Coupled differential equations

The Eik/TDDM approximation can be computationally implemented with a procedure based on a local interaction picture for the density matrix, and on its propagation in a relax-and-drive perturbation treatment with a relaxing density matrix as the zeroth-order contribution and a correction due to the driving effect of nuclear motions. This allows for an efficient computational procedure for differential equations coupling functions with short and long time scales, and is of general applicability. 

This latter differential equation, coupled with the algebraic expression relating qout to h, may be solved to determine the fluid height as a function of time. The initial fluid height is 50.93 m, based upon the specified initial fluid volume and tank diameter. This serves as the initial condition to be used for the integration. The E-Z Solve syntax is ... 

Currently, computing the structure of bluff-body stabilized flames has become a subject of intense activity. The general objective of numerical studies is to describe the phenomenon by solving the fundamental differential equations coupled with turbulence and combustion closures. Since there are many possible approaches, more or less substantiated, the reported results are often contradictory. Apparently, this is caused by the lack of basic understanding of the physico-chemical phenomena accompanying flame stabilization and spreading. 

Box21.6 Solution of Two Coupled First-Order Linear Inhomogeneous Differential Equations (Coupled FOLIDEs)... 

Obviously, we have to deal with a partial differential equation coupled to our ordinary differential equation [Eq. (1)]. It can be easily shown that... 

The reference time, i f and concentration, Cp. f, are chosen for a specific application (e.g., in a flow reactor, the mean residence time and feed concentration, respectively). Equation 5.2.C-6 now permits a solution for the amount of poison, /Cpia, to be obtained as a function of the bulk concentration, Cp, and the physicochemical parameters. In a packed bed tubular reactor, Cp varies along the longitudinal direction, and so Eq. 5.2.C-6 would then be a partial differential equation coupled to the flowing fluid phase mass balance equation—these applications will be considered in Part Two—Chapter 11. 

The rate of an overall reaction is a composite of the rates of the elementary reactions in the mechanism, which form a set of ordinary differential equations coupled through the concentrations of chemical species, and can be expressed as the following initial value problem ... 

Solving Differential-Algebraic Equation DAE Systems Many of the previously mentioned numerical methods lead to large sets of differential equations coupled with sets of nonlinear algebraic equations. These are the... 

Equations (9.2) represent a set of second-order differential equations coupled in time, which can be solved numerically on a large computer for N 1000 and a certain small time period, if the pair potential is not too complicated. Here, we restrict ourselves to the application of the Lennard-Jones pair potential of the form... 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

There is no general, simple solution to this set of coupled differential equations, and thus one will usually have to resort to numerical teclmiques [42, 43] (see also chapter A3.4). 

Hence, in order to contract extended BO approximated equations for an N-state coupled BO system that takes into account the non-adiabatic coupling terms, we have to solve N uncoupled differential equations, all related to the electronic ground state but with different eigenvalues of the non-adiabatic coupling matrix. These uncoupled equations can yield meaningful physical... 

Substitution of Eq. (12) into the Schrodinger equation leads to a system of coupled differential equations similai to Eq. (5), but with the following differences the potential matrix with elements... 

Here, dj = cos(y,j) and sy = sin(Yy). The three angles are obtained by solving the following three coupled first-order differential equations, which follow from Eq. (19) [36,84,85] ... 

Equation (26) is a set of partial first-order differential equations. Each component of the Curl forms an equation and this equation may or may not be coupled to the other equations. In general, the number of equations is equal to the number of components of the Curl equations. At this stage, to solve this set of equation in its most general case seems to be a fomiidable task. 

In Section V.B, we discussed to some extent the 3x3 adiabatic-to-diabatic transformation matrix A(= for a tri-state system. This matrix was expressed in terms of three (Euler-type) angles Y,y,r = 1,2,3 [see Eq. (81)], which fulfill a set of three coupled, first-order, differential equations [see Eq. (82)]. 

The quaternions obey coupled differential equations involving the angular velocities tJi, tbody frame (i.e. tJi represents the angular velocity about the first axis of inertia, etc.). These differential equations take the form... 

The procedure we followed in the previous section was to take a pair of coupled equations, Eqs. (5-6) or (5-17) and express their solutions as a sum and difference, that is, as linear combinations. (Don t forget that the sum or difference of solutions of a linear homogeneous differential equation with constant coefficients is also a solution of the equation.) This recasts the original equations in the foiin of uncoupled equations. To show this, take the sum and difference of Eqs. (5-21),... 

Greater detail in the treatment of neutron interaction with matter is required in modem reactor design. The neutron energy distribution is divided into groups governed by coupled space-dependent differential equations. 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

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