near future. |