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| The purpose of this work is to find local symmetries and exact solutions of the following model given by Perumpanani et al.:
\frac{\partial u}{\partial t} = f(u)+ u_{xx} \\
\frac{\partial v}{\partial t} = k (v_{x}u_{x}+v u_{xx})
This model described benign tumour growth where \textit{u(x,t)} represents the concentrations of the tumour cells, \textit{v(x,t)} the connective tissue, and \textit{x} and \textit{t} are the space and time coordinates. The model studies the averaged behaviour of the tumour cells in the direction of expansion only, disregarding all the variations in the plane perpendicular to the direction growth.
Symmetry groups have several different applications in the context of nonlinear differential equations. In particular, they are used to obtain exact solutions of partial differential equations. We calculate the symmetries of the model applying Lie's classical method. Then, we construct the optimal system, to determine those solutions that cannot be derived from others under the action of a symmetry group. Using the optimal system, we obtain the variables and similarity solutions. This, we allow us to transform our system of partial differential equations into a system of ordinary differential equations. From these reductions we find exact solutions of the model. |
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