Higher-Order Differentiators

The higher-order differential equations, especially those of order 2, are of great importance because of physical situations describable by them. 

Equation (8-14) shows that starts from 0 and builds up exponentially to a final concentration of Kcj. Note that to get Eq. (8-14), it was only necessaiy to solve the algebraic Eq. (8-12) and then find the inverse of C (s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher order differential equations with Laplace transforms. 

Numerical solution of higher order differential equations is accomplished most conveniently by first converting them into an equivalent set of first order equations. Thus the second order equation... 

Modeling of some systems leads to higher order differential equations of the form... 

Any reaction scheme can be described by the set of first- or higher-order differential equations. These can be solved exactly for only relatively simple schemes. For more complicated sets, it is necessary to use either analog computers or digital numerical methods. 

The arbitrary sign in the last equations may be eliminated at the expense of ending up with a higher order differential equation [101]. Let us consider Eqn. (3.3.6) as an example, neglecting for a while the Brownian force X q,t) that will be reinstated later. If we time differentiate and multiply by Tq, then add the result to the starting equation to get rid of the memory integral, we may write ... 

The method described here is general and can be applied to higher-order differential equations. The method provides an attractive alternative to the use of particular solutions obtained using trial solutions based on the form of the function f x) and, in some cases, on the form of the homogeneous solution. ... 

Y. Bayazitoglu and J. Higenyi, Higher Order Differential Equations of Radiative Transfer Py Approximation, AIAA Journal, 14, p. 424,1979. 

Since the right-hand side is a higher order differential, the required boundary condition is... 

Shvartsburg, A. A., Mashkevich, S.V., Smith, R.D., Feasibility of higher-order differential ion mobility separations using new asymmetric waveforms. J. Phys. Chem. A 2006,110, 2663. 

Over the last decade, scientific and engineering interests have been shifting from canventional ion mobility spectrometry (IMS) to field asymmetric waveform ion mobility spectrometry (FAIMS). Differential Ion Mobility Spectrometry Nonlinear Ion Transport and Fundamentals of FAIMS explores this new analytical technology that separates and characterizes ions by the difference between their mobility in gases at high and low electric fields. It also covers the novel topics of higher-order differential IMS and IMS with alignment of dipole direction. 

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