Stability with respect to the initial data

Stability with respect to the initial data. Let us investigate the stability of scheme (II) with homogeneous boundary conditions by the method of separation of variables. For this, we proceed as usual. This amounts to applying to scheme (II) with homogeneous boundary conditions the identities... 

The energy inequality method. An investigation of difference schemes for the string vibration equation may be carried out by means of the energy inequality method (see Section 1). Here we restrict ourselves to stability with respect to the initial data with regard to the problem... 

In this chapter we study the stability with respect to the initial data and the right-hand side of two-layer and three-layer difference schemes that are treated as operator-difference schemes with operators in Hilbert space. Necessary and sufficient stability conditions are discovered and then the corresponding a priori estimates are obtained through such an analysis by means of the energy inequality method. A regularization method for the further development of various difference schemes of a desired quality (in accuracy and economy) in the class of stability schemes is well-established. Numerous concrete schemes for equations of parabolic and hyperbolic types are available as possible applications, bring out the indisputable merit of these methods and unveil their potential. 

In conformity with the superposition principle ( is a linear operator), the stability of the Cauchy problem with respect to the right-hand side follows from the uniform stability with respect to the initial data... 

Inequality (12) expresses the property of continuous dependence which is uniform in h and t of the Cauchy problem (4) upon the input data. Here and below the meaning of this property is stability. A difference scheme is said to be absolutely stable if it is stable for any r and h (not only for all sufficiently small ones). It is fairly common to distinguish the notion of stability with respect to the initial data and that with respect to the right-hand side. Scheme (4) is said to be stable with respect to the initial data if a solution to the homogeneous equation... 

Quite often, the reader can encounter the statement that stability with respect to the initial data implies stability with respect to the right-hand side . How is one to understand the nature of this assertion ... 

It is worth noting here that condition (21) is sufficient for uniform stability with respect to the initial data. 

On the other hand, (28) yields 2/ +i < i / (i)- Comparison of these inequalities gives (27), from which the stability with respect to the initial data immediately follows. 

Stability with respect to the initial data in Ha- Stability of scheme (1) with respect to the initial data is investigated with further estimation of a solution of problem (la). 

Stability with respect to the initial data in Hb- Let us write down the second energy identity for scheme (la) assuming B also to be a self-adjoint operator 5 = 5 > 0. At the first stage we take the inner product of (la) and 2ry ... 

Stability with respect to the initial data. Since the conditions a > A/4 and S > 0 of Theorem 1 are fulfilled, we have for a solution of problem (65) with (p = 0 the estimate... 

Substituting (72) into (63) and taking into account (78), we conclude that (72) is just the solution of problem (63). As we stated in (71), on the strength of stability with respect to the initial data we have for ... 

Stability with respect to the initial data. Recall the definition of stability with respect to the initial data and the right-hand side. Scheme (1) is said to be stable with respect to the initial data if for problem (la) the a priori estimate holds ... 

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