Iterative numerical integration

In unseeded polymerizations, the number of particles increases and they grow in size by polymerization and imibition of monomer, so that both N and r are functions of time. The value of N is computed by iterative numerical integration of Equation 4, whilst the value of r must be calculated from the rate of polymerization Since the particles are spheres,... 

The general model assumes instantaneous equilibria in the boundary layer of all solution species except C02> It uses a different diffusivity for each species. It accounts for the finite-rate, reversible reaction of CO2 and H2O to give IT " and HCO3- by iterative, numerical integration of a second-order, nonlinear differential equation and a set of nonlinear algebraic equations. 

The first of the two sequential steps of the iterative numerical integration is described. As indicated above, the electromigration is the only transport phenomena considered in this stage that changes the concentration of the ions. Therefore, the mass conservation equation for a volume element of the soil (1 < / < A,) is given by... 

In the second approach, reaction curves are calculated with sets of preset parameters by iterative numerical integration from a preset staring point. Such calculated reaction curves are fit to a reaction curve of interest the least sum of residual squares indicates the best fitting (Duggleby, 1983, 1994 Moruno-Davila, et al., 2001 Varon, et al., 1998 Yang, et al., 2010). In this approach, calculated reaction curves still utilize reaction time as the paedictor variable and become discrete at the same intervals as the reaction curve of interest. Clearly, there is no transformation of data from a reaction curve in this approach. 

With any enzyme, iterative numerical integration of the differential rate equation(s) from a starting point with sets of preset parameters can be universally applicable regardless of the complexity of the kinetics. Thus, the second approach exhibits better imiversality and there are few technical challenges to kinetic analysis of reaction curve via NLSF. In fact, however, the second approach is occasionally utilized while the first approach is widely practiced. 

Numerieal and eomputational problems associated with the implementation of the approach for routine use fall in two main categories (a) numerical integration and (b) enforcement of the orthogonality and renormalization of the numerical orbitals during the iteration steps. Many different integration schemes have been considered in the past, some of which will be detailed in the section 3.2. As concerns orthonormalization, at... 

Thus Y1 is obtained not as the result of the numerical integration of a differential equation, but as the solution of an algebraic equation, which now requires an iterative procedure to determine the equilibrium value, Xj. The solution of algebraic balance equations in combination with an equilibrium relation has again resulted in an implicit algebraic loop. Simplification of such problems, however, is always possible, when Xj is simply related to Yi, as for example... 

In all considered above models, the equilibrium morphology is chosen from the set of possible candidates, which makes these approaches unsuitable for discovery of new unknown structures. However, the SCFT equation can be solved in the real space without any assumptions about the phase symmetry [130], The box under the periodic boundary conditions in considered. The initial quest for uy(r) is produced by a random number generator. Equations (42)-(44) are used to produce density distributions T(r) and pressure field ,(r). The diffusion equations are numerically integrated to obtain q and for 0 < s < 1. The right-hand size of Eq. (47) is evaluated to obtain new density profiles. The volume fractions at the next iterations are obtained by a linear mixing of new and old solutions. The iterations are performed repeatedly until the free-energy change... 

There are many analytical chemistry textbooks that deal with the chemical equilibrium in fairly extensive ways and demonstrate how to resolve the above system explicitly. However, more complex equilibrium systems do not have explicit solutions. They need to be resolved iteratively. In kinetics, there are only a few reaction mechanisms that result in systems of differential equations with explicit solutions they tend to be listed in physical chemistry textbooks. All other rate laws require numerical integration. 

Digital simulation is a powerful tool for solving the equations describing chemical engineering systems. The principal difficulties are two (1) solution of simultaneous nonlinear algebraic equations (usually done by some iterative method), and (2) numerical integration of ordinary differential equations (using discrete finite-difference equations to approximate continuous differential equations). 

