Gülnihal Meral

Differential quadrature solution of nonlinear reaction-diffusion equation with relaxation-type time integration

This paper presents the combined application of differential quadrature method (DQM) and finite-difference method (FDM) with a relaxation parameter to nonlinear reaction–diffusion equation in one and two dimensions. The polynomial-based DQM is employed to discretize the spatial partial derivatives by using Gauss–Chebyshev–Lobatto points. The resulting system of ordinary differential equations is solved, discretizating the time derivative by an explicit FDM. A relaxation parameter is used to position the solution from the two time levels, aiming to increase the convergence rate with a moderate time step to the steady state and also to obtain stable solution. Numerical experiments are given to illustrate the scheme for one-dimensional Fisher-type problems and also for two-dimensional reaction–diffusion boundary-value problems. The agreement of the solution with the exact solution is very good in two-dimensional case while some other numerical schemes may result in some unwanted oscillations in the computed solution. Optimal value of the relaxation parameter is obtained numerically to prevent the use of very small time steps and to achieve stable solutions. The DQM with a relaxation-type finite-difference time integration scheme exhibits superior accuracy at large time values for the problems tending towards a steady state.

Mathematical Analysis and Numerical Simulations for a System Modeling Acid-Mediated Tumor Cell Invasion

This work is concerned with the mathematical analysis of a model proposed by Gatenby and Gawlinski (1996) in order to support the hypothesis that tumor-induced alteration of microenvironmental pH may provide a simple but comprehensive mechanism to explain cancer invasion. We give an intuitive proof for the existence of a solution under general initial conditions upon using an iterative approach. Numerical simulations are also performed, which endorse the predictions of the model when compared with experimentally observed qualitative facts.

DRBEM solution of the acid-mediated tumour invasion model with time-dependent carrying capacities

It is known that the pH level of the extracellular tumour environment directly effects the progression of the tumour. In this study, the mathematical model for the acid-mediated tumour cell invasion consisting of a system of nonlinear reaction diffusion equations describing the interaction between the density of the tumour cells, normal cells and the concentration of protons produced by the tumour cells is solved numerically using the combined application of dual reciprocity boundary element method (DRBEM) and finite difference method. The space derivatives in the model are discretised by DRBEM using the fundamental solution of Laplace equation considering the time derivative and the nonlinearities as the nonhomogenity. The resulting systems of ordinary differential equations after the application of DRBEM are then discretised using forward difference. Because of the highly nonlinear character of the model, there arises difficulties in solving the model especially for two-dimensions and the boundary-only nature of DRBEM discretisation gives the advantage of having solutions with a lower computational cost. The proposed method is tested with different kinds of carrying capacities which also depend on time. The results of the numerical simulations are compared among each case and seen to confirm the expected behaviour of the model.

Mathematical analysis and numerical simulations for the HSP70 synthesis model

The heat shock proteins (HSPs) protect the other proteins in the process of folding under stressful conditions such as oxygen deprivation, hypothermy or presence of alcohol. They have also an important role in tumour invasion. In this paper, the existence, uniqueness and permanence properties for the solution of the mathematical model which focuses on the synthesis of HSP70 is proved. Moreover the numerical simulations are performed by using FDM, namely a combination of backward and forward Euler methods, and the results confirm the expected behaviour of the solution.