The mathematical modelling of tumour growth - an implicit method

In this paper, the semi-linearization approach is presented. The semi-linearization approach is a numerical method for solving differential equations or systems of differential equations, which actually gives a set of points as a result. Firstly, one semi-linearization technique is presented and examined, an implicit method. It is used for the model equations in order to solve them with respect to the concentration and density. In addition, the stability of the equations is examined in the implicit case and it is proved that the implicit finite-difference method is unconditionally stable and thus very reliable. Furthermore, chemotherapy is applied to the model of equations and a treatment factor is introduced. It is a periodic treatment factor and it is subtracted from the concentration differential equation. Then an explicit finite-difference method is applied to the system of equations with chemotherapy and again the stability of the system is examined and very useful results and graphical representations are obtained.