Forward difference operator

Depletion is handled by solving for exposure using a forward difference operator, both for the single assembly perturbations and the second-order reconstructed power density responses. Since this approach yields group constants accurate through second-order in exposure, there is no need to consider sensitivities with respect to exposure explicitly within the GPT functional. 

One can apply this forward difference operator to Ay to obtain A y that is,... 

D = differential operator I = integral operator E = shift operator A = forward difference operator V = backward difference operator 6 = central difference operator p = averager operator. 

The relationship between forward difference operators and differential operators can now be developed. Combine Eqs. (3.45) and (3.17) to obtain... 

The higher-order forward difference operator, A, A, ..can be obtained by raising the first forward difference operator to higher powers ... 

Expansion of the exponential terms and rearrangement yields the following equations for the second and third forward difference operators ... 

Eqs. (3,53), (3.57), and (3.58) express the forward difference operators in terms of infinite series of differential operators. In order to complete the set of relationships, equations that express the differential operators in terms of forward difference operators will also be derived. To do this, first rearrange Eq. (3.52) to solve for... 

The relationships between forward difference operators and differential operators, which are summarized in Table 3.2, enable us to develop a variety of formulas expressing derivatives of functions in terms of forward finite differences and vice versa. As was demonstrated in Sec. 4.2, these formulas may have any degree of accuracy desired, provided that a sufficient number of terms are retained in the manipulation of these infinite series. A set of expressions, parallel to those of Sec. 4,2, will be derived using the forward finite differences. 

The forward difference operators, A, A. . ., are replaced by their equivalent in terms of differential operators [Eqs. (3.57) and (3.58)], and the remainder term becomes... 

Application of the elastic forward modeling operator to the initial model with the background Lame velocities results in nothing more than the incident elastic field in the background model, u (r, w). The difference between this incident field and the observed total field dx, gives us the scattered elastic field (with a minus sign) ... 

The derivative (D) being approximated by the finite-difference operator (FD) to within a truncation error (TE) (or, discretization error). The foregoing mathematical consideration provides an estimate of the accuracy of the discretization of the difference operators. It shows that TE is of the order of (Ax)2 for the central difference, but only O(Ax) for the forward and backward difference operators of first order. Equations (4.41) and (4.42) involve 2 or 3 nodes around node i at x , leading to 2- and 3-point difference operators. Considering additional Taylor series expansions extending to nodes i + 2 and i - 2 etc., located at x + 2Ax and x. — 2Ax, etc., respectively, one may derive 4- and 5-point difference formulas with associated truncation errors. Results summarized in Table 4,8 show that a TE of O(Ax)4 can be achieved in this manner. The penalty for this increased accuracy is the increased complexity of the coefficient matrix of the resulting system of equations. 

Unfortunately, the way one specifies a file path is different on the three different operating systems. On UNIX systems, a path begins with a forward slash and each directory is separated by additional slashes. On Windows systems, a path begins with the drive letter (e.g., c ) and uses backslashes to separate directories. On Macintoshes, the path begins with the name of the hard disk, and colons separate the name of each successive folder. Examples of fully qualified path names on UNIX, Windows, and Macintosh systems are shown below. 

To facilitate the development of explicit and implicit methods, it is necessary to briefly consider the origins of interpolation and quadrature formulas (i.e., numerical approximation to integration). There are essentially two methods for performing the differencing operation (as a means to approximate differentiation) one is the forward difference, and the other is the backward difference. Only the backward difference is of use in the development of eiqjlicit and implicit methods. 

In order to describe different time series models compactly, it is necessary to introduce the z- ov forward shift operator It is defined as... 

Using these parameters, the concentration-time profiles at different operating conditions were also predicted. For example the effect of catalyst loading on the concentration-time profile is presented in Fig. 4.16, while the model predictions for the effect of particle size are shovm in Fig. 4.17. In all these cases the model predictions were found to agree well with the experimental observations. The temperature dependence of the rate parameters ks to kg is shown in Fig. 4.18. From these Arrhenius plots the values of activation energies for the forward and reverse heterogeneous reactions were evaluated as 38.5, 28.894 and 72.1, 46.03 kJ/mol respectively. 

With these introductory concepts in mind, let us proceed to develop the backward, forward, and central difference operators and the relationships between these and the differential operators. 

The complete set of relationships between central difference operators and differential operators is summarized in Table 3.3. These relationships will be used in Chap. 4 to develop a set of formulas expressing the derivatives in terms of central finite differences. These formulas will have higher accuracy than those developed using backward and forward finite differences. 

In hyperspherical coordinates, the wave function changes sign when <]) is increased by 2k. Thus, the cotTect phase beatment of the (]) coordinate can be obtained using a special technique [44 8] when the kinetic energy operators are evaluated The wave function/((])) is multiplied with exp(—i(j)/2), and after the forward EFT [69] the coefficients are multiplied with slightly different frequencies. Finally, after the backward FFT, the wave function is multiplied with exp(r[Pg.60]

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