BVPsymmetry · Reductions and exact solutions of boundary value problems with moving boundaries by means of symmetry based methods

Reductions and exact solutions of boundary value problems with moving boundaries by means of symmetry based methods

The following results towards the project objectives were obtained during the project (period 23/09/2013-22/06/2015): A definition of conditional invariance for nonlinear multidimensional boundary-value problems (BVPs) with a wide range of boundary conditions was derived, and an algorithm for how to construct all possible conditional symmetries for the given class of BVPs was established. It was shown that the symmetry of multidimensional BVPs depends essentially on the domain geometry and appropriate domains can be described using the governing equation(s) of BVP in question. To demonstrate the efficiency of the definition and algorithm, specific nonlinear two-dimensional BVPs (with both Dirichlet and Neumann boundary conditions) arising in heat transfer theory and population dynamics were tested. As a result known results were recovered and new ones were obtained. Furthermore the theoretical foundations developed were applied systematically to more complicated BVPs (including multidimensional problems and those with moving boundaries). A class of the (1+2)-dimensional nonlinear BVPs modelling the processes of heat transfer was examined in details. As result, an exhaustive Lie and conditional symmetry classification being derived and the symmetries obtained for constructing reductions and exact solutions were applied. In particular, we have found an exceptional exponent for the power diffusivity when BVPs with non-vanishing flux on the boundary admits additional Lie symmetry operators. In order to demonstrate the applicability of the symmetries derived, they were used for reducing nonlinear problems with power diffusivities (which are widely used in real world applications) to (1+1)-dimensional problems. The properties and exact/numerical solutions of the problems obtained were analysed. A simplified nonlinear Keller-Segel model used for modelling some processes in biology and medicine (especially tumour growth) was also examined. It was shown that the Cauchy and Neumann problems in two-dimensional space-time can be exactly solved, and the corresponding exact solutions were constructed. While the result cannot be generalised to the multidimensional case, new results were derived for the above problems in three-dimensional space-time using Lie symmetry operators. In particular, invariance under infinite-dimensional Lie algebras was derived and this property was used in reducing the three-dimensional problems to two-dimensional ones. A class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity, which are widely used in real world models describing the image processing, turbulent gas flows etc., was under study by symmetry-based methods. In particular, the results of the Lie symmetry classification for the reduction to lower dimensionality, and a search for exact solutions of the nonlinear (1+2)-dimensional equation (often called the Perona-Malik equation), were implemented. Moreover, the results obtained were applied to a classical boundary-value problem with moving boundary modelling turbulent gas flows. In particular it has been proved that the exact solutions can be found provided the moving boundary is expanded in time according to exponential and power laws. A well-known two-phase model of solid tumour growth was also studied. Because this model is a very complicated nonlinear BVP with moving boundary, a simplified biologically meaningful (1+1)-dimensional model neglecting tumour cell viscosity was used. Lie and conditional symmetry classifications were performed for the governing system of the given BVP and they were generalised on the general two-component reaction-diffusion system. The symmetries obtained were applied in order to reduce the given BVP to one-dimensional one in order to construct exact solutions. Properties of the solutions obtained were investigated in order to explore their biological meaning. Moreover, these solutions were generalised to multidimensional cases under the assumption of spherical-symmetric growth of the solid tumour. Three papers based on the results obtained within the project have been prepared and submitted to international scientific journals (see details in Section 2). Preparing the joint monograph with Prof. JR King is under way and the main part will be finished during the Return Phase in Kyiv. The preliminary title is ‘Symmetries of Nonlinear Evolution Equations and Their Applications for Solving Real World Problems’. The monograph is based on the results obtained in this project and those derived earlier (in particular published in joint papers). The most important results obtained can be summarised as follows: - A new definition of conditional invariance for nonlinear multidimensional boundary-value problems with a wide range of boundary conditions was derived and successfully tested. - The Lie and conditional symmetry classification of a class of the (1+2)-dimensional nonlinear boundary-value problems modelling the processes of heat transfer was derived. - New reductions of (1+2)-dimensional nonlinear boundary value problems with power conductivity to two-dimensional problems were found and analyzed. - Exact solutions of (1+1)-dimensional nonlinear boundary value problems for a simplified Keller-Segel system were constructed and analyzed. - The Lie symmetry classification of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity was performed. - Symmetries of a classical problem with moving boundary modelling turbulent gas flows were found and exact solutions were constructed and analysed. - The Lie and conditional symmetry classifications were performed for the governing system of a well-known (1+1)-dimensional model of solid tumour growth. - Symmetries and exact solutions of a nonlinear moving boundary problem modelling problem solid tumour growth were constructed under some assumptions. The results obtained have been presented at 17 talks at international conferences, symposia, seminars, and workshops (see details in Section 2). The Fellow’s outreach activities were realised by the following activities: - Creation of a project site on Facebook with the address - https://www.facebook.com/BVPsymmetry/info?tab=overview containing a lot information about this project for the attention of researchers, students and the general public. - Participation in the University Open Day, June 2015 (private discussions with youth aiming to enter the University). - The Fellowship has enabled the study of the organizational structure of research and teaching in the UK. In particular this led to relevant proposals presented to the Ministry of Education and Science of Ukraine during several meetings with the Minister SM Kvit and to publication of an article in the influential national-wide newspaper ‘Dzerkalo Tyzhnia’ (Weekly Mirror) http://gazeta.dt.ua/science/nauka-v-ukrayini-osobliviy-shlyah-rozvitku-chi-glibokiy-zanepad.html

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