The third boundary value problem

We suppose that the boundary surface S is equipotential, that is, 

Inasmuch as the boundary surface is an equipotential surface for both potentials Ui and U2, their difference is also constant on this surface and correspondingly we can write 

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  

Inside the volume the potential obeys Poisson s equation 

On the level surface S, surrounding this volume, we have 

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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. 

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. 

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