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Nonlinear terms

It is well noted that, in contiast to the two-state equation [see Eq. (26)], Eq. (25) contains an additional, nonlinear term. This nonlinear term enforces a perturbative scheme in order to solve the required x-matrix elements. [Pg.697]

Another subject with important potential application is discussed in Section XIV. There we suggested employing the curl equations (which any Bohr-Oppenheimer-Huang system has to obey for the for the relevant sub-Hilbert space), instead of ab initio calculations, to derive the non-adiabatic coupling terms [113,114]. Whereas these equations yield an analytic solution for any two-state system (the abelian case) they become much more elaborate due to the nonlinear terms that are unavoidable for any realistic system that contains more than two states (the non-abelian case). The solution of these equations is subject to boundary conditions that can be supplied either by ab initio calculations or perturbation theory. [Pg.714]

A convergence justification of the nonlinear terms 5 p rrf )ij to the term 5 p m )ij can be done by monotonicity arguments. The details are omitted here. [Pg.334]

Fig. 7. Load versus deflection for (a) perfectly brittle fracture and (b) slight nonlinearity. Terms are defined ia text. Fig. 7. Load versus deflection for (a) perfectly brittle fracture and (b) slight nonlinearity. Terms are defined ia text.
Keeping the pre-factor of this new nonlinear term free, we can summarize ... [Pg.862]

Let us generalize the differential equation (6-127) by adding a nonlinear term associated with x cos 21, obtaining... [Pg.381]

The principal difficulty with these equations arises from the nonlinear term cb. Because of the exponential dependence of cb on temperature, these equations can be solved only by numerical methods. Nachbar has circumvented this difficulty by assuming very fast gas-phase reactions, and has thus obtained preliminary solutions to the mathematical model. He has also examined the implications of the two-temperature approach. Upon careful examination of the equations, he has shown that the model predicts that the slabs having the slowest regression rate will protrude above the material having the faster decomposition rate. The resulting surface then becomes one of alternate hills and valleys. The depth of each valley is then determined by the rate of the fast pyrolysis reaction relative to the slower reaction. [Pg.42]

We can readily derive the higher-order expressions by simply retaining the nonlinear terms that arise from perturbed wavefunctions in the inhomogeneous equations. Thus,... [Pg.154]

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... [Pg.250]

Here summation is done over all types of interactions, including inductive, resonance and mixed (which is possible due to nonlinear term). The o constants denote electronic (Hammett-type) constants, whereas 5 and Hb denote steric and H-bonding interaction increments which mostly occur in the ortho position. [Pg.369]

Figure 4.8 Potential-dependent reaction energies for water dissociation to form OH, O, and H over Pt(l 11). (a) Energy curves based on the full charge model the nonlinearity of these plots expresses the capacitance of the interface, (b) Differences of the curves indicate the reaction energies. The nonlinear terms cancel almost completely. The dashed lines indicate predictions made from the linear model, whereas the solid lines are predictions made from a fuU solvation/ charge-based model [Rossmeisl et al., 2006]. Figure 4.8 Potential-dependent reaction energies for water dissociation to form OH, O, and H over Pt(l 11). (a) Energy curves based on the full charge model the nonlinearity of these plots expresses the capacitance of the interface, (b) Differences of the curves indicate the reaction energies. The nonlinear terms cancel almost completely. The dashed lines indicate predictions made from the linear model, whereas the solid lines are predictions made from a fuU solvation/ charge-based model [Rossmeisl et al., 2006].
We next expand the nonlinear term about the steady state value hs (also our initial condition by choice) to provide 1... [Pg.35]

The first-order transfer and exit rate constants can be replaced by nonlinear terms dependent on the amount or concentration of drug in a particular compartment. For instance, saturable metabolism of drug in compartment 1 (the central compartment) would result in the Michaelis-Menten equation... [Pg.77]

In the development given by Wood and Charles (W11), formulation B was used and the network elements were modeled by Eq. (74). Clearly the only nonlinear terms are those in Eq. (38) introduced by the network element models. By replacing Eq. (74) by the following equation,... [Pg.156]

One would notice that there are a number of nonlinear terms in the above constraints, specifically in the contaminant balance constraints. The linearisation technique used to remove these nonlinearities is that proposed by Quesada and Grossman (1995), the general form of this linearization technique can be found in Appendix A. During the application of the model to the illustrative examples,... [Pg.160]

The amount of water used for a cleaning operation is also now variable and is dependent on the size of the processing unit. Constraint (8.6) is reformulated to account for this and takes the form of constraint (8.60). Constraint (8.60) shows that the amount of water used is related to the size of the processing unit through a proportionality factor. Once again there is a nonlinear term present in constraint... [Pg.189]

The application of the maximum outlet concentration condition (Savelski and Bagajewicz, 2000) allows for the simplification of constraint (9.5) and subsequent linearisation of two nonlinear terms present in the resulting constraint. This is done as follows. [Pg.210]

