Decoupling equations

Decoupling equations is one of the most important tools for reducing complex mathematical models. A model witii many dependent variables, e.g. different velocities, concentrations of chemical substances, and temperature, can be simplified substantially if we look for variables that are not dependent on the others. In Equations (5.1) (5.4), the coupling occurs mainly through the velocity in all the equations and through temperature. The reaction rate in Equation (5.1) depends on temperature, and the temperature in Equation (5.2) depends on file reaction rate through the heat of the reaction. Viscosity and density of file fluid also depend on temperature and chemical composition, and as a consequence all reactions must be solved simultaneously. On the contrary, if the heat of the reaction is low, tire temperature variation will be small, the fluid density and viscosity can be assumed constant, and the flow may be calculated in advance. In such a case, the chemical reactions can then be added to a precalculated velocity distribution. 

Coupling may also occur via boundary conditions, e.g. the reaction rate in a catalyst pellet depends on the concentration and the temperature of the fluid surrounding the pellet. At steady state, when coupling between equations occnrs throngh boundary conditions, an exact or approximate analytical solution can be calculated with boundary conditions as variables, e.g. the effectiveness factor for a catalyst particle can be formulated as an algebraic fimction of surface concentrations and temperature. The reaction rate in the catalyst can then be calculated using the effectiveness factor when solving the reactor model. However, this is not possible for transient problems. The transport in and out of the catalyst also depends on the accumulation within the catalyst, and the actual reaction rate depends on the previous history of the particle. 

A very efficient method is to precalculate steady-state solutions and save the results in look-up tables. Interpolation in multidimensional look-up tables is most often much faster than iterative solutions of non-linear equations. It is also possible to use look-up tables for initial-value differential equations using time as one variable in the look-up table. 

Decoupling equations may also be used to enhance the numerical stability of stationary problems even when the equations have strong coupling. You can start by solving, e g., the flow field and then continue with species, reactions, and heat, and finally solve the flow field, species, and heat with updated properties for concentrations, temperature, density, and viscosity. This control of the iterations allows for a more stable approach to the final solution. 

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Since G0 is diagonal, the matrix [I + G0 G0 is also diagonal, and happily, we have two decoupled equations in (10-42). 

These equations (14) and (15) determine the scalar and vector potentials in terms of p and J. When p and J are zero, these equations become wave equations with wave velocity c = y/l/pe. That is, A and are solutions of decoupled equations, where they are related by the wave operator... 

The decoupled equations can be integrated to any given order. Equation (7) simply confirms that the cj are the time-dependent initial conditions of the problem. The system is assumed to be in an initial well-defined stationary... 

Chebyshev Method for Calculating State-to-State Reaction Probabilities from the Time-Independent Wavepacket Reactant-Product Decoupling Equations. 

The analytical solution of the decoupled equation (69) is discussed in reference [32], The non-adiabatic coupling between the lower sheet e and upper sheet e+ is calculated from... 

Equation (6.29) is an easy equation to solve because D is a diagonal matrix and equation (6.29) yields the following decoupled equation for Zj, the i row of Z matrix (du is the diagonal element of D matrix)... 

Therefore, the elements of dC) are linear combinations of the elements of dO, and the elements of dX are linear combinations of the elements of dX. Because D is a diagonal matrix, Eq. [26c] represents a set of n decoupled equations. Each equation relates an element of dO to an element of dX, that is, ... 

We now define (wi p) as the solutions of the decoupled equations obtained from (5.6) by omitting all coupling terms ... 

Substituting Equation (15.31) into Equation (15.20), and making use of the orthogonality relationships a set of N decoupled equations is obtained (in reality only if the displacement matrix satisfies certain conditions (Castellani et al., 2000)) ... 

The variational solution of the decoupled equation of motion (Eq. Ills) for the Wigner distribution function might serve as a starting point for further studies of exchange and correlation in the dielectric function. Its connection with several other approximations has been examined, showing that many of them are particular cases or additional approximations to this variational approach. The improvement upon the RPA from dynamical exchange effects, and the fact that all checked sum rules are satisfied, gives... 

Thus, the transformation equation (A.41) has converted the original equation (A.39) into n completely decoupled equations, each of which has a solution of the form... 

In the time domain, the RPD scheme partitions the full time-dependent (TD) wave function into a sum of reactant component r) and all product components ( p, p = 1,2,3,...) that satisfy the following decoupled equations ... 

Althorpe SC, Kouri DJ, Hoffman DK (1997) Further partitioning of the reactant-product decoupling equations of state-to-state reactive scattering and their solution by the time-independent wave-packet method. J Chem Phys 107(19) 7816... 

This condition is called the Lorentz condition and defines the Lorentz gauge. Using this condition in (3.19) provides us finally with a set of decoupled equations ... 

Eliminating E ox H from these equations will yield decoupled equations for H and E ... 

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