Abstract: A block backward differentiation formula of uniform
order eight is proposed for solving first order stiff initial value
problems (IVPs). The conventional 8-step Backward Differentiation
Formula (BDF) and additional methods are obtained from the same
continuous scheme and assembled into a block matrix equation which
is applied to provide the solutions of IVPs on non-overlapping
intervals. The stability analysis of the method indicates that the
method is L0-stable. Numerical results obtained using the proposed
new block form show that it is attractive for solutions of stiff problems
and compares favourably with existing ones.
Abstract: A systems approach model for prostate cancer in prostate duct, as a sub-system of the organism is developed. It is accomplished in two steps. First this research work starts with a nonlinear system of coupled Fokker-Plank equations which models continuous process of the system like motion of cells. Then extended to PDEs that include discontinuous processes like cell mutations, proliferation and deaths. The discontinuous processes is modeled by using intensity poisson processes. The model incorporates the features of the prostate duct. The system of PDEs spatial coordinate is along the proximal distal axis. Its parameters depend on features of the prostate duct. The movement of cells is biased towards distal region and mutations of prostate cancer cells is localized in the proximal region. Numerical solutions of the full system of equations are provided, and are exhibit traveling wave fronts phenomena. This motivates the use of the standard transformation to derive a canonically related system of ODEs for traveling wave solutions. The results obtained show persistence of prostate cancer by showing that the non-negative cone for the traveling wave system is time invariant. The traveling waves have a unique global attractor is proved also. Biologically, the global attractor verifies that evolution of prostate cancer stem cells exhibit the avascular tumor growth. These numerical solutions show that altering prostate stem cell movement or mutation of prostate cancer cells lead to avascular tumor. Conclusion with comments on clinical implications of the model is discussed.