Fundamental Solution via Invariant Approach for a Brain Tumor Model and its Extensions

²School of Computational and Applied Mathematics, DST-NRF Centre of Excellence in Mathematical and Statistical Sciences; Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa and School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052 Australia

We firstly show how one can use the invariant criteria for a scalar linear (1 + 1) parabolic partial differential equations to perform reduction under equivalence transformations to the first Lie canonical form for a class of brain tumor models. Fundamental solution for the underlying class of models via these transformations is thereby found by making use of the well-known fundamental solution of the classical heat equation. The closed-form solution of the Cauchy initial value problem of the model equations is then obtained as well. We also demonstrate the utility of the invariant method for the extended form of the class of brain tumor models and find in a simple and elegant way the possible forms of the arbitrary functions appearing in the extended class of partial differential equations. We also derive the equivalence transformations which completely classify the underlying extended class of partial differential equations into the Lie canonical forms. Examples are provided as illustration of the results.