Abstract: In this plenary talk we focus on the development of low dimensional approximations to coupled nonlinear systems of partial differential equations (PDE) describing phase transformations. The methodology is explained on the example of nonlinear ferroelastic/thermoelastic dynamics. We start from the general three-dimensional Falk-Konopka model and with the center manifold reduction obtain a Ginzburg-Landau-Devonshire one-dimensional model. The Chebyshev collocation method is applied for the numerical analysis of this latter model, followed by the application of an extended proper orthogonal decomposition. Finally, we present several numerical results where we demonstrate performance of the developed methodology in reproducing hysteresis effects occurring during phase transformations and provide a survey of related methodologies and applied mathematical problems arising in this context.
Current project is a joint work with O. Tsviliuk and L. Wang.

Brief Biography of the Speaker:
Roderick Melnik is a Full Professor at the Wilfrid Laurier University in Waterloo, Canada. He is a Tier I Canada Research Chair in Mathematical Modelling. Before moving to Canada, Professor Melnik held senior professorial and research positions in the USA, Europe, and Australia. He was also a visiting fellow at the Isaac Newton Institute of the University of Cambridge , at the Institute for Mathematics and its Applications of the University of Minnesota and other research institutions in Europe, North America, and Australia. Professor Melnik's major results are in the development, analysis and applications of mathematical models based on partial differential equations and computational mathematics, focusing on coupled dynamic phenomena, systems, and processes. The areas of his research contributions include computational physics, applied numerical analysis, chemistry, and biology, non-smooth control, and stochastic differential equations. Over the past years, some of his main contributions have been to the development and applications of mathematical models in the area nano- and bionano- sciences with particular emphasis on the analysis of coupled multiscale phenomena, processes, and systems. This includes his contributions to the analysis of coupled effects in low-dimensional nanostructures, such as quantum dots, in bio-inspired and in biological systems.