Dynamics of Gompertzian tumour growth under environmental fluctuations

We investigate the effect of correlated additive and multiplicative Gaussian white noise oil the Gompertzian growth of tumours. Our results are obtained by Solving numerically the time-dependent Fokker-Planck equation (FPE) associated with the stochastic dynamics. In Our numerical approach we have adopted B-spline functions as a truncated basis to expand the approximated eigenfunctions. The eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under Study. We perform simulations to analyze various aspects, of the probability distribution. of the tumour cell populations in the transient- and steady-state regimes. More precisely, we are concerned mainly with the behaviour of the relaxation time (tau) to the steady-state distribution as a function of (i) of the correlation strength (lambda) between the additive noise and the multiplicative noise and (ii) as a function of the multiplicative noise intensity (D) and additive noise intensity (alpha). It is observed that both the correlation strength and the intensities of additive and multiplicative noise, affect the relaxation time.

Stochastic analysis of seepage under water-retaining structures

This paper investigated the problem of confined flow under dams and water retaining structuresusing stochastic modelling. The approach advocated in the study combined a finite elementsmethod based on the equation governing the dynamics of incompressible fluid flow through aporous medium with a random field generator that generates random hydraulic conductivity basedon lognormal probability distribution. The resulting model was then used to analyse confined flowunder a hydraulic structure. Cases for a structure provided with cutoff wall and when the wall didnot exist were both tested. Various statistical parameters that reflected different degrees ofheterogeneity were examined and the changes in the mean seepage flow, the mean uplift forceand the mean exit gradient observed under the structure were analysed. Results reveal that underheterogeneous conditions, the reduction made by the sheetpile in the uplift force and exit hydraulicgradient may be underestimated when deterministic solutions are used.