The first boundary value problem

Suppose that the potential U p) is known on the boundary surface that is, 

Then their difference Uj. on this surface becomes zero  

Equation (1.84) form Dirichlet s boundary value problem, which can be either exterior or internal one. Fig. 1.8a, and it has several important applications in the theory of the gravitational field of the earth. It is worth to notice that in accordance with Equation (1.83) we can say that along any direction tangential to the boundary surface, the component of the field is also known, since = dU/dt. Consequently, the boundary value problem can be written in terms of the field as 

This first case vividly illustrates the importance of the boundary condition. Indeed, Poisson s equation or the system of field equations have an infinite number of solutions corresponding to different distributions of masses located outside the volume. Certainly, we can mentally picture unlimited variants of mass distribution and expect an infinite number of different fields within the volume V. In other words, Poisson s equation, or more precisely, the given density inside the volume V, allows us to find the potential due to these masses, while the boundary condition (1.83) is equivalent to knowledge of masses situated outside this volume. It is clear that if masses are absent in the volume V, the potential C7 is a harmonic function and it is uniquely defined by Dirichlet s condition. 

---

Solution estimation for difference bormdary-value problems by the elimination method. In tackling the first boundary-value problem difference equation (21) has the tridiagonal matrix of order TV — 1... 

Example 4. The first boundary-value problem for the heat conduction equation ... 

In such a setting it is required to find the values of the parameter A such that these homogeneous equations have nontrivial solutions y(x) 0. In contrast to the first boundary-value problem, here the parameter A enters not only the governing eqnation, but also the boundary conditions. The introduction of new sensible notations... 

Example 4 Consider the first boundary-value problem... 

In such a setting problem (9) turns out to be equivalent to the first boundary-value problem... 

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. 

We concentrate primarily on the first boundary-value problem associated with equation- (3) in the rectangle D = Q < x < l,0[Pg.300]

In the forthcoming example the first boundary-value problem is posed for the heat conduction equation in the rectangle Go = 0 < < /, 0 <... 

Example 2 The statement of the first boundary-value problem for the parabolic equation with mixed derivatives in the parallelepiped Go = 0 < a < L, a =1,2,..., p is... 

The original problem. We begin by placing the first boundary-value problem for for the heat conduction equation in which it is required to find a continuous in the rectangle Dt = 0 < a < 1, 0 < f < T solution to the equation... 

---