Space derivatives

The Fock representation of the space derivatives, (d/dqk)2 and the time derivatives (0/0 ), involved in H0, and on the right side of Eq. (8-154), require special attention. We start with Eq. (8-141) with F replaced by the appropriate derivative. For instance we rewrite Eq. (8-142) as ... 

Substitution of this relation in the vector meson kinetic term (i.e., the replacement of Ffll/(p) by Fpv(v)) gives the following four derivative operator with two time derivatives and two space derivatives [40] ... 

The second space derivative of the concentration at each enzyme layer may be approximated by the difference between the two gradients on each side of the layer ... 

Study 1 Broad range of odorants. A group of odorants that varied widely in quality and structure were arranged on the basis of odor similarity in a two dimensional space shown in Figure 1 (4). The space derived is two-dimensional with an affectively rather pleasant subset falling to the left and a rather... 

For most problems one needs to know how the components of the energy-flux vector are related to the space derivatives of the temperature. For conductive heat transfer the necessary relations are of the form... 

Estimate 3.3. We shall obtain here some estimates for the time and the higher space derivatives and infer the existence-uniqueness of a global classical solution for the auxiliary problem (3.2.34)-(3.2.36). Denote... 

Equation (316) should be compared with eqn. (44). It is second order because it involves the second space derivative V2, partial because of the three space dimensions and time (independent variables), inhomogeneous because the term J (r, t) is taken to be independent of p(r, t), linear because only first powers of the density p appear, and self adjoint in efic/p(r, t), the importance of which we shall see in the next section [491, 499]. The homogeneous equation corresponding to eqn. (316) has a solution p0 (r, t), which satisfies the same boundary conditions as p... 

For mi > 0 the time derivative was approximated by a forward difference ratio and the space derivatives by central difference ratios. It is known (E4, F2) that the solution to the difference approximation to the heat conduction equation will not be stable unless the ratio (Ay/Az 2) < 5. Stability implies that small perturbations introduced by rounding off or truncation are damped out instead of being magnified. Taking this ratio to be 5, the difference approximation to Eq. (99)... 

Ehrlich (El) uses the Crank-Nicholson (C16) finite-difference procedure for the integration of the diffusion equation, with a three-point approximation of the space derivatives on either side of the moving... 

Others have examined the geometry of the chromate and dichromate ions and compared then to models of the silica surface (12). Again, the uncertainty is high because we have very little idea what the silica surface really looks like. Hydroxyl spacings derived from a particular face of cristobalite or tridymite may have little relevance to amorphous, high surface area silica gel. 

Substitution in (128) shows that Sotime derivatives of the scalars can be expressed in terms of the scalars and their space derivatives. In other words, the Cauchy data are just the pair of complex functions cf)(r, 0), 0(r, 0) that verify the condition (126). The system therefore has two degrees of freedom with a differential constraint that is conserved naturally under the time evolution. 

Next, we have to define the boundary and the initial conditions. For the zero flux sensors (Fig. 2.9), the first space derivatives (i.e., fluxes) of all variables at the transducer/gel boundary (point x = 0) are zero ... 

A good option will be to use a second order approximation for the space derivatives as follows... 

Transient problems begin with an initial condition and march forward in time in discrete time steps. We have discussed space derivatives, and now we will introduce the time derivative, or transient, term of the differential equation. Although the Taylor-series can also be used, it is more helpful to develop the ED with the integral method. The starting point is to take the general expression... 

The species balance relation Eq. 13.2-8 is transformed to a difference equation using the forward difference on the time derivative and the backward difference on the space derivative. The finite difference form of the x-momentum equation (Eq. 13.2-25) is obtained by using the forward difference on all derivatives, and is solved by the Crank-Nicolson method. The same is true for the energy equation (Eq. 13.2-26). 

Analogous to the forward-difference method previously discussed, it is only first-order accurate in At. The only formal difference with respect to the forward-difference equation (8-10) appears to be the fact that the space derivative is evaluated at time tn+i, not at time tn. 

This expresses the rate of change of concentration with time at given coordinates (t, x, y, z) in terms of second space derivatives and three different diffusion coefficients. It is theoretically possible for D to be direction-dependent (in anisotropic media) but for a solute in solution, it is equal in all directions and usually the same everywhere, so (2.3) simplifies to... 

The Method of Lines or MOL is not so much a particular method as a way of approaching numerical solutions of pdes. It is described well by Hartree [295] as the replacement of the second-order (space) derivative by a finite difference that is, leaving the first (time) derivative as it is, thus forming from, say, the diffusion equation a set of ordinary differential equations, to be solved in an unspecified manner. Thus, a system such as (9.17) on page 151, can be written in the general vector-matrix form... 

The difference equation may also be formulated by computing the space derivatives in terms of the temperatures at the p + 1 time increment. Such an... 

Figure 8. Chemistry space occupation for reference databases, using a 6 dimensional BCUT chemistry space derived using DiverseSolutions and 6 bins per axis.

The classical example of the instability that results is the difference formulation given by Richardson (R2) for the heat-conduction equation. He proposed using a central-difference formula [as in Eq. (5-15)] for the derivative with respect to time, together with the usual central-difference formula for the space derivative. The resulting equation has a truncation error of the third order in both time and space steps, but the solution is unstable for any length of step. Thus, this natural and accurate formulation is not available for use if more than a few steps are to be taken. 

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