Final Report Summary - BVPSYMMETRY (Reductions and exact solutions of boundary value problems with moving boundaries by means of symmetry based methods)
This project is a natural continuation (Return Phase) of the project 328563 with the same title, which was conducted at the University of Nottingham in 2013—2015.
The development of new theoretical foundations and algorithms for the reduction of nonlinear boundary value problems (especially those with moving boundaries) to problems of lower dimensionality and the construction of exact solutions of the problems in question are the main aims of the project. An applied goal is to compare the analytical results derived with those obtained by means of the appropriate numerical techniques for some physically and biologically motivated problems.
The most important results obtained can be summarised as follows:
Lie symmetries, reductions and exact solutions of a (1+2)-dimensional nonlinear model of the Keller-Segel type were constructed. As a result, the relevant Neumann and Cauchy problems were exactly solved under appropriate restrictions on boundary conditions and initial profiles.
Lie and Q-conditional symmetries of a well-known boundary-value problem with a free boundary describing the tumour growth processes were studied. Using the symmetry reductions, some exact solutions of the problem in question were derived and their biological interpretation established. The results were generalized on the multidimensional case under assumption that the tumour possesses a spherical symmetry.
A generalization of the algorithm for the symmetry reduction, which was worked out earlier, on the boundary-value problems with more complicated boundary conditions (including a set of moving boundaries) was derived. To demonstrate the efficiency of the algorithm, a specific boundary-value problem (including multidimensional case) describing melting and evaporation of materials was studied.
The results obtained have been published in two papers and a preprint, and presented at seven talks at international conferences, seminars, and workshops (see details in Section 2).
The project site on Facebook with the address
https://www.facebook.com/BVPsymmetry/info?tab=overview(opens in new window)
contains an appropriate information about this project for the attention of researchers, students and the general public.
The development of new theoretical foundations and algorithms for the reduction of nonlinear boundary value problems (especially those with moving boundaries) to problems of lower dimensionality and the construction of exact solutions of the problems in question are the main aims of the project. An applied goal is to compare the analytical results derived with those obtained by means of the appropriate numerical techniques for some physically and biologically motivated problems.
The most important results obtained can be summarised as follows:
Lie symmetries, reductions and exact solutions of a (1+2)-dimensional nonlinear model of the Keller-Segel type were constructed. As a result, the relevant Neumann and Cauchy problems were exactly solved under appropriate restrictions on boundary conditions and initial profiles.
Lie and Q-conditional symmetries of a well-known boundary-value problem with a free boundary describing the tumour growth processes were studied. Using the symmetry reductions, some exact solutions of the problem in question were derived and their biological interpretation established. The results were generalized on the multidimensional case under assumption that the tumour possesses a spherical symmetry.
A generalization of the algorithm for the symmetry reduction, which was worked out earlier, on the boundary-value problems with more complicated boundary conditions (including a set of moving boundaries) was derived. To demonstrate the efficiency of the algorithm, a specific boundary-value problem (including multidimensional case) describing melting and evaporation of materials was studied.
The results obtained have been published in two papers and a preprint, and presented at seven talks at international conferences, seminars, and workshops (see details in Section 2).
The project site on Facebook with the address
https://www.facebook.com/BVPsymmetry/info?tab=overview(opens in new window)
contains an appropriate information about this project for the attention of researchers, students and the general public.