| Abstract: |
| In this study, we present the mathematical model of a tumor invasion model described by \cite{kubo,kubo2}
\begin{eqnarray}\label{mapule}
\left.
\begin{array}{rl}
\displaystyle{\frac{\partial \theta}{\partial t} = d_\theta \frac{\partial^2 \theta}{\partial x^2} - \gamma \frac{\partial}{\partial x} (\theta\frac{\partial v}{\partial x} ) + \mu_1 \theta(1-\theta-v)},\
\displaystyle{\frac{\partial v}{\partial t} = -\eta m v + \mu_2 v(1-\theta-v)},\
\displaystyle{\frac{\partial m}{\partial t}= d_m \frac{\partial^2 m}{\partial x^2} + \alpha \theta - \beta m},
\end{array}\right\}
\end{eqnarray}
where $d_{\theta},\, d_m, \, \mu_1, \, \mu_2, \, \eta, \, \alpha$ and $ \beta$ are positive constants and $\theta$, is the density tumour cells, $v,$ is the extra cellular matrix density and $m$ is the degradation enzymes $\theta$, $v$ and $m$ depend on position and the time on a smooth bounded domain. The purpose of this paper is to apply the techniques of Lie symmetry to the model and present the group invariant solutions of the system of second order partial differential equations describing a tumor invasion model. The similarity solutions obtained, are presented in the general form.
\begin{thebibliography}{99}
\bibitem{kubo}
{\sc A.~Kubo, A. and Y.~Miyata,} (2019) International Journal of Mathematical and Computational Methods, 4, 10--16.
\bibitem{kubo2}
{\sc A.~Kubo, A. and Y.~Miyata,} (2015) WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE, 15, 101--111.
\end{thebibliography}
\end{document} |
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