Group classification of coupled partial differential equations with applications to flow in a collapsible channel and diffusion processes

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.