BVP Symmetry

undaries are widely used in
mathematical modeling a huge number of processes, which arise in physics, biology and
industry. While these processes can be very different from a formal point of view, they
have the common peculiarity: unknown moving boundaries. The most important subclass
of such boundary value problems (BVPs) is the Stefan problems, in which the movement
of unknown boundaries is governed by the well-known Stefan boundary conditions. BVPs
with moving boundaries, particularly the multidimensional Stefan problems, are the main
object of the proposed project. Developing new theoretical foundations and algorithms for reduction of such BVPs to those of lower dimensionality and construction of exact solutions of BVPs in question is
the main aim of the project. Applied goal is to compare the analytical results derived with
those obtained by means of the appropriate numerical techniques in the case of a wide
range of the physically and biologically motivated problems. Moreover such comparison
will demonstrate the real interdisciplinary aspect of the proposal. The novel idea of the
project is to develop the algorithms mentioned above using such symmetry based
methods as the classical Lie-Ovsiannikov method, the Bluman-Cole method of non-
classical symmetry, conditional symmetry method and their recent extensions. The main results to be achieved: new definitions of (generalized) conditional invariance
for BVPs with a wide range of boundary conditions will be derived; algorithms for how to
construct all possible conditional symmetries for the given class of BVPs will be
determined; new analytical results (conditional symmetries, reductions and exact
solutions of BVPs) will be established by application of the algorithm to a wide range of
nonlinear BVPs modeling the tumour growth processes and melting-evaporation
processes. Keywords for the general public: tumour growth, melting & evaporation of metals, heat transfer, diffusion process, turbulent flow, moving surface, mathematical modelling, differential equation. Keywords for specialists: nonlinear heat equation, nonlinear reaction-diffusion equation, nonlinear boundary value problem, nonlinear problem with moving boundary, exact solution, Lie symmetry, conditional symmetry. Roman Cherniha

Principal collaborator and Scientist-in-Charge: Prof.