Third boundary value problem

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

The statement of the third boundary-value problem on eigenvalues is... 

Example 2. The third boundary-value problem. Given the same grid u)j as in Example 1, we now consider the difference boundary-value problem of the third kind... 

Example 5 Consider now the third boundary-value problem (9). As in Example 2 of Section 1 it will be convenient to introduce the space = 0 , of the dimension A +1 consisting of all grid functions defined on the uniform ... 

It is easily seen from (19) and (20) that no values of yu(x) at the vertices of the rectangle appear in this matter. This feature has some influence on a proper choice of 7 . For the third boundary-value problem and the scheme of accuracy 0 h ) (see Section 5) the boundary 7 consists of all the nodes on the boundary of the rectangle including its vertices. 

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 third boundary-value problem. For the moment, the statement of the problem is... 

Locally one-dimensional schemes find a wide range of applications in solving the third boundary-value problem. If, for example, G is a rectangle of sides /j and or a step-shaped domain, then equations (21) should be written not only at the inner nodes of the grid, but also on the appropriate boundaries. When the boundary condition du/dx = cr u- -v[ is imposed on the side = 0 of the rectangle 0 < < / , a = 1,2, the main idea... 

Fig. 1.8. (a) Dirichlet s problem, (b) Neumann s problem, (c) the third boundary value problem. 

Thus, boundary conditions (1.89) and (1.90) define the potential within the volume V up to some constant. Correspondingly, the third boundary value problem can be formulated as ... 

Maximum principle for the third kind boundary-value problem. The... 

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). 

Remark The third difference boundary-value problem for Poisson s equation can always be represented in the form (38), equation (38) being satisfied for all X E and conditions (39) being valid. Here, in addition, D > > 0 on 7,. 

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. 

The first situation involves two algebraic equations, the second involves an algebraic equation (the mixed phase) and a first-order ordinary differential equation (the unmixed phase), and the third situation involves two coupled differential equations. Countercurrent flow is in fact more compHcated than cocurrent flow because it involves a two-point boundary-value problem, which we will not consider here. 

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