Solitary wave solutions for the magma equation: symmetry methods and conservation laws - Mindu, Nkululeko (My University, 2015)
The magma equation which models the migration of melt upwards through the Earth’s mantle is considered. The magma equation depends on the permeability and viscosity of the solid mantle which are assumed to be a function of the voidage . It is shown using Lie group analysis that the magma equation admits Lie point symmetries provided the permeability and viscosity satisfy either a power law, or an exponential law for the voidage or are constant. The conservation laws for the magma equation for both power law and exponential law permeability and viscosity are derived using the multiplier method. The conserved vectors are then associated with Lie point symmetries of the magma equation. A rarefactive solitary wave solution for the magma equation is derived in the form of a quadrature for exponential law permeability and viscosity. Finally small amplitude and large amplitude approximate solutions are derived for the magma equation when the permeability and viscosity satisfy exponential laws.
A dissertation submitted for the degree of Masters of Science, School of Computational and Applied Mathematics, University of Witwatersrand, Johannesburg, 2014.
Keywords: Magmas.; Symmetry.; Conservation laws.