Variation of parameters

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. 

The sensitivity of analyses may be inereased by variation of parameters what provides the rise of the eompleteness of gaseous extraetion - t°, v, for hydrophilie organie substanees - the presenee of salting-out agents. It has been determined that the most effeetive extraetion of aliphatie aleohols is a eharaeteristie feature for high eoneentration alkaline solutions. 

The assumptions that go into the analysis often are the outgrowth of a thorough understanding of perfect or near perfect thermodynamic behavior. Some assumptions are based upon the near constancy of certain parameters for wide ranges of commercial practice. Linear variation of parameters is also used at times for simplification. 

The use of this theory in studies of nonlinear oscillations was suggested in 1929 (by Andronov). At a later date (1937) Krylov and Bogoliubov (K.B.) simplified somewhat the method of attack by a device resembling Lagrange s method of the variation of parameters, and in this form the method became useful for solving practical problems. Most of these early applications were to autonomous systems (mainly the self-excited oscillations), but later the method was extended to... 

Agitated reactor (possibly with catalyst particles) Catalytic and noncatalytic Reactions, polymerizations (special agitator required) High transport rates, convenient to operate, easy variation of parameters, most versatile Catalyst erosion... 

Basket-type reactor (CSTR) for gas-phase reactions) High temperature, high pressure catalytic processes High transport rates, easy variation of parameters Limited particle size, high equipment cost, difficult to operate under a wide range of conditions without creating flow maldistribution... 

Microreactors Low conversion, catalytic reactions Simple design, transport rates can be increased by external recycling Limited ease of variation of parameters, maldistribution of flow can be prohibitive... 

Acoustic cavitation is as a result of the passage of ultrasound through the medium, while hydrodynamic cavitation occurs as the result of the velocity variation in the flow due to the changing geometry of the path of fluid flow. In spite of this difference in the mechanisms of generation of two types of cavitation, bubble behavior shows similar trends with the variation of parameters in both these types of cavitation. The two main aspects of bubble behavior in cavitation phenomena are ... 

Regarding the global sensitivity analysis, the results indicate that the variation of the model output is highly sensitive to the variations of parameters used in fish and root compartments. The higher concentration of Pb in fish than in potato, leaf, root, milk, and beef (Fig. 6) reflects that the variation of the model output is more sensitive to variations of fish parameters than of potato, leaf, root, milk, and beef parameters. 

The group of all rotations about an axis is a continuous group of order 1, whose parameter may be chosen to be the angle of rotation, 0 taking values in the interval [—7r, 7r]. A group like this, where the domain of variation of parameters is finite, is called a closed group. 

Larger scattering of the properties of quark stars is obtained by Andersen and Strickland (2002), who used the same Hard-Dense-Loop approach, but with wider variation of parameters. Their models of quark stars are represented in Fig. 10. 

Release profiles were characterized by fitting the time-exponent equation (equation (2)) to the data. Equation (2), which also includes equation (1) empirically decribes release profiles of matrices deviating from equation (1). Release exponents greater than 0.5 indicate time-dependent variation of parameters in equation (1) which leads to a more constant drug delivery... 

It will be seen from Table 5 that the variation of parameters with atomic number is by no means always monotonic. The explanation of this must again be connected with variation of coordination number along the series of lanthanides. If the strongly coordinating organic ligands are reasonably assumed to exert their full ligancy, then any variation must be in the extent of inner-sphere hydration of the various species, as it is rather unlikely that there is a... 

To solve the diffusion equation and obtain the appropriate rate coefficient with these initial distributions is less easy than with the random distribution. As already remarked, the random distribution is a solution of the diffusion equation, while the other distributions are not. The substitution of Z for r(p(r,s) — p(r, 0)/s) is not possible because an inhomogeneous equation results. This requires either the variation of parameters or Green s function methods to be used (they are equivalent). Appendix A discusses these points. The Green s function g(r, t r0) is called the fundamental solution of the diffusion equation and is the solution to the... 

To solve a second-order inhomogeneous ordinary differential equation, either the Green s function method or the variation of parameters method can be used. Consider the self-adjoint equation... 

The variation of parameters method uses the two linearly independent solutions of the homogeneous equation (332), y y (x) and y2 (x) (which, of course, also appear in the Green s function) and so... 

The difference between eqn. (330) and eqn. (338) is that y andy2 in eqn. (338) are forced to satisfy the boundary conditions prior to using these independent solutions, while in eqn. (330), the independent solutions do not satisfy the boundary conditions. The form of eqns. s33) and (337) are very similar and shows that the Green s function method and the variation of parameters method of solution are equivalent. 

The performed calculations demonstrate that a type of the asymptotic solution of a complete set of the kinetic equations is independent of the initial particle concentrations, iVa(0) and 7Vb(0). Variation of parameters a and (3 does not also result in new asymptotical regimes but just modifies there boundaries (in t and k). In the calculations presented below the parameters 7Va(0) = 7Vb(0) =0.1 and a = ft = 0.1 were chosen. The basic parameters of the diffusion-controlled Lotka-Volterra model are space dimension d and the ratio of diffusion coefficients k. The basic results of the developed stochastic model were presented in [21, 25-27],... 

The phenomena of ignition and extinction of a flame are typical examples of discontinuous change in a system under smooth variation of parameters. It is natural that they have played a substantial role in the formation of one of the branches of modern mathematics—catastrophe theory. In Ya.B. s work it is clearly shown that steady, time-independent solutions which arise asymptotically from non-steady solutions as the time goes to infinity are discontinuous. It is further shown that transition from one type of solution to the other occurs when the first ceases to exist. The interest which this set of problems stirred among mathematicians is illustrated by I. M. Gel fand s... 

All substances described are prepared by the same technique, with slight variations of parameters, which are given separately. 

There are several available methods for disrupting cells or tissues. The operational conditions can be optimized through the systematic variation of parameters such as medium composition, time, temperature, stirring rate, and size and shape of the blades. Selection of a suitable procedure... 

Berty reactor High-pressure, high-temperature petroleum and chemical operations Can provide intense mixing and high transport rates Not useful for low-pressure operations Ease of variation of parameters can be limited... 

Convenience of operation and ease of variation of parameters poor... 

The simulations generally fit the experimental curves satisfactorily and uniquely within the indicated variation of parameters. Sometimes at longer delay times the fit is worse due to phase mismatch so the earlier delay time data is more heavily weighted. The analysis procedures and examples of the quality of fits obtainable have been thoroughly described (9.10). 

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