Particular solution parameters

Method of Variation of Parameters This technique is applicable to general linear difference equations. It is illustrated for the second-order system -2 + yx i + yx = ( )- Assume that the homogeneous solution has been found by some technique and write yY = -I- Assume that a particular solution yl = andD ... 

A is a linear transformation matrix which allows one to transform from one "best-fit solution to another (from the unprimed set to the primed set mentioned above). In order to specify a particular solution (and a unique transformation matrix A to get to that solution from some arbitrary solution), four E and C parameters may ) be chosen and assigned specific values, as long as they are chosen in such a way that the transformation from our arbitrary solution is completely defined and finite, i.e., the four parameters must be chosen so that the elements of A, fly, are completely defined and the determinant of A must be nonzero in order that A- exist, [flu fl22 — i2 ai 9 0]. These require-... 

The latter equation can be interpreted to mean that the third Stokes parameter does not vary with time in a circularly polarized beam of light. The particular solution (87) gives the cyclic theorem (9) self-consistently [11-20]. 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

Equations that have solutions, subject to particular boundary conditions, only Tor certain specific parameters occurring in them. In differential equations, the complete solution includes the characteristic solution and the particular solution. Ihe c haracteristic solution is obtained from the roots of the characteristic equation, and defines the transient or lime response of the system. The particular solution is obtained from the forcing function or input signal and defines the steady-state response. 

Temperature is another parameter that can be used for optimal analysis on RP columns. The overall resolution of a particular solute matrix can be improved and analysis time can be reduced because temperature affects every term in the resolution equation ... 

We choose to carry out only few numerical experiments to select the solution parameters. Detailed optimization of the solution parameters is difficult and often expensive computationally, so we do not recommend it. Finally, we must validate the model. Though detailed experimental data for the velocity and pressure profiles are not available for this particular RFR, we can employ the data on the overall pressure drop across the bed to validate the model to some extent. We find that the predicted overall pressure drop across the bed (10 kPa) shows good agreement with the available data. 

Clearly, the values for e° are different on silica and alumina. They are very different from those observed on carbon, which is partly, but not entirely due to the different conventions used in this case by Colin et al. [351]. Likewise, the solute parameters Ai and S will be different for different adsorbents. Hence, eqn.(3.72) gives a consistent description for one particular stationary phase. A new set of parameters will have to be established for each new adsorbent. 

As shown by Eq. (1), the resolution of components in a liquid chromatographic separation is dependent on (1) their relative retention on a particular chromatographic system and (2) their peak widths. To optimize these parameters for maximum resolution, a clear understanding of their nature and the factors that affect them is necessary. Although the retention time of a component adequately describes the amount of time a particular solute takes to elute from a chromatographic system, a more useful parameter describ-... 

When nsolutions with respect to a set of n unknowns, and each particular solution depends on certain assumptions which were made about the values of the remaining m-n parameters. When n-m, Eq. 5.30 has one exact solution, if it exists when det(A) 0. On the other hand, when n> m, the solution of Eq. 5.30 can be obtained in two fundamentally different ways. 

The parameters of the annealing protocol are largely determined empirically and are characteristic of the general topology of the particular solution space. But, in theory, there is a lower bound on the value of Ic- The expected distance... 

Its general solution is a one-parameter family of curves y = y(x, c), but in physical problems a particular solution is needed and this is picked out by the initial condition... 

Snyder has discussed liquid solvent characterization on several occasions (29,30,38,39). One of the very interesting points is that when several solvents have essentially the same solvent power as described by the Hildebrand solubility parameter (40), those solvents often do not dissolve a particular solute to the same extent. This is attributed to the fact that the Hildebrand solubility parameter does not, and cannot by regular solution theory. 

The second virial coefiicient A2, which is related to the Flory dilute solution parameters by Eq. (3.121), is a measure of solvent-polymer compatibility. Thus, a large positive value of A% indicates a good solvent for the polymer favoring expansion of its size, while a low value (sometimes even negative) shows that the solvent is relatively poor. The value of A2 will thus tell us whether or not the size of the polymer coil, which is dissolved in a particular solvent, will be perturbed or expanded over that of the unperturbed state, but the extent of this expansion is best estimated by calculating the expansion factor a. As defined by Eqs. (3.123) and (3.124), a represents the ratio of perturbed dimension of the polymer coil to its unperturbed dimension. 

In more concentrated solutions, additional mutual interactions must be considered, which can only be described in terms of ion-specific parameters. We will not do that here, but instead use an expression that, again, does not require any species-specific parameters, yet tends to yield a reasonably good description for the average behavior of more concentrated solutions (even though it may not represent any particular solution very well). This is the so-called Davies expression,... 

A model for calculating viscosities of concentrated polymer solutions has been formulated and used successfully to predict viscosities of alkyd resin solutions in both pure aromatic solvents and in mixtures of hydrocarbons and oxygenated materials. It was also found to describe viscosity trends in polystyrene-diethylbenzene solutions accurately. The formulation explicitly accounts for the observation that concentrated solution viscosities increase markedly with decreasing compatibility between resin and diluent. The proposal of an empirical relationship which interprets the viscosity enhancement in poorer solvents in terms of increased chain-chain interactions is of interest. The model contains three constants which are fixed for a particular resin and are independent of diluent type. These are the Mark-HouuAnk constant, the parameter in the Martin viscosity equation, and the constant relating the postulated clustering to the solution thermodynamics of a particular solution. 

---