Coupling equation

The central equations of electromagnetic theory are elegantly written in the fonn of four coupled equations for the electric and magnetic fields. These are known as Maxwell s equations. In free space, these equations take the fonn ... 

With these simplifications, and with various values of the as and bs, van Laar (1906-1910) calculated a wide variety of phase diagrams, detennining critical lines, some of which passed continuously from liquid-liquid critical points to liquid-gas critical points. Unfortunately, he could only solve the difficult coupled equations by hand and he restricted his calculations to the geometric mean assumption for a to equation (A2.5.10)). For a variety of reasons, partly due to the eclipse of the van der Waals equation, this extensive work was largely ignored for decades. 

Alexander M H and Manolopoulos D E 1987 A stable linear reference potential algorithm for solution of the quantum close-coupled equations in molecular scattering theory J. Chem. Phys. 86 2044-50... 

Here all couplings are ignored except the direct couplings between the initial and final states as in a two-level atom. The coupled equations to be solved are... 

B2.2.9.1 CLOSE-COUPLING EQUATIONS FOR ELECTRON-ATOM (ION) COLLISIONS... 

Numerical solution of this set of close-coupled equations is feasible only for a limited number of close target states. For each N, several sets of independent solutions F.. of the resulting close-coupled equations are detennined subject to F.. = 0 at r = 0 and to the reactance A-matrix asymptotic boundary conditions,... 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

For electronic transitions in electron-atom and heavy-particle collisions at high unpact energies, the major contribution to inelastic cross sections arises from scattering in the forward direction. The trajectories implicit in the action phases and set of coupled equations can be taken as rectilinear. The integral representation... 

Miller W H 1969 Coupled equations and the minimum principle for collisions of an atom and a diatomic molecule, including rearrangements J. Chem. Phys. 50 407... 

Stechel E B, Walker R B and Light J C 1978 R-matrix solution of coupled equations for inelastic scattering J. Chem. Phys. 69 3518... 

While the presence of sign changes in the adiabatic eigenstates at a conical intersection was well known in the early Jahn-Teller literature, much of the discussion centered on solutions of the coupled equations arising from non-adiabatic coupling between the two or mom nuclear components of the wave function in a spectroscopic context. Mead and Truhlar [10] were the first to... 

Substitution of Eq. (3) into the molecular Schrodinger equation leads to a system of coupled equations in a coupled multistate electronic manifold... 

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),... 

To extraet from this set of coupled equations relations that ean be solved for the eoeffieients , whieh embodies the desired wavefunetion perturbations /k ", one eolleets together all terms with like power of X in the above general equation (in doing so, it is important to keep in mind that Ek itself is given as a power series in X). 

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the flmd. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-HiU, New York, 1958 and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949 vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation hL I L p gp At cjl... 

The couphng equation is a vapor mass balance written at the vent system entrance and provides a relationship between the vent rate W and the vent system inlet quahty Xq. The relief system flow models described in the following section provide a second relationship between W and Xo to be solved simultaneously with the coupling equation. Once W andXo are known, the simultaneous solution of the material and energy balances can be accomplished. For all the preceding vessel flow models and the coupling equations, the reader is referred to the DIERS Project Manual for a more complete and detailed review. 

Analytieal solutions to equation 4.32 for a single load applieation are available for eertain eombinations of distributions. These coupling equations (so ealled beeause they eouple the distributional terms for both loading stress and material strength) apply to two eommon eases. First, when both the stress and strength follow the Normal distribution (equation 4.38), and seeondly when stress and strength ean be eharaeterized by the Lognormal distribution (equation 4.39). 

Figure 4.28 Derivation of the coupling equation for the case when both loading stress and material strength are a Normal distribution...

Substituting the given parameters for stress and strength in the coupling equation (equation 4.38) gives ... 

Comparing answers with that derived from the coupling equation, the Standard Normal variate is ... 

There are many different solutions for X1 and X2 to this pair of coupled equations, but it proves possible to find two particularly simple ones called normal modes of vibration. These have the property that both particles execute simple harmonic motion at the same angular frequency. Not only that, every possible vibrational motion of the two particles can be described in terms of the normal modes, so they are obviously very important. 

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

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