Stability with respect to coefficients

Stability of difference schemes with respect to coefficients. In solving some or other problems for a differential equation it may happen that coefficients of the equation are specified not exactly, but with some error because they may be determined by means of some computational algorithms or physical measurements, etc. Coefficients of a homogeneous difference scheme are functionals of coefficients of the relevant differential equation. An error in determining coefficients of a scheme may be caused by various 

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VVe say that a scheme is stable with respect to coefficients (costable) if a solution of the boundary-value problem has slight variations under small perturbations of the scheme coefficients. In order to avoid misunderstanding, we focus our attention on the scheme with coefficients a, d, ip 

We estimate the difference z = y — y in terms of perturbed coefficients. Substituting y = z + y into (2) and taking into account (1), we get 

To evaluate a solution of problem (4)-(7) on the basis of (6)—(8), we rely on the estimate obtained in Section 3 for (4)-(7), making it possible to establish 

We shall need the estimates for y and from Section 6 such as 

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Here we give the general formulation of the concept of stability with respect to coefficients for the difference scheme in Section 1 by having recourse to an operator equation of the first kind... 

It is natural to require a solution of problem (11) to depend continuously not only on perturbations of the right-hand side, but also on perturbations of the operator A (for instance, on the coefficients of a difference operator). As in the case of difference schemes arising in Section 1, this property of operator equations is to be understood as stability with respect to coefficients or co-stability of an operator equation. 

Stability of the first kind operator equations with respect to coefficients. 

The amplitude of the periodic orbits is therefore determined by the linear stability with respect to perturbations transverse to the orbit. In this sense, the leading term in expression (2.13), obtained by setting C = 0, treats the dynamics transverse to the orbit at the level of the harmonic approximation. The nonlinear stability properties appear thus as anharmonic corrections to the dynamics transverse to the orbit. These anharmonicities contribute to the trace formula by corrections given in terms of series in powers of the Planck constant involving the coefficients C , which can be obtained as Feynman diagrams [14, 31]. 

The cyclic voltammetry of 1,2,3,4,5-pentachloro- and of decachloroferrocene have been studied in acetonitrile. A plot of redox potential E(f) vs. op (ap is the Hammett para coefficient for the chloro substituent) shows that the neutral molecules are stabilized with respect to the corresponding ferrocenium cations by 0.16 to 0.12 V per Cl260. 

Various methods are employed to estimate and compare foam stability with respect to foam column destruction. Most often they are reduced to determination of the lifetime of the foam column (or part of it) up to its complete disappearance. Several earlier works [13,14] dealing with the stability of flotation foams (froths) involved the use of a coefficient of stability... 

The first applications of CMRs have concerned high temperature reactions. The employed inorganic membranes, characterized by higher chemical and thermal stability with respect to polymeric membranes, still today suffer from some important drawbacks high cost, limited lifetime, difficulties in reactor manufacturing (delamination of the membrane top-layer from the support due to the different thermal expansion coefficients). 

Since water acts as a plasticizer, when the polymer absorbs water, the strength and stiffness decrease, but the toughness increases. Although, if the polymer is reinforced with carbon fiber, less water is absorbed since the carbon fiber replaces some of the polymer and will consequently absorb less water. Also, addition of the fiber will increase the dimensional stability with respect to temperature change, since the fiber has a much lower thermal expansion coefficient. On the other hand, a molding that exhibits preferred orientation can create warping, which occurs due to stresses arising from the differences in thermal expansion in the transverse and flow direction. 

Zielinska et al. (1996) and Kelly and Holdren (1995) have summarized the stability in canisters of organics, some of which are U.S. EPA designated HAPs (hazardous air pollutants). Kelly and Holdren propose that for compounds whose stability in canisters is not known, estimates can be made based on species of similar physical and chemical characteristics. These characteristics include their vapor pressure, polarizability, water solubility, Henry s law coefficient in water, and estimated lifetimes with respect to reactions in air and in the aqueous phase. 

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