Green s formula

In this section we define trace spaces at boundaries and consider Green s formulae. The statements formulated are applied to boundary value problems for solids with cracks provided that inequality type boundary conditions hold at the crack faces. 

Moreover, the following generalized Green s formula holds ... 

Using Green s formula (2.194), it follows from this that... 

Assume that the solution W of (2.265) is sufficiently smooth. We use Green s formula... 

Let us now obtain a complete system of boundary conditions fulfilled at Lc provided that the simplified nonpenetration condition (3.185) holds. We assume the solution x G iX is smooth enough and use Green s formulas for smooth functions (see Section 1.4),... 

By Green s formula, (3.228) gives the boundary value problem... 

Some difference formulae. In the sequel, when dealing with various difference expressions, we shall need the formulae for difference differentiating of a product, for summation by parts and difference Green s formulae. In this section we derive these formulae within the framework similar to the appropriate apparatus of the differential calculus. Similar expressions were obtained in Section 2 of Chapter 1 in studying second-order difference operators, but there other notations have been used. It performs no difficulty to establish a relationship between formulae from Section 2 of Chapter 1 and those of the present section. 

By introducing the inner product in the usual way and applying Green s formulae we get the inequalities A > c Aq and A > c Aq. In conformity with Chapter 2, Section 5,... 

It seems clear that Green s formulas are certainly true in the case when the operator A is defined in such a way. Moreover, A = A > 0. All this provides the sufficient background for the possible applications of the general stability theory outlined in Chapter 6, within the framework of which the scheme concerned is unconditionally stable for a > 0.5. 

By Green s formula it is straightforward to verify the inequalities CjR < A < Cji with constants c, = y and c. = A -f 2y incorporated. By exactly the same reasoning as in the preceding examples ATM requires ng(e) iterations, where... 

GREEN S FORMULA AND THE RELATIONSHIP BETWEEN POTENTIAL AND BOUNDARY CONDITIONS... 

This relationship is called the second Green s formula and it represents Gauss s theorem when the vector X is given by Equation (1.98). In particular, letting ij/ — constant we obtain the first Green s formula ... 

Here Lqp is the distance between points q and p. Note that G q, p) is called a Green s function. There are an infinite number of such functions and all of them have a singularity at the observation point p. Inasmuch as the second Green s formula has been derived assuming that singularities of the functions U and G are absent within volume V, we cannot directly use this function G in Equation (1.99). To avoid this obstacle, let us surround the point by a small spherical surface S and apply Equation (1.99) to the volume enclosed by surfaces S and S, as is shown in Fig. 1.9. Further we will be mainly interested by only cases, when masses are absent inside the volume V, that is. 

Taking into account Equations (1.101 and 1.103), the volume integral in the second Green s formula vanishes and we obtain... 

As in the previous section it is natural to start from the Green s formula... 

Integrating Eq. (12.11) in triangle APB in a counterclockwise direction, and applying Green s formula and Eqs. (12.9), one arrives at the required boundary condition... 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

Similar to scalar field problems, in order to obtain an integral representation for the momentum eqns. (10.63) and (10.64) for the flow field (u, p), Green s formulae for the momentum equations (Theorems (10.2.1) and (10.2.2)) are used together with the fundamental singular solution of Stokes equations, i.e.,... 

We now can apply Green s formula (9.17) to the perturbed electric field, 6Ey, and Green s function G . Repeating the derivations conducted in the previous subsection, we arrive at the formula analogous to (9.20) ... 

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