Application of fixed point-collocation method for solving an optimal control problem of a parabolic-hyperbolic free boundary problem modeling the growth of tumor with drug application

In this paper, employing a fixed point-collocation method, we solve an optimal control problem for a model of tumor growth with drug application. This model is a free boundary problem and consists of five time-dependent partial differential equations including three different first-order hyperbolic equations describing the evolution of cells and two second-order parabolic equations describing the diffusion of nutrient and drug concentration. In the mentioned optimal control problem, the concentration of nutrient and drug is controlled using some control variables in order to destroy the tumor cells. In this study, applying the fixed point method, we construct a sequence converging to the solution of the optimal control problem. In each step of the fixed point iteration, the problem changes to a linear one and the parabolic equations are solved using the collocation method. The stability of the method is also proved. Some examples are considered to illustrate the efficiency of method.