On a moving boundary problem associated with a mathematical model of breast cancer

This paper is associated with a nonlinear parabolic moving boundary problem  raised from the mathematical modeling of the behavior of the breast avascular cancer tumors at their first stage. This model is a modification of the previous works. Using the weak form of the proposed problem, the uniqueness of the solution is proved. Based on the finite difference method, a variable time step approach is proposed to solve the problem, numerically. It is shown that the numerical approach preserves the positivity of the solution and is unconditionally stable. To show the robustness and ability of the numerical method, the numerical and exact solutions are discussed and compared for two examples with the exact solutions.