Nonlinear coupled equations

Wlien working with any coordinate system other than Cartesians, it is necessary to transfonn finite displacements between Cartesian and internal coordinates. Transfomiation from Cartesians to internals is seldom a problem as the latter are usually geometrically defined. However, to transfonn a geometry displacement from internal coordinates to Cartesians usually requires the solution of a system of coupled nonlinear equations. These can be solved by iterating the first-order step [47]... 

The trial wave functions of a Schrodinger equation are expressed as determinant of the HF orbitals. This will give coupled nonlinear equations. The amplitudes were solved usually by some iteration techniques so the cc energy is computed as... 

The computational problem, then, is determination of the cluster amplitudes t for aU of the operators included in tlie particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wave functions expressed as dctcnninants of the HF orbitals. This generates a set of coupled, nonlinear equations in the amplitudes which must be solved, usually by some iterative technique. With the amplitudes in hand, the coupled-cluster energy is computed as... 

Equations (5) and (8) represent a system of coupled nonlinear equations, which can be solved for

boundaiy conditions, which are discussed in Appendix A. The various contributions to the free energy are also calculated in Appendix A. 

Langevin systems of coupled nonlinear equations have been used recently in the modeling of galactic evolution, in the framework of the so-called... 

The principal numerical problem associated with the solution of (7) is that lengthy calculations are required to integrate several coupled nonlinear equations in three dimensions. However, models based on a fixed coordinate approach may be used to predict pollutant concentrations at all points of interest in the airshed at any time. This is in contrast to moving cell methods, wherein predictions are confined to the paths along which concentration histories are computed. 

To determine the values of the amplitudes, one must solve this set of coupled nonlinear equations iteratively. A simple starting approximation for tf and on the left-hand sides of the equations may be obtained by setting all the amplitudes on the right-hand side to zero. Hence, for the T amplitudes we have... 

Only three of the parameters entering (91)—(93) create the system of coupled nonlinear equations which can be represented in the following form,... 

Equation (6) combines the mass balance and chemical equilibrium constraints for iron. It also shows that the chemistry of iron is coupled to that of other elements because the fugacities of sulfur, hydrogen, and oxygen are included in equation (6). In general, the chemistry of all the elements is coupled, and the mass balance equations form a set of coupled, nonlinear equations that are solved iteratively. An initial guess is assumed for the activity... 

The algebra involved in solving for the molar extent of reaction in general chemical equilibrium calculations can be tedious, especially if several reactions occur simultaneously, because of the coupled, nonlinear equations that arise. It is frequently possible, however, to make judicious simplifications based on the magnitude of the equilibrium constant. This is demonstrated in the next illustration. 

This partitioning allows for partial decoupling of the open-shell CC equations. The equations for the (m, n) sector involve only S elements from sectors (fc, 1) with k very large system of coupled nonlinear equations is separated into smaller subsystems, which are solved consecutively first, the equations for are iterated to convergence the 5(1,0)... 

The most general form of holonomic constraint is nonlinear in the particle positions. Even the simple bond-stretch constraint is nonlinear. Consequently, Eq. [39] is in general a system of / coupled nonlinear equations, to be solved for the / unknowns (7). This nonlinear system of equations must be contrasted with the linear system of equations Eqs. [10] and [11] (which is also in general part of the method of undetermined parameters) used in the analytical method to solve for the Lagrangian multipliers and their derivatives. A solution of Eq. [39] can be achieved in two steps ... 

When the surface is taken as ideal, that is, flat and homogeneous, the physical quantities depend only on the distance a from the surface. The surface imposes boundary conditions on the polymer order parameter fix) and electrostatic potential fix). In thermodynamic equihhrium, all charge carriers in solution should exactly balance the surface charges (charge neutrality). The Poisson-Boltzmann Equation (55), the self-consistent field Equation (56), and the boundary conditions uniquely determine the polymer concentration profile and the electrostatic potential. In most cases, these two coupled nonlinear equations can only he solved numerically. 

According to Golovin, Neponmyaschy and Pismen (1994), near the instability threshold, the nonlinear evolution and interaction between the two modes can be described by a system of two coupled nonlinear equations, namely ... 

This is a system of (m x n) quadratic equations for v. This system must be solved in combination with the equation of state, and thus, we are finally left with a system of (m x n + 1) coupled nonlinear equations for the reduced density, P, and the Vy. 

The calculation of the net ionic current at any desired location and time requires a self-consistent computation of the local concentration of the ions, flux, and the electric potential given by the combination of Equations 8.1, 8.4, and 8.5, for a prescribed dielectric heterogeneity, charge decoration on the pore, and the presence of polymer molecules. The above coupled nonlinear equations constitute the PNP formalism. For general situations, and for monitoring the time evolution of transients, it is necessary to resort to numerical procedures in solving these nonlinear coupled equations. 

All the terms in this equation are analytical functions of the parameter z except for A(z), which is the numerical solution of Equation (6.181). Equation (7.27) is a system of coupled nonlinear equations with removable singularity at z = -1 that may cause instability of the numerical solution. Note, however, that the nonlinearity of this equation is totally due to the factor 9 z) [see Equation (6.193)], which is related to the normalization condition for the vector U. Thus, for a given mode y the solution can be written in the form... 

These coupled nonlinear equations eire the key governing equations. It is possible to proceed further. First observe that... 

In addition to these three coupled nonlinear equations, we have two auxiliary equations from the moment closure approximation and the overall mass balance on the reaction medium,... 

Chapters 10 through 13 are devoted to the solution of nonlinear differential equations using numerical techniques. The chapters build in complexity from differential equations in single variables to coupled systems of nonlinear differential equations to nonlinear partial differential equations. Both initial value and boundary value differential equations are discussed with many examples. The discussion emphasizes the reuse of computer code developed in previous chapters in solving systems of coupled nonlinear equations. 

A final example in this section is that of a large system of coupled nonlinear equations. For this consider the equation set ... 

This example illustrates the time dependent solution of 5 coupled nonlinear equations with the automatic step selection algorithm used by flie odeivsQ integration routine. In this example, the automatic step selection routine is found to generate a solution at 8105 time points. However, about 1000 of these time steps are used in the first 0.01 time interval to really get the solution started with acceptable accuracy. Following this initial interval the odeibseQ routine generates an average of about 285 time steps per cycle to achieve the accuracy specified in the code. 

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