Spline finite difference for solution of a mathematical model in the initiation angiogenesis

The Numerov-spline method is proposed for a mathematical model of the endothelial, pericyte and macrophage cell's roles in the initiation of tumor angiogenesis with initial-boundary values. The two time-level explicit method based on the cubic spline is developed for the solution of the problem. The cubic spline for approximating the second order derivative in a space variable x and finite difference for time variable t has been used. Stability of the proposed method has been analysed. Finally figures of endothelial, Pericyte and macrophage cells in the capillaries are presented. Numerical results are given to examine the accuracy and efficiency of the proposed method.

Keywords:

Cubic spline , Numerov type , Finite difference , Capillary formation , Angiogenesis