Implicit Equation

An implicit differential equation is an equation in which the dependent variable is expressed as a function of its derivative  

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To derive equations for the order-by-order contributions to the eigenvalue X, the implicit equation for the eigenvalue is first rewritten as... 

Since angular momentum is conserved, equation (A3.11.192) may be rearranged to give the following implicit equation for the time dependence of r ... 

This is an implicit equation which can only be solved by iteration. One might use a spreadsheet or a programmable calculator to solve for P from this equation. An alternative is to read P from Figure 6.26 or 6.27. 

This implicit equation is usually employed to derive values of rtn corresponding to the minima of G. Graphical methods for this purpose have been discussed by Drickamer and Frank [37]. The transition temperature is defined for nj, = 5, and this corresponds according to Eq. (13) to ... 

The integrated form of substrate utilization in an enzyme catalyzed batch bioreactor is given by the implicit equation... 

Taking the average values for the constant C as given above, we calculated the value of from (the implicit) Equation 14. 

Equations (18.26) and (18.28) are implicit equations for the occupation probability n, since it appears both on the left- and on the right-hand side. They can be solved by simple numerical procedures. In the cases considered here there is always one unique solution. 

Substituting Equation (6.35) and taking 7b0t = 7 XJ gives an implicit equation for 7) ... 

This gives an implicit equation in the choked pressure ratio T, (shorthand for T ch/T s) ... 

Combining the two equations for steam density and pressure gives an implicit equation requiring a numerical root estimation. 

In some cases, the characteristic velocity can cause difficulties in solution, owing to the presence of an implicit equation. In this the appropriate value of L or Gn satisfying the value of hn generated by the differential material balance equation must be found by root finding algorithms increasing computation time required. 

Roots of implicit equations and extrema of functions the Newton method... 

Explicit different schemes show poor stability properties (Mitchell, 1969). In terms of the central difference operator, it may be shown that an accurate implicit equation is... 

The third relation needed is Eq. (73) the carbide was immersed in liquid uranium metal. The fourth was a compromise between the Young equation (1) and Eq. (75) in that cos a of the latter was implicitly equated to unity, so that... 

There are several other possibilities for robustly estimating the central value. Well known are M-estimators for location (Huber 1981). The basic idea is to use a function iji that defines a weighting scheme for the objects. The M-estimator is then the solution of the implicit equation... 

Starting with the basic model assumptions, the analytical heater model is developed in several steps [126]. The equations include common model equations such as the Shichman-Hodge model [127], the LEVELS model [128] and the BSlMS.vS model [129]. Only selected components of these partly complex models were taken to yield a set of equations that is suitable for modelling the transistor heater. The variables and parameters have been defined in accordance to standard notations. First, a model has been estabhshed that describes the unheated transistor, then, temperature dependencies have been introduced, and, finally, the electrothermal coupHng to the microhotplate has been considered. The result is an implicit equation, which can be iteratively solved. The considered model will be compared to measurement data in Sect. 4.4.4. 

The explicit methods considered in the previous section involved derivative evaluations, followed by explicit calculation of new values for variables at the next point in time. As the name implies, implicit integration methods use algorithms that result in implicit equations that must be solved for the new values at the next time step. A single-ODE example illustrates the idea. 

It does not appear that the ICSEs can be solved by self-consistent iteration, however. In Eq. (68), CSE(2) is expressed in a form that affords the 2-RDM as an explicit functional of the 2-, 3-, and 4-RDMs, but no analogous formulation of ICSE(l) or ICSE(2) is possible, since the 1- and 2-RDMCs appearing in these equations are always acted upon by or g (cf. Eqs. (66) and (67)). Thus the ICSEs are implicit equations for the cumulants. 

The only function of interest in the given context is w(Ar). The stability question is then answered if the rate, w(A), has been found to be positive or negative at any value of k or wavelength A of the perturbation. The validity of this argument is due to the linearized differential equations, for which we know their solutions can be superposed. Negative w(A) means that 0- O for t- o°. Insertion of Eqns. (11.16) and (11.17) into the transport equation and the boundary condition yields an implicit equation for w(k). If we use the following transformations to express w and tin terms of the characteristic parameters Dv and v of the system, namely... 

The critical behavior is, however, the same there is a Kosterlitz-Thouless (KT) transition at the phase boundary Ku between a disorder dominated, pinned and a free, unpinned phase which terminates in the fixed point K = 6/p2. One can derive an implicit equation for Ku by combining (23a) and (23b) to a differential equation... 

We see that a calculation of Ar involves a Laplace transform of the time-dependent friction kernel. This may typically be determined in a molecular dynamics (MD) simulation, where the autocorrelation function of the random force (R(O)R(t)) may be determined, which then allows us to determine (f) using the fluctuation-dissipation theorem in Eq. (11.58). Note that Eq. (11.85) is an implicit equation for Ar that in general must be solved by iteration. In the absence of friction we see from Eq. (11.85)... 

The design charts and the examples provided in this paper illustrate the simple procedure of solving a common ultrafiltration problem. In general, when P, Q and R fall beyond the covered ranges, additional charts can be readily prepared by solving the implicit equations presented in this paper. 

As for location, M-estimators of scale can be defined as the solution of an implicit equation [3], Then, again, an initial scale estimate is needed, for which the MAD is usually taken. Simultaneous M-estimators of location and scale can also be considered, but they have a smaller breakdown value, even in small samples [5],... 

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