Numerical methods integration

Numerical integration methods are widely used to solve these integrals. The Gauss-Miihler method [28] is employed in all of the calculations used here. This method is a Gaussian quadrature [29] which gives exact answers for Coulomb scattering. 

Once the diabatic potential energy surfaces relevant to describing a process, the integration of the sources of external potential (nuclear dynamics) can be done in real space using numerical integration methods. 

If linear diffusion is considered and the influence of comproportionation/dispro-portionation reactions is neglected, the solution for the response of this process in SWV is given as a system of recursive formulae by Lovric [54] using the numerical integration method proposed by Oldmstead and Nicholson [26]. 

The distributed nature of the tubular plug flow reactor means that variables change with both axial position and time. Therefore the mathematical models consist of several simultaneous nonlinear partial differential equations in time t and axial position z. There are several numerical integration methods for solving these equations. The method of lines is used in this chapter.1... 

TIME Time interval for which moles of reaction are calculated in rate programs, automatically set in the time-step algorithm of the numerical integration method... 

Coppens [147] analyzed the difficulties of the numerical integration method as follows. In the centrosymmetric case, the total electron population P is given by... 

In Fig.20 is shown a comparison between the observed and calculated conversion versus time curves, calculation being done by using the following equation and numerical integration method. 

Another reason for the choice of the title is the above-mentioned introduction of the Xa-method and the MS-Xa method by Slater and coworkers. There are, however, in particular two other reasons for choosing the title. The first is the formulation of the Density Functional Theory by Hohenberg and Kohn in 1964 [19], which today is probably one of the most quoted papers in electronic structure calculations. This basic work was followed by another important paper in 1965 by Kohn and Sham [20], where they showed how one could use the method for practical calculations and introduced the Kohn-Sham, KS, exchange potential. Exactly the same expression for the exchange potential had previously been derived by Caspar [21], This exchange potential is therefore often known as the Caspar-Kohn-Sham, GKS, potential. Another very important reason for choice of the title is the introduction of the three dimensional numerical integration method by Ellis and Painter in 1968-1970 [22-24]. This... 

In the next step, the dipole matrix element, Eq. (3), for Cl Z =17) and Mn Z 25) atoms were calculated with the DV-integration method for all possible dipole transitions. The obtained results for the square of the matrix element are shown in Table 1. For comparison, the nonrelativistic atomic HFS calculations were carried out by the use of the computer code of Herman and Skillman (HS) [35] and the dipole matrix elements corresponding to Eq. (3) were evaluated by the conventional numerical integration method. The calculated values are also listed in Table 1 and compared with the DV values. 

Purves RD. Optimum numerical integration methods for estimation of area-under-the-curve and area-under-the-moment-curve. J Pharmacokinet Biopharm 1992 20 211-26. 

In recent years, the research area of construction of numerical integration methods for ordinary differential equations that preserve qualitative properties of the analytic solution was of great interest. Here we consider Hamilton s equations of motion which are linear in position p and momentum q... 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of the Born-Oppenheimer approximation. 

Many processes in the pharmaceutical sciences are dynamic. Thus, models of these processes may commonly involve differential equations, which must be numerically integrated at each step in the optimization procedure. A variety of numerical integration methods can be used, and some of these are discussed later. 

After selecting and describing an appropriate model, choosing a numerical integration method (if necessary)... 

Performing Simulations As illustrated earlier, the mathematical representation of a PBPK model generally comprises a system of coupled ordinary differential and algebraic equations. These equations are normally amenable to solution via numerical integration methods on the computer. 

Spreadsheet Summary In the first experiment in Chapter 11 of Applications of Microsoft Excel in Analytical Chemistry, numerical integration methods are investigated. These methods are used to determine the charge required to electrolyze a reagent in a controlled-potential coulometric determination. A trapezoidal method and a Simpson s rule method are studied. From the charge, Faraday s law is used to determine the amount of analyte. 

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