The spectral method is used for direct numerical simulation (DNS) of turbulence. The Fourier transform is taken of the differential equation, and the resulting equation is solved. Then the inverse transformation gives the solution. When there are nonlinear terms, they are calculated at each node in physical space, and the Fourier transform is taken of the result. This technique is especially suited to time-dependent problems, and the major computational effort is in the fast Fourier transform. [Pg.59]

As mentioned, this equivalence is a consequence of the fluctuation-dissipation theorem (the general basis of linear response theory [51]). In (12.68), we have dropped nonlinear terms and we have not indicated for which state Variance (rj) is computed (because the reactant and product state results only differ by nonlinear terms). We see that A A, AAstat, and AAr x are all linked and are all sensitive to the model parameters, with different computational routes giving a different sensitivity for AArtx. [Pg.453]

Then this system of simultaneous linear algebraic equations can be solved using the subroutine GAUSS developed in Section 3.3. Because I have dropped the nonlinear term, I must always use a delx sufficiently small to ensure that all the dely values are indeed much smaller than the y values. [Pg.34]

The discretized equations of the finite volume method are solved through an iterative process. This can sometimes have difficulty converging, especially when the nonlinear terms play a strong role or when turbulence-related quantities such as k and s are changing rapidly, such as near a solid surface. To assist in convergence a relaxation factor can be introduced ... [Pg.341]

Here Ho includes not only the quadratic potential term but also some contribution from nonlinear terms a la the Hartree-Fock or mean-field approximation. Introducing a Green function for Ho... [Pg.280]

For weak electric fields the magnitude of the induced polarization is linearly proportional with the amplitude of the electric field. Yet, when the field strength increases, the linear relationship no longer holds, and nonlinear terms have to be taken into account. In this case, the induced dipole moment and polarization can be expressed up to second order in the electric field as11... [Pg.523]

As expected from the presence of nonlinear terms in the boundary conditions, no analytical solution for the problem defined by equations (1)—(6) is available to our knowledge, so a numerical strategy is applied here. As seen in ref. [22], the problem given by the differential equation (1) with boundary conditions (4)—(6) can be recast in the form of an integral equation for cm(/o, t) ... [Pg.152]

The hardness h are intimately related to the linear and nonlinear electronic responses as shown explicitly in Equation 24.18. In particular, h is simply the inverse of the linear polarizability it is well known in chemistry that a hard atom has a low polarizability. The nonlinear terms hn/, could allow to better quantify the hardness/softness and polarizability relations (see Section 24.2.2). Note that for an atom in a molecule, the contribution of a2 has to be considered as well in Equation 24.12 through Equation 24.18. On the other hand, Equation 24.18 shows that all the polarizabilities can be formulated in terms of the linear one, if the derivatives hn, which are function of p, are known ... [Pg.338]

As the intensity of the electromagnetic field, E co), increases the nonlinear terms become increasingly significant and, consequently, more easily measurable. [Pg.202]

So far we have looked at examples where all the nonlinearity is in the derivative terms, i.e., the right-hand sides of the ODE. Quite often the model of a system will give an ODE which contains nonlinear terms inside the time derivative itself. For example, suppose the model of a nonlinear system is... [Pg.174]

We have previously mentioned that the nonlinear nature of the polymerization reactor gives rise to a nonlinear immersion and as a consequence, it is impossible to construct its corresponding exponential holder. To avoid this problem, it is necessary to analyze the mathematical structure of wiss, to conclude that the nonlinear term is mainly due to the function that describes the effect gel. A suitable solution is given by finding a simpler mathematical function to satisfactorily describe the gel effect phenomena (we should recall that the most common gel effect functions such as (47), are actually given by empirical correlations). We propose the following function for gt to represent the diffusional limitation of the polymerization reaction... [Pg.109]


See other pages where Nonlinear terms is mentioned: [Pg.354]    [Pg.389]    [Pg.37]    [Pg.149]    [Pg.153]    [Pg.445]    [Pg.460]    [Pg.37]    [Pg.16]    [Pg.352]    [Pg.57]    [Pg.285]    [Pg.336]    [Pg.322]    [Pg.37]    [Pg.122]    [Pg.62]    [Pg.63]    [Pg.67]    [Pg.247]    [Pg.160]   
See also in sourсe #XX -- [ Pg.194 , Pg.246 ]

See also in sourсe #XX -- [ Pg.6 ]




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Dynamical nonlinear terms

Group Values and Nonlinear Correction Terms for Estimation of Solid Heat Capacity with the Goodman et al ethod

Linearising nonlinear terms

Nonlinear Short-Term Scheduling Model

Nonlinear coupling terms

Nonlinear polarization source term

Nonlinear polarization source term equation

Nonlinear terms Newton iteration

Small nonlinear terms, effect

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