Department of Mathematics
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Browsing Department of Mathematics by Subject "Advection-diffusion equation"
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Item Application of Coloured Noise as a Driving Force in the Stochastic Differential Equations(2010) Charles, Wilson M.In this chapter we explore the application of coloured noise as a driving force to a set of stochastic differential equations(SDEs). These stochastic differential equations are sometimes called Random flight models as in A. W. Heemink (1990). They are used for prediction of the dispersion of pollutants in atmosphere or in shallow waters e.g Lake, Rivers etc. Usually the advection and diffusion of pollutants in shallow waters use the well known partial differential equations called Advection diffusion equations(ADEs)R.W.Barber et al. (2005). These are consistent with the stochastic differential equations which are driven by Wiener processes as in P.E. Kloeden et al. (2003). The stochastic differential equations which are driven by Wiener processes are called particle models. When the Kolmogorov’s forward partial differential equations(Fokker-Planck equation) is interpreted as an advection diffusion equation, the associated set of stochastic differential equations called particle model are derived and are exactly consistent with the advection-diffusion equation as in A. W. Heemink (1990); W. M. Charles et al. (2009). Still, neither the advection-diffusion equation nor the related traditional particle model accurately takes into account the short term spreading behaviour of particles. This is due to the fact that the driving forces are Wiener processes and these have independent increments as in A. W. Heemink (1990); H.B. Fischer et al. (1979). To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this chapter. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection-diffusion equation.Item Stochastic Particle Models for Transport Problems in Coastal Waters(2005) Charles, Wilson M.; Heemink, Arnold W.; Van den Berg, E.In this paper transport processes in coastal waters are described by stochastic differential equations (SDEs). These SDEs are also called particle models (PMs). By interpreting a Fokker-Planck equation associated with the SDE as an advection diffusion equation (ADE), it is possible to derive the underlying PM which is exactly consistent with the ADE. Both the ADE and the related classical PM do not take into account accurately the short term spreading behaviour of particles. In the PM this shortcoming is due to the driving noise in the SDE which is modelled as a Brownian motion and therefore has independent increments. To improve the behaviour of the model shortly after the release of pollution we develop an improved PM forced by a coloured noise process representing the short-term correlated turbulent velocity of the particles. This way a more accurate and detailed short-term initial spreading behaviour of particles is achieved. For long-term simulations both the improved and classical PMs are consistent with the ADE. However, the improved PM is relatively easier to handle numerically than a corresponding ADE. In this paper both models are applied to a real life pollution problem in the Dutch coastal waters.