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Extra resources for Computational Mdlg. of Multi-scale Non-Fickian Dispersion in Porous Media
Example text
A particle moving through a porous medium is such an example. In this case, advection gives rise to the drift term and hydrodynamic dispersion and microdiffusion give rise to the “diffusive” term. In the population dynamics, the diffusive term is a direct result of noise in the proportionality constant. Therefore it is important to study Ito process further in order to apply it in modeling situations. (t ) is called the drift coefficient and (t ) the diffusion coefficient and they can depend on Y(t) and/or B(t).
The jumps are the discontinuities of the first kind and any other discontinuity is called a discontinuity of the second kind. Obviously a function can only have countable number of jumps in a given range. From the mean value theorem in calculus, it can be shown that we can differentiate a function in a given interval only if the function is either continuous or has a discontinuity of the second kind during the interval. Stochastic calculus is the calculus dealing with often non-differentiable functions having jumps without discontinuities of the second kind.
1), and the only discernible pattern is that as time progresses, the position tends to wander away from the starting position at the origin. In other words, if the statistical variance over realisations for a fixed time is evaluated, this Stochastic Differential Equations and Related Inverse Problems 47 increases gradually – a property referred to as time varying variance. The use of the Wiener process in a modelling situation to represent the noise in the system should be carefully thought through.