Combining Eq. (24) with Eqs. (12) (14), Chen obtained f(a) for given N and c by the following numerical analysis. Firstly, a zero-th approximation to U(a) was calculated by numerical integration of Eq. (24) using a properly chosen zero-th approximation to f(a), secondly the calculated U(a) was used to obtain a first-order approximation to f(a) by solving the differential equation, Eq. (12), with Eqs. (13) and (14), and finally the process was iterated until the mean-square relative difference between the two successive approximations toT(a) became less than a prescribed small value. 

A Jacobi or Gauss-Seidel iteration on (6) will provide us with the coordinates of the steady state, or it will cycle indefinitely, depending on the slope of the functions. On the other hand, determining the trajectory by numerical integration of equation (5) will lead to a stable steady state or to a limit cycle depending on the slope of the functions. There is thus an obvious formal similarity between the two situations. However, the steepness corresponding to the transition from a punctual to a cyclic attractor is much smaller in the first case (in which the cyclic attractor is an iteration artifact) as in the second case (in which the cyclic attractor is close to the real trajectory). 

Despite the non-isothermal nature of the film blowing process we will develop here an isothermal model to show general effects and interactions during the process. In the derivation we follow Pearson and Petrie s approach [20], [19] and [21]. Even this Newtonian isothermal model requires an iterative solution and numerical integration. Figure 6.21 presents the notation used when deriving the model. 

Since T depends on BUR, we must first specify BUR and iterate until a solution of R(z) is found that agrees with the choice of BUR. Hence, we must integrate eqn. (6.119) numerically with each choice of BUR. After the correct value of BUR has been found, we numerically integrate eqn. (6.120). Figures 6.23 and 6.24 present solutions for a fixed value of Z = 20 and a fixed value of B = 0.1, respectively. 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

This procedure had converged in 4 or 5 iterations to four significant figures for all cases tried in this study. The accuracy of the calculations depends on the time increment At because the finite difference approximations become more accurate as At gets smaller. A summary of some iteration results and a comparison between this technique and the numerical integration with Gear s method will be presented after the following discussion on the stability of the temperature equation. 

Roothaan s Self-Consistent-Field Procedure.—While numerical integration techniques may be used to solve the Hartree-Fock equations in the case of atoms by the iterative method described above, the lower symmetry of the nuclear field present in molecules necessitates the use of an expansion for the determination of the molecular orbitals by a method developed by Roothaan.81 In Roothaan s approach, it is assumed that each molecular orbital may be adequately represented by a linear expansion in terms of some (simpler) set of basis functions xj, i.e. 

Other details of molecular orbital calculations were as follows A numerical basis set used was Is atomic orbital for H, 1 s-2p for C and O, and 1 s-3p for A1 and Si. The formal charges of oxide elements in iterations were +3e for Al, +4e for Si, —2e for O and those of the other molecule elements were neutral. The DV sampling points for numerical integration were 500-1000 per atom. For molecular orbitals, point group was not taken into account. 

The usual kinetic equations for the calculation of quantum yields and thermal fade rate constants have been extended by taking into account the information contained in the experimentally recorded absorbance vs. time curves recorded under continuous irradiation and by adding additional kinetic terms representing photodegradation or other mechanistic complications. The extraction of the rate constants and quantum yields from the experimental curve requires numerical integration and iterative calculations.176... 

Normalized variables and nondimensional parameters may be identified from equation (114) [73], and an iterative method of solution may be devised [50] and applied [73] to obtain an approximate formula for — Po o Po o- This formula exhibits the attributes of the burning velocity for the heterogeneous regime that were discussed in Section 11.6.1. Thus a more formal development leads to the qualitative results that have been inferred from physical reasoning. It is instructive to have seen a representative sequence of approximations involved in a formal development. The iterative method [50] has been applied successfully [77] with fewer approximations, through use of techniques for numerical integration, to derive... 

Integral Equation Solutions. As a consequence of the quasi-steady approximation for gas-phase transport processes, a rigorous simultaneous solution of the governing differential equations is not necessary. This mathematical simplification permits independent analytical solution of each of the ordinary and partial differential equations for selected boundary conditions. Matching of the remaining boundary condition can be accomplished by an iterative numerical analysis of the solutions to the governing differential equations. 

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