Numerical Reduced Models Based on Fitting

1 Calculation of Temporal Concentration Changes Using Difference Equations 

One possible approach for storing the solution of akinetic model is according to the following general algorithm  

Time step At is selected to achieve good resolution of the characteristic time-scale of the system. 

Several thousand, spatially homogeneous simulations are carried out with a series of initial concentrations and/or temperatures, which are typical for the circumstances of applications of the final intended model. 

A function G is fitted to the data and can then be used to predict the change in concentration after time step At Y(t+A0 = G(Y(t)). 

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Oran et al. [218,219] developed a global parameterized model which describes the chemical induction time as a function of temperature and pressure. Parameters of the induction time function were determined for stoichiometric hydrogen and methane in air mixtures. The parameters were fitted to numerical results obtained from the simulations based on detailed reaction mechanisms. This technique allowed a 22-times faster calculation of the induction time and reduced the simulation time in a onedimensional model by a factor of 7.5. The fitted model was used in two-dimensional shock-wave simulations. 

It is possible to derive a classical mechanical 7i from the quantum mechanical Heff spectroscopic fit model. The relationship between the exact, full dimensional quantum mechanical H and the approximate, reduced dimensional Heff fit model is, at best, numerically complicated. The 7i derived from Heff is based on an accurate numerical representation of a spectrum. It is vastly simpler than an exact classical mechanical 7i derived from the exact quantum mechanical H. 

In the other study mentioned [18], the principal objectives were to show that reduced residence times were possible without loss of resolution and to illustrate a computer interfaced, user-based approach to the data analyses. A cation exchange resin having a nominal diameter of 15 jum was used as packing in a 10 mm-I.D. stainless steel column. Non-Gaussian dispersion behaviour in the calibration standards was fitted to a numerical algorithm based on the Pearson Type VII model. The detector turbidity response at 254 nm was generated experimentally and fitted to two separate polynomial forms. 

Advanced EMR methods may be used to conduct quantitative measurements of nuclear hyperfine interaction energies, and these data, in turn, may be used as a tool in molecular design because of their direct relation to the frontier orbitals. The Zeeman field dependence of hyperfine spectra enables one to greatly improve the quantitative analysis of hyperfine interaction and assign numeric values to the parametric terms of the spin Hamiltonian. Graphical methods of analysis have been demonstrated that reduce the associated error that comes from a multi-parameter fit of simulations based on an assumed model. The narrow lines inherent to ENDOR and ESEEM enable precise measures of peak position and high-resolution hyperfine analyses on even powder sample materials. In particular, ESEEM can be used to obtain very narrow lines that are distributed at very nearly the zero-field NQI transition frequencies because of a quantum beating process that is associated with... 

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