Second boundary value problem

In the case of the second boundary-value problem with dv/dn = 0, the boundary condition of second-order approximation is imposed on 7, as a first preliminary step. It is not difficult to verify directly that the difference eigenvalue problem of second-order approximation with the second kind boundary conditions is completely posed by... 

Thus, in order to determine the potential we have to know at the surface S both the potential and its normal derivative. At the same time, as follows from the first and second boundary value problems, in order to find the potential it is sufficient to know only one of these quantities on S. This apparent contradiction can be easily resolved by the appropriate choice of Green s function and we will illustrate this fact in the two following sections. 

Note that we allow the boundary regions B it), B it) to be time-depen-dent. In fact, it emerges that if they are not, the boundary value problem is relatively trivial in the sense of being closely related to the corresponding elastic problem (Sect. 1.2.1). The first and second boundary value problems are particular examples of this. 

We have also allowed the boundary regions to depend on the component being specified. This is required to cover examples such as frictionless contact. These are not the most general boundary conditions that can be conceived. For example the case of frictional contact is not even covered by this scheme. However, we shall see in Chap. 3 that, at least in the plane case, such problems can be handled by methods similar to those used in the frictionless case. The fact is that the methods outlined in later chapters for attacking viscoelastic boundary value problems are all indirect, in the sense that they focus on the boundary quantities, with the aim of determining these quantities everywhere on B. Once this is done, the task of determining any quantity in the interior of the medium is in principle easy. Indeed, it is either a first or second boundary value problem as defined above. 

This includes the special cases of the first and second boundary value problems. The time Fourier transform (FT) of (1.8.9, 11) are given by (1.8.19). The boundary conditions (1.8.15) similarly transformed, read... 

In this chapter, we will consider the first and second boundary value problems and then mixed problems, where the contacts are limited to certain, finite, time-dependent regions and the stresses are zero elsewhere on the boundary. Only limiting friction and zero friction problems will be discussed. 

In the second boundary value problem, we have displacements, or rather their tangential derivatives, specified at all times and for every point on the A -axis, so that the left-hand side of the displacement equation (3.1.3c) is known. The integral over first sight a hindrance to achieving a reduction to a sim-... 

I. Method of Solution. The method of solution is based on the viscoelastic Kolosov-Muskhelishvili equations, adapted to a half-space. Explicit solutions to the first and second boundary value problems are presented in detail. In these cases no restrictions on material behaviour are necessary. In the case of mixed boundary value problems where surface friction is present, it is necessary to make the proportionality assumption. Limiting frictional contact problems are... 

Also as formulated, the FD method appears to be applicable only to differential equations (or sets of differential equations) of even order (second, fourth, sixth, etc.). If one has a third order differential equation for example, this eould be expressed as one first order equation and one second order equation. However, what does one do with the first order equation, as there will be only one boundary eon-dition A prototype of this problem is the question of whether the FD method developed here ean be used with a single first order differential equation with only one boundary eondition. Such a problem is in fact an initial value problem and can be solved by die techniques of Chapter 10. However, the question remains as to whether the formulism of the FD method can be used for such a problem Several authors have suggested that the way to handle such a problem is to simply take another derivative of the given first order differential equation and convert it into a second order differential equation. For the second boundary value one then uses the original first order differential equation at the second boundary. 

Equations (11.111) - (11.113) define a boundary value problem for a pair of simultaneous second order differential equations in and x, subject... 

Proof. We consider a parabolic regularization of the problem approximating (5.68)-(5.72). The auxiliary boundary value problem will contain two positive parameters a, 5. The first parameter is responsible for the parabolic regularization and the second one characterizes the penalty approach. Our aim is first to prove an existence of solutions for the fixed parameters a, 5 and second to justify a passage to limits as a, d —> 0. A priori estimates uniform with respect to a, 5 are needed to analyse the passage to the limits, and we shall obtain all necessary estimates while the theorem of existence is proved. 

One approach to second-order boundary value problems is a matrix formulation. Given... 

In the present section a direct method for solving the boundary-value problems associated with second-order difference equations will be the subject of special investigations. 

The second-order difference equations. The Cauchy problem. Boundary-value problems. The second-order difference equation transforms into a more transparent form... 

For the second-order difference equations capable of describing the basic mathematical-physics problems, boundary-value problems with additional conditions given at different points are more typical. For example, if we know the value for z = 0 and the value for i = N, the corresponding boundary-value problem can be formulated as follows it is necessary to find the solution yi, 0 < i < N, of problem (6) satisfying the boundary conditions... 

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

Example 1. The third boundary-value problem for an ordinary second-order differential equation ... 

In this way, the third kind difference boundary-value problem (2)-(4) of second-order approximation on the solution of the original problem is put in correspondence with the original problem (1). 

Difference Green s function. Further estimation of a solution of the boundary-value problem for a second-order difference equation will involve its representation in terms of Green s function. The boundary-value problem for the differential equation... 

We call the nodes, at which equation (1) is valid under conditions (2), inner nodes of the grid uj is the set of all inner nodes and ui = ui + y is the set of all grid nodes. The first boundary-value problem completely posed by conditions (l)-(3) plays a special role in the theory of equations (1). For instance, in the case of boundary conditions of the second or third kinds there are no boundary nodes for elliptic equations, that is, w = w. 

In the case of the second eigenvalue boundary-value problem (8) the space Hh = Clh comprises all the functions defined on the grid ujh, the inner product (,) on Hh is to be understood in the sense (14) and the operator A is defined as a sum... 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

The difference boundary-value problem associated with the difference equation (7) of second order can be solved by the standard elimination method, whose computational algorithm is stable, since the conditions Ai 0, Ci > Ai -f Tj+i are certainly true for cr > 0. 

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