Mid-point rule

The first order interpolation with two neighboring nodes leads to the mid point rule for each imaginary node (i 1/2). The derivative will be of the second order... 

We simply need to find an expression for fi+1/2 If we use linear interpolation, since the point is in the middle of the interval, we arrive to the mid point rule, and the integral between two limits Xi and xi+i is given by... 

The representative of each size, usually the center of each cell = (, -1/2 -I- i-hi/2)/2, is called pivot or grid point. Note that these wiU be the points where the particle distributions are considered to be represented by a point mass. They may also be seen as the quadrature points using the mid point rule, which is of second order. This type of partitioning of the spatial domain is known as a cell centered representation of the grid. The integration of the governing PBE (12.308) over each cell yields a semi-discrete system which can be written as ... 

Ehase Inversion Temperatures It was possible to determine the Phase Inversion Temperature (PIT) for the system under study by reference to the conductivity/temperature profile obtained (Figure 2). Rapid declines were indicative of phase preference changes and mid-points were conveniently identified as the inversion point. The alkane series tended to yield PIT values within several degrees of each other but the estimation of the PIT for toluene occasionally proved difficult. Mole fraction mixing rules were employed to assist in the prediction of such PIT values. Toluene/decane blends were evaluated routinely for convenience, as shown in Figure 3. The construction of PIT/EACN profiles has yielded linear relationships, as did the mole fraction oil blends (Figures 4 and 5). The compilation and assessment of all experimental data enabled the significant parameters, attributable to such surfactant formulations, to be tabulated as in Table II. 

Thus by measuring the distance of the mark on the lead plate from the mid point of the fuse the velocity of detonation of the explosive under test may be calculated. In practice, it is simple to construct a rule by which the velocity can be measured directly from the lead plate. 

RULE 3 Intensities of the multiplet are asymmetrical about the mid-point of the signal, that corresponds to the origin of the multiplet and is equal to the chemical shift. 

As the interaction potential at the mid point of the slitlike pore is the deepest in the pore of w < 0.6 nm and it is not seriously different from the double minima even for w > 0.6 nm, molecular potential for a molecule in the micropore can be approximated by the potential at the mid-point of the slitlike pore. The molecular potential indicates the presence of the inherent micropore of the volume of Wl for each adsorptive. Here the inherent micropore must have sufficiently strong molecular field in comparison with the thermal energy at a measuring temperature. The inherent micropore volume Wl for vapour molecules is almost equal to the micropore volume Wq obtained from N2 adsorption at 77 K (Gurvitch rule) Wl for a supercritical gas which depends on the molecule-surface interaction can be evaluated as the saturated amount of adsorption from the Langmuir plot and Wl is less than Wo in general. 

This is a second-order Runge-Kutta method (Finlayson, 1980), sometimes called the midpoint rule. The first step is an approximation of the solution halfway between the beginning and ending time, and the second step evaluates the right-hand side at that mid-point. The error goes as (At), which is much smaller than that achieved with the Euler method. The second-order Runge-Kutta methods (there are several) also have a stability limitation. 

For E less than 0.4 the errors are less than about 2%, a tolerable error for analysis of most of our data. The r ot R of the quadratic function f(R), which Is obtained when El Is substituted for the Integral In Equation 41, Is given by Equatlon lA (Appendix A). We used this equation for all of our calculations including those with E, >0.4, since sample calculations performed iteratively and with I evaluated by Simpson s rule, gave final values of R differing by less than 1% from those found by using the mid-point approximation for I. 

Geometric construction rules are fixed position, horizontal, vertical, end point, and mid point. In Figure 7-20a choice of four rules is offered by the navigation. One of them should be selected, in contrast to dimensional construction rules. Following this, the straight-line segment can be extended using the rule that its end point will be the mid point of the extended entity. 

Elsewhere, in the mid-IR, photon energy is sufficient to modify the quantized terms vib and iJjo in expression 10.2. This is therefore a vibration-rotation spectrum, that is, several tens of rotational transitions accompany each vibrational transition. For the simplest molecules it is possible to interpret particular aspects of the absorption bands. Experience and theory have enabled rules of the permitted transitions to be drawn up. Small molecules as carbon monoxide and hydrogen chloride (Figure 10.5) have been intensely studied from this point of view. 

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