Method of variation

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

Method of Variation of Parameters This technique is applicable to general linear difference equations. It is illustrated for the second-order system -2 + yx i + yx = ( )- Assume that the homogeneous solution has been found by some technique and write yY = -I- Assume that a particular solution yl = andD ... 

Variable Coejftcients The method of variation of parameters apphes equally well to the linear difference equation with variable coefficients. Techniques are therefore needed to solve the homogeneous system with variable coefficients. 

Comparatively little space will therefore be devoted to some rather recent approaches, such as the plasma model of Bohm and Pines, the two-body interaction method developed by Brueckner in connection with nuclear theory, Daudel s loge theory, and the method of variation of the second-order density matrix. This does not mean that these methods would be less powerful or less impor-... 

When the spectrum of R happens to be unknown in advance, it seems Worthwhile adapting iterative methods of variational type for determination of the correction. Unfortunately, in this book we have no possibility to say more about other combinations arising time and again in such matters. 

In Section II, the basic equations of OCT are developed using the methods of variational calculus. Methods for solving the resulting equations are discussed in Section III. Section IV is devoted to a discussion of the Electric Nuclear Bom-Oppenhermer (ENBO) approximation [41, 42]. This approximation provides a practical way of including polarization effects in coherent control calculations of molecular dynamics. In general, such effects are important as high electric fields often occur in the laser pulses used experimentally or predicted theoretically for such processes. The limits of validity of the ENBO approximation are also discussed in this section. 

Although in principle an exact solution to the Schrodinger equation can be expressed in the form of equation (A.13), the wave functions and coefficients da cannot to determined for an infinitely large set. In the Hartree-Fock approximation, it is assumed that the summation in equation (A.13) may be approximated by a single term, that is, that the correct wave function may be approximated by a single determinantal wave function , the first term of equation (A.13). The method of variations is used to determine the... 

Parameters ve, a , x and dispersion values, decay kinetics of trapped electrons in water-alkaline (6M NaOH) glass... 

T = 77 K. The parameters were calculated using the method of variation of log v . 

The method of variation of the irradiation intensity gives as well as Po " 4-dimethyl-aminoazobenzene in n-hexane irradiation at 366 nm or 405 nm, both of which are wavelengths of the 7i —> 7U band, the results are 0F.- z = 0.17 and (j)2 i = 0.25. Albini et al.,"" who calculated the extent of Z —> E isomerization by an extrapolation of this conversion back to zero time of irradiation, found somewhat higher values for 4-diethylamino-... 

Fig. 2 presents the gain spectrum for the laser structure with Xi = 1.8 nm obtained using the method of variation of exciting stripe length. The stripe length L was 50 pm for these measurements, the pumping power density was - 47.5 GW/cm. It is seen that amplification occurs in a wide interval of 80 nm and the gain value reaches 85 cm. ... 

For simple-shaped electrodes (plane, cylinder, and sphere), the problem can be solved analytically [1-9]. For more complex-shaped electrodes, numerical methods are used. For the numerical solution of the set of equations (11), both iteration methods (Ref. 36, for example) and the method of variation of inequalities are used. The latter allows one to reduce the initial problem to an equivalent problem in the region with fixed boundaries [37, 38]. 

THE TREATMENT OF A TIME-DEPENDENT PERTURBATION BY THE METHOD OF VARIATION OF CONSTANTS... 

The same probabilities are given directly by our application of the method of variation of constants. The probability of transition to states of considerably different energy as the result of a small perturbation acting for a short time is very small, and we have neglected these transitions. Our calculation shows that the probability of finding the system in the state B depends on the value of t in the way given by Equation 41-12, varying harmonically between the limits 0 and 1. 

Let us consider a system composed of five harmonic oscillators, all with the same characteristic frequency v, which are coupled with one another by weak interactions. The set of product wave functions 4 (a) k(6) (c) (d) k(e) can be used to construct approximate wave functions for the system by the use of the method of variation of constants (Chap. XI). Here (a), ... 

By changing the composition of the solvent, we modify x and therefore also the contrast length b (x). This is the method of variation of contrast.8,9... 

This example can be solved most easily by use of equation 63 in the tables following. However, it is given here as an example of the method of variation of parameters. 

In fact, there are a few generalized approaches utilizing the method of variation. In this particular context, the familiarity with the molecular structure is of the prime importance. The various possible approaches in designing newer drugs by applying variation of a prototype are quite numerous. Once the molecular structure of the compound in question is drawn on the drawing board, one takes into consideration such information as the following ... 

The application of the method of variation, depending on the considerations enumerated above, is exploited in two different manners to evolve a better drug. The two main approaches for this goal can be indicated as ... 

Enumerate the various cardinal objectives of The Methods of Variation" giving appropriate examples. 

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