Generalizing Temporal Integration

The solution of (2.56) with the concurrent substitution of time marching t = nAt leads 

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The direct evaluation of 6pt in Eq. (2.17) involves in general temporal integrations of operators [cf. Eqs. (B.3) and (B.4)] which are often numerically too expensive to be practically useful. To facilitate this problem, we... 

In the usual derivations of the Klein-Kramers equation, the moments of the velocity increments, Eq. (68), are taken as expansion coefficients in the Chapman-Kolmogorov equation [9]. Generalizations of this procedure start off with the assumption of a memory integral in the Langevin equation to finally produce a Fokker-Planck equation with time-dependent coefficients [67]. We are now going to describe an alternative approach based on the Langevin equation (67) which leads to a fractional IGein-Kramers equation— that is, a temporally nonlocal behavior. 

Figure 7.1 Integrative biomarker index as star plots. The integrated biomarker response is obtained from the sum of areas of each biomarker in the test battery. For the spatial survey, it can be observed that sites 1 and 4 are more stressed or affected than the other sites. The general response of biomarkers can be appraised in temporal settings where changes are observed between sites over time.

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