Parameter-dependent initial value

Sensitivities of Parameter-Dependent Initial Value Problems... 

Three sets of data, sieved to yield constant-temperature triplets, were used to obtain the initial estimates of the parameters (kinetic constants) in these rate expressions. Arrhenius plots of these constants yielded the initial estimates of the activation energy and frequency factor of each of the temperature-dependent parameters. The initial values of the Arrhenius parameters were then inserted into the all-up rate expressions, including all the Arrhenius parameters. 

You need to specify two parameters the equilibrium value of the internal coordinate and the force constant for the harmonic potential. The equilibrium restraint value depends on the reason you choose a restraint. If, for example, you would like a particular bond length to remain constant during a simulation, then the equilibrium restraint value would probably be the initial length of the bond. If you want to force an internal coordinate to anew value, the equilibrium internal coordinate is the new value. 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

We can find from expressions Eq. (7) and Eq. (8) the functional dependence between the values of structure s parameter p, the value of electron s entry point into the structure xq and yo and the values of components of its entry s initial velocity Vqx and Voy at which electron remains in structure, or abandons it ... 

The initial values are Sj(0,p) = 0, j = 1, 2, 3 and 4. (Note that the initial values of the concentrations y and y2 do not depend upon the parameters investigated.) To solve the extended system of differential equations the following main program is used ... 

These starting values are used as initial guesses for fitting the model to industrial data and the preexponential factors are changed to obtain the best fit. This is done because the kinetic parameters depend upon the specific characteristics of the catalyst and of the gas oil feedstock. This complexity is caused by the inherent difficulties with accurate modeling of petroleum refining processes in contradistinction to petrochemical processes. These difficulties will be discussed in more details later. They are clearly related to our use of pseudocomponents. But this is the only realistic approach available to-date for such complex mixtures. 

The results in Fig. 6.37 show that a given gas can behave with respect to the matrix either as a donor or as an acceptor of electrons depending on the initial value of the WF of the polymer. Because the gas type and the doping concentration remained constant throughout those experiments, the only variable parameter was the initial work function value (WFjnit) of the polymer. Thus, the slope of the... 

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