Variation of parameters method

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. 

TABLE 8.1 P- Particular Trial Solutions for the Variation of Parameters Method ... 

The Variation of Parameters method is based on the premise that the particular solutions are linearly independent of u(x) and v(x). We start by proposing... 

As is evident in Example 3.12, challenging integration problems can arise when the variation of parameter method is used. 

In Section 3.4, we employed the undetermined coefficients (equivalent— annihilation) or variation of parameters methods to solve the nonhomoge-... 

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 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... 

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... 

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... 

Several modifications of the method are described in the literature (Artursson and Karlsson 1991 Hidalgo and Borchardt 1989 and many others). Modifications include cell culture medium, time of cultivation and frequence of medium change, variations of trypsinization methods and others. In an industrial environment cell cultivation methods are maintained over many years constant to reduce variability and ensure constant results in quality assessment protocols. Additionally to quality control parameters like TEER and permeability markers expression levels of major enzymes and transporters are checked. 

In Refs. 244 and 248 the possibility to control friction has been discussed in model systems described by differential equations. Usually, in realistic systems, time series of dynamical variables rather than governing equations are experimentally available. In this case the time-delay embedding method [258] can be applied in order to transform a scalar time series into a trajectory in phase space. This procedure allows one to find the desired unstable periodic orbits and to calculate variations of parameters required to control friction. 

Semiempirical molecular orbital (SEMO) methods have been used widely in computational studies [1,2]. Various reviews [3-6] describe the underlying theory, the different variations of SEMO methods, and their numerical results. Semiempirical approaches normally originate within the same conceptual framework as ab initio methods, but they overlook minor integrals to increase the speed of the calculations. The mistakes arising from them are compensated by empirical parameters that are introduced into the outstanding integrals and standardized against reliable experimental or theoretical reference data. This approach is successful if the semiempirical model keeps the essential physics and chemistry that describe the behavior of the process. 

Now, following the method of the variation of parameters,4 assume C to be time dependent and insert Eq. (3.28) into Eq. (3.27). The result, after some rearrangement, is... 

Qualitative methods have not long been developed due to a lack of precise determination of the function form. However, owing to a progress in the field of differential geometry, a precise definition of the differential type (form) of a function has become possible (Chapter 2). The appearance of a good definition of the function form has enabled Thom and other mathematicians to examine changes in the function form in relation to variations of parameters on which the investigated function depended. 

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