Laplace operators

With the Laplace operator V. The diffusion coefficient defined in Eq. (62) has the dimension [cm /s]. (For correct derivation of the Fokker-Planck equation see [89].) If atoms are initially placed at one side of the box, they spread as ( x ) t, which follows from (62) or from (63). 

Gradient operator Laplace operator Dot product Cross product Divergence operator Curl operator Vector transposition Complex conjugate... 

The difference approximation of the Laplace operator. We begin by defining a difference analog of the Laplace operator in the plane x =... 

The next step is to calculate the error of approximation of the Laplace operator (2) by the difference operator (5). Since for a = 1,2... 

Approximation of the Laplace operator on an irregular cross pattern. 

We now consider a difference approximation of the Laplace operator on an irregular cross pattern. In the two-dimensional case (p = 2) such a pattern consists of the five points... 

Thus, on any irregular pattern the Laplace operator is approximated to first order by the difference operator A specified by formula (13). 

Remark Quite often, the Dirichlet problem is approximated by the method based on the difference approximation at the near-boundary nodes of the Laplace operator on an irregular pattern, with the use of formulae (14) instead of (16) at the nodes x G However, in some cases the difference operator so constructed does not possess several important properties intrinsic to the initial differential equation, namely, the self-adjointness and the property of having fixed sign, For this reason iterative methods are of little use in studying grid equations and will be excluded from further consideration. 

In this section we reveal some properties of difference operators approximating the Laplace operator in a rectangle and derive several estimates for difference approximations to elliptic second-order operators with variable coefficients and mixed derivatives. 

The eigenvalue problem for the Laplace operator in the rectangle Go subject to the first kind boundary conditions... 

This problem can be solved by the method of separation of variables. The eigenvalue problem for the difference Laplace operator i. y = - -... 

We list below the properties of the difference Laplace operator acting from... 

By inserting in (1) the difference operator A in place of the Laplace operator we are led to the system of differential-difference equations... 

We begin our exposition with a discussion of examples that make it possible to draw fairly accurate outlines of the possible theory regarding these questions and with a listing of the basic results together with the development desired for them. Common practice involves the Laplace operator as the operator R in the case of difference elliptic operators A. The present section is devoted to rather complicated difference problems of the elliptic type. Here and below it is supposed that the domain of interest is a p-dimensional parallelepiped G = 0 < < / , a = 1,2,..., p) with... 

It is worth noting here that the same estimate for no( ) was established before for ATM with optimal set of Chebyshev s parameters, but other formulas were used to specify r] in terms of and A - If R = —A, where A is the difference Laplace operator, and the Dirichlet problem is posed on a square grid in a unit square, then... 

In this one-dimensional flat case the Laplace operator is simpler than in the case with spherical symmetry arising when deriving the Debye-Huckel limiting law. Therefore, the differential equation (B.5) can be solved without the simplification (of replacing the exponential factors by two terms of their series expansion) that would reduce its accuracy. We shall employ the mathematical identity... 

Here S Nft = 0 because of the electroneutrality condition.) Equation (1.3.10) is substituted into Eq. (1.3.6) and the Laplace operator is expressed in polar coordinates (for the spherically symmetric problem) ... 

As a last example in this section, let us consider a sphere situated in a solution extending to infinity in all directions. If the concentration at the surface of the sphere is maintained constant (for example c — 0) while the initial concentration of the solution is different (for example c = c°), then this represents a model of spherical diffusion. It is preferable to express the Laplace operator in the diffusion equation (2.5.1) in spherical coordinates for the centro-symmetrical case.t The resulting partial differential equation... 

P(Qol, t) is the conditional probability of the orientation being at time t, provided it was Qq a t time zero. The symbol — F is the rotational diffusion operator. In the simplest possible case, F then takes the form of the Laplace operator, acting on the Euler angles ( ml) specifying the orientation of the molecule-fixed frame with respect to the laboratory frame, multiplied with a rotational diffusion coefficient. Dr. Equation (44) then becomes identical to the isotropic rotational diffusion equation. The rotational diffusion coefficient is simply related to the rotational correlation time introduced earlier, by tr = 1I6Dr. 

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