Group classification of coupled partial differential equations with applications to flow in a collapsible channel and diffusion processes
Date
2010-07-09T09:02:48Z
Authors
Molati, Motlatsi
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Abstract
The main purpose of this work is to perform the symmetry classification of the coupled
systems of partial differential equations modelling flow in a collapsible tube and diffusion
phenomenon. The idea of symmetry is exploited in the two models of paramount importance
in applications: the one-dimensional model of flow through a collapsible tube and
the coupled diffusion system. These models contain several unknown functions (tube
law, friction, diffusion coefficients etc.) the forms of which are specified via the method
of group classification.
The solution procedure to carry out the group classification involves the use of two
approaches. In both of them the first part of the classification procedure deals with
the execution of the usual approach of Lie to obtain the general form of the symmetry
generator and the determining equations for the underlying models. The second part is
concerned with the utilization of the structure of the low-dimensional Lie algebras and the
Lie algebras of higher dimension to find the symmetry operators admitted by the given
models. The procedure is continued until the functional forms of the unknown functions
are completely specified. The latter part is different for each approach. In one approach
no use of the equivalence transformations is required whereas for the other approach the
equivalence group is a necessary component. In the former the classification with respect
to the subalgebras of three- and four-dimensional Lie algebras is used. Thereafter the
symmetry analysis of every case arising from the classification is performed in order to
determine the maximal Lie symmetry algebra. On the other hand the equivalence group
is used to obtain the canonical forms of the symmetry operators which satisfy the models
under consideration. The canonical forms of the low-dimensional Lie algebras and the
Lie algebras of higher dimension provide a means to specify the arbitrary functions of
the models.
The secondary aim is to employ the symmetries to construct the invariant solutions
wherever applicable. The solution of the optimal system problem allows the classification
of all the invariant solutions, i.e. solutions that are left unchanged by subalgebras of the symmetry Lie algebra. The invariant solutions of a given equation satisfy an equation
with a reduced number of the independent variables.