Algorithms based on DQM with new sets of base functions for solving parabolic partial differential equations in (2+1)(2+1) dimension

This paper deals with the numerical computations of two space dimensional time dependent parabolic partial differential equations by adopting adopting an optimal five stage fourth-order strong stability preserving Runge Kutta (SSP-RK54) scheme for time discretization, and three methods of differential quadrature with different sets of modified B-splines as base functions, for space discretization: namely i) mECDQM: (DQM with modified extended cubic B-splines); ii) mExp-DQM: DQM with modified exponential cubic B-splines, and iii) MTB-DQM: DQM with modified trigonometric cubic B-splines. Specially, we implement these methods on convection-diffusion equation to convert them into a system of first order ordinary differential equations,in time which can be solved using any time integration method, while we prefer SSP-RK54 scheme. All the three methods are found stable for two space convection-diffusion equation by employing matrix stability analysis method. The accuracy and validity of the methods are confirmed by three test problems of two dimensional convection-diffusion equation, which shows that the proposed approximate solutions by any of the method are in good agreement with the exact solutions

Bifurcation in cell cycle dynamics regulated by p53

We study the regulating mechanism of p53 on the properties of cell cycle dynamics in the light of the proposed model of interacting p53 and cell cycle networks via p53. Irradiation (IR) introduce to p53 compel p53 dynamics to suffer different phases, namely oscillating and oscillation death (stabilized) phases. The IR induced p53 dynamics undergo collapse of oscillation with collapse time \Delta t which depends on IR strength. The stress p53 via IR drive cell cycle molecular species MPF and cyclin dynamics to different states, namely, oscillation death, oscillations of periods, chaotic and sustain oscillation in their bifurcation diagram. We predict that there could be a critical \Delta t_c induced by p53 via IR_c, where, if \Delta t < \Delta t_c the cell cycle may come back to normal state, otherwise it will go to cell cycle arrest (apoptosis)