First- order difference method

The use of difference methods offers a means whereby a detailed picture of ionic hydration can be obtained 22). For neutron diffraction, the first-order isotopic difference method (see Section III,A) provides information on ionic hydration in terms of a linear combination of weighted ion-water and ion-ion pair distribution functions. Since the ion-water terms dominate this combination, the first-order difference method offers a direct way of establishing the structure of the aquaion. In cases for which counterion effects are known to occur, as, for example, in aqueous solutions of Cu + or Zn +, it is necessary to proceed to a second difference to obtain, for example, gMX and thereby possess a detailed knowledge of both the aquaion-water and the aquaion-coun-terion structure. 

Successful application of the first-order difference method depends on availability of isotopes that have significantly different coherent scattering lengths. Table I lists the 6 s for nuclei of the ions that can be studied by the method. 

The first-order difference method of neutron diffraction (ND) can be carried out on Li+, K+, Ag, and T1+ (Table I). The sodium ion can be investigated at an approximate level by total X-ray diffraction and... 

First-order difference neutron scadermg methods for die analysis of concentrated solutions of anions and... 

A first-order difference approximation for the axial derivative, 9(/0 z)/9j, is consistent with the first-order convergence of Euler s method. The convected-mean concentration is calculated from the dimensionless version of Equation (8.4) ... 

The principle of the perfectly-mixed stirred tank has been discussed previously in Sec. 1.2.2, and this provides essential building block for modelling applications. In this section, the concept is applied to tank type reactor systems and stagewise mass transfer applications, such that the resulting model equations often appear in the form of linked sets of first-order difference differential equations. Solution by digital simulation works well for small problems, in which the number of equations are relatively small and where the problem is not compounded by stiffness or by the need for iterative procedures. For these reasons, the dynamic modelling of the continuous distillation columns in this section is intended only as a demonstration of method, rather than as a realistic attempt at solution. For the solution of complex distillation problems, the reader is referred to commercial dynamic simulation packages. 

Derivative of intensity against structure parameters and thickness can be obtained using the first order perturbation method [31]. The finite difference method can also be used to evaluate the derivatives. Estimates of errors in refined parameters can also be obtained by repeating the measurement. In case of CBED, this can also be done by using different... 

The generation of a first-order difference by the use of anomalous scattering methods (either X-ray or neutron) offers the advantage that only a single sample is required. However, little use has been made of this possibility so far. On the other hand, extended X-ray absorption fine structure (EXAFS) spectroscopy has been used to investigate, in a limited way, gMo( ) and the existence of inner-sphere complexing 69). 

There are three major problems for the Euler method. First, the accuracy is poor, since the method is based upon Eq. (F.16), in which only a first-order difference expression is used. The errors in the method are proportional to At. Second, stability is difficult to achieve for many problems. The only way to have a stable Euler method is to use a small enough time step-size, but you may not know what value is sufficient. Furthermore, a value that is sufficient at the beginning may not be sufficient later on, and it may take an excessively long time to finish the computation. Third, to validate the results it is necessary to solve the problem at least twice, with different time-steps. The method can, however, be programmed in Excel, as Figures A.3 and A.4 demonstrate. 

These equations are directly suitable for solution by Euler s method. Use a first-order difference approximation for the time derivatives, for example,... 

Consider for the moment, data which is collected hourly. To obtain the smoothed data using the method of differencing, each observation is replaced by the difference of itself and the observation obtained x hours previously. First order differences involve obtaining the difference between the "current" observation and the previous observation second order... 

All factors on the right-hand side of Eq. (10.3-20) are constant. This equation is a linear first-order difference equation and can be solved by the calculus of finite-difference methods (Gl. M1). The final derived equations are as follows. 

Nevertheless the higher order of CFD is evident. For slower reaction k = 10, figure 3.1b) we see a strong dependence on the approximation of the boundary condition (3.3). If the boundary condition is approximated with a simple first order difference, the CFD looses it s high approximation order, the observed order is 1.37 for the SFD and 1.2 for the CFD. Only if the boundary condition is approximated with second order accuracy, the order of CFD is 2.88 compared to 1.95 for the SFD. A careful implementation of the boundary condition is therefore absolutely necessary if the compact method is to be used. 

Different methods are presented for determining the time constant of a calorimeter treated as an inertial object of first order. The methods used to determine the dynamic parameters of calorimeters that are inertial objects of order higher than one are also discussed. All these numerical methods, algorithms and listing programs are described in detail in [67]. 

The zero-order optimization method is similar to the first-order optimization method with the main difference being in the disposition of variables. Derivatives of the variables are used in the first-order optimization method whereas variables themselves are used in the zero-order optimization method. 

Four assessment functions are proposed as the optimization criteria, by first-order optimization method to deal with quantitive assessment. Different functions can lead to different optimization results and their optimization results are compared with each other to indentify the most effective one. 

ABSTRACT Runway overrun is one of the main accident types in airline operations. Nevertheless, due to the high safety levels in the aviation industry, the probability of a runway overrun is small. This motivates the use of structural reliability concepts to estimate this probability. We apply the physically-based model for the landing process of Drees and Holzapfel (2012) in combination with a probabilistic model of the input parameters. Subset simulation is used to estimate the probability of runway overrun for different runway conditions. We also carry out a sensitivity analysis to estimate the influence of each input random variable on the probability of an overrun. Importance measures and parameter sensitivities are estimated based on the samples from subset simulation and concepts of the First-Order Reliability Method (FORM). 

All the simulations reported in this chapter are fully resolved. A first-order Euler method is used for time-stepping the equations. The Laplacian terms in the reaction-diffusion equations are approximated with a 9-point finite-difference formula, which to leading order, eliminates the underlying 4-fold symmetry of square grid [13]. (For spiral waves in excitable media, anisotropies in the grid have a far greater effect on solutions than do anisotropies in the boundaries.)... 

The eigenvectors are normalized such that the first element is identical for all the three computational method. Therefore, the eigenvectors only differ in the second and the third elements. The absolute value of the errors for the first-order perturbation method and the iterative method is given by... 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

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