Variability, coefficient

Thus, the value of a definite integral depends on the limits a, b, and any selected variable coefficients in the func tion but not on the dummy variable of integrations. Symbolically... 

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. 

So far we have established an estimate for the rate of convergence in a very simple problem. It is possible to obtain a similar result for this problem by means of several other methods that might be even much more simpler. However, the indisputable merit of the well-developed method of energy inequalities is its universal applicability it can be translated without essential changes to the multidimensional case, the case of variable coefficients, difference schemes for parabolic and hyperbolic equations and other situations. 

This chapter presents the theory of homogeneous difference schemes for the solution of equations with variable coefficients... 

HOMOGENEOUS SCHEMES FOR SECOND-ORDER EQUATIONS WITH VARIABLE COEFFICIENTS... 

We will elaborate on this for rather complicated cases and turn to the equation with variable coefficients... 

In practice the use of truncated schemes in the case of equation (1) with variable coefficients necessitates carrying out calculations of multiple integrals on each interval of the grid. Replacing those integrals by finite sums we are able to create more simpler schemes of accuracy 0[h ) and 0 h ), whose coefficients can be expressed through the values of k, q and /... 

In this section we reveal some properties of difference operators approximating the Laplace operator in a rectangle and derive several estimates for difference approximations to elliptic second-order operators with variable coefficients and mixed derivatives. 

Equations with variable coefficients. The Dirichlet problem for the elliptic equation in the domain G + F = G comes next ... 

For the heat conduction equation with a variable coefficient k x) we might have... 

Homogeneous Difference Schemes for Time-Dependent Equations of Mathematical Physics with Variable Coefficients... 

A scheme for the governing equation with variable coefficients. The... 

Example 4 By having recourse to problem (23) associated with the heat conduction equation with variable coefficients for the same choice of the operators A, Ri and R as in Example 1 for the two-layer economical scheme (24) we concentrate on the primary scheme (36)... 

LOS for equations with variable coefficients. One way of covering equations with variable coefficients is connected with possible constructions of locally one-dimensional schemes and the main ideas adopted for problem (15). It sufficies to point out only the necessary changes in the formulas for the operators Lc, and Aq., which will be used in the sequel, and then bear in mind that any locally one-dimensional scheme can always be written in the form (21)-(23). Several examples add interest and help in understanding. 

Iterative methods of successive approximation are in common usage for rather complicated cases of arbitrary domains, variable coefficients, etc. Throughout the entire section, the Dirichlet problem for Poisson s equation is adopted as a model one in the rectangle G = 0 < x < l, a = 1,2 with the boundary P ... 

Adopting those ideas, some progress has been achieved by means of MATAI in tackling the Dirichlet problem in an arbitrary complex domain G with the boundary T for an elliptic equation with variable coefficients ... 

On solving difference equations for problems with variable coefficients. 

In the preceding sections this trend of research was due to serious developments of the Russian and western scientists. Specifically, the method for solving difference equations approximating an elliptic equation with variable coefficients in complex domains G of arbitrary shape and configuration is available in Section 8 with placing special emphasis on real advantages of MATM in the numerical solution of the difference Dirichlet problem for Poisson s equation in Section 9. 

One of the most important issues is concerned with a smaller number of the iterations performed in the numerical. solution of equations with variable coefficients. It was shown in Section 7 that the number of the iterations required during the course of ATM is proportional to where Cj and are the smallest and the greatest values of coefficients, respectively. The operator R in question can be put in correspondence with the operator A with variable coefficients such that... 

From here it is easily seen that MATM offers more advantages not only in an arbitrary domain, but also in the case of variable coefficients. 

The notation is meant to suggest that the frequency is variable and depends on the propagator matrix elements. The following criteria have proved valuable in choosing the variable coefficients of eq. IV.5 (1) at low temperature, the VQRS reference should weight the region around the potential minimum most heavily, and (2) at high temperature, our approximation should approach the classical limit ... 

The use of confidence intervals is one way to state the required precision. Confidence limits provide a measure of the variability associated with an estimate, such as the average of a characteristic. Table I is an example of using confidence intervals in planning a sampling study. This table shows the interrelationships of variability (coefficient of variation), the distribution of the characteristic (normal or lognormal models), and the sample frequency (sample sizes from 4 to 365) for a monitoring program. 

The same idea can be developed in the case of a non-Euclidean metric such as the city-block metric or L,-norm (Section 31.6.1). Here we find that the trajectories, traced out by the variable coefficient kj are curvilinear, rather than linear. Markers between equidistant values on the original scales of the columns of X are usually not equidistant on the corresponding curvilinear trajectories of the nonlinear biplot (Fig. 31.17b). Although the curvilinear trajectories intersect at the origin of space, the latter does not necessarily coincide with the centroid of the row-points of X. We briefly describe here the basic steps of the algorithm and we refer to the original work of Gower [53,54] for a formal proof. 

Summarizing all that has been said above concerning the structures of the octachloroditechnetates ( + 2.5), it may be concluded that their true composition is described by a formula with variable coefficients, namely M 6M"3-.,t(H30) [[Tc3Cl8] nH20, where x and n vary from 0 to 3. The substitution of some of the M ions by H30+ ions is possible by virtue of the similarity of the properties of the hydroxonium cation and the alkali metal cations both in solution and in the crystalline state [85,86],... 

Substituting (4.11) and its derivatives into (4.10) and equating terms of the same order of 10, we obtain the chain of linear differential equations of the second order with variable coefficients ... 

The general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of c, as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function Cj(x,y,z) is essentially independent of time t, i.e. in the steady-state. Then equation (15) reduces to ... 

Ordinary Points of a Linear Differential Equation. We shall have occasion to discuss ordinary linear differential equations of the second order with variable coefficients whose solutions cannot he obtained in terms of Lhe elementary functions of mathematical analysis, la such cases one of the standard procedures is to derive n pair of linearly independent solutions in the form ofinfinite series and from these series to compute tables of standard solutions. With the aid of such tables the solution appropriate to any given initial conditions may then he readily found. The object of this note is to outline briefly the procedure to he followed in these instances for proofs of the theorems... 

The first-order linear equation [Eq. (6.44)] could have a time-variable coefficient that is, 0) could be a function of time. We will consider only linear second-order ODEs that have constant coefficients (tj, and ( are constants). 

Therefore, the dlffusivity is the only variable coefficient in Eq. (12), which can be integrated as follows ... 

The types of problems capable of being solved with the following regression methods are defined by the manner in which each of the regression methods works. The basic premise is that a method is given input variables (bioactivities and descriptors) and in turn the method produces output variables (coefficients and a constant). These methods work best when information about the system of interest is known and inferences can be made about the problem being solved. This can only be done if there is confidence that a relationship exists between the known input data and the unknown output data before these methods are utilized if there is no relationship, then the model will be useless. 

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