Douglas equation

In 1924, the Russian astronomer Numerov (transliterating his own name as Noumerov), published a paper [421] in which he described some improvements in approximations to derivatives, to help with numerical simulations of the movement of bodies in the solar system. His device has been adapted to the solution of pdes, and was introduced to electrochemistry by Bieniasz in 2003 [108]. The method described by Bieniasz is also called the Douglas equation in some texts such as that of Smith [514], where a rather clear description of the method is found. With the help of the Numerov method, it is possible to attain fourth order accuracy in the spatial second derivative, while using only the usual three points. The first paper by Bieniasz on this method treated equally spaced grids, and was followed by another on unequally spaced grids [107], The method makes it practical to use higher-order time derivative approximations without the complications of, say, the (6,5)-point scheme described above, which makes the solution of the system of equations a little complicated (and computer time consuming). 

An economic constitutive relationship that has a functional form postulated to take into account capital, labor, and their relationship to process plant size is the Cobb-Douglas equation ... 

All the early work was concerned with atoms, with Sir William Hartree regarded as the father of the technique. His son, Douglas R. Hartree, published the definitive book, The Calculation of Atomic Structures, in 1957, and in this he derived the atomic HF equations and described numerical algorithms for their solution. Charlotte Froese Fischer was a research student working under the guidance of D. R. Hartree, and she published her own definitive book. The Hartree—Fock Method for Atoms A Numerical Approach in 1977. The Appendix lists a number of freely available atomie structure programs. Most of these can be obtained from the Computer Physics Communications Program Library. 

Sir William Hartree developed ingenious ways of solving the radial equation, and they are documented in Douglas R. Hartree s book (1957). By the time this book was published, the SCF method had been well developed, and its connection with the variation principle was finally understood. It is interesting to note that Chapter 2 of Douglas R. Hartree s book deals with the variation principle. 

Douglas, A. S., Proc. Cambridge Phil. Soc. 52, 687, "A method for improving energy-level calculations for series electrons." Inclusion of a polarization potential in the Hartree-Slater-Fock equation. 

Crank-Nicholson and Douglas sehemes with improved interfaee conditions have been applied to transform Eq. (2.9) into a finite-differenee equation. 

The Jafarey, Douglas, and McAvoy equation design and control... 

Jafarey. Douglas, McAvoy s control equation, 126-129 Jenny limiting composition. 66 Jeronimo and Sawistowski, 279, 297, 320 Johnson and Fair regime transition, 332... 

Equation (5.11) represents a straight line in the diagram of fractional temperature rise versus reactor feed temperature. We show three such lines in Fig. 5.21. All lines intersect the temperature rise curve at least once (at a low temperature not shown in Fig. 5.21). It therefore appears that the reactor FEHE can have one, two, or three steady-state solutions for this particular set of reaction kinetics. Furthermore, the intermediate steady state, in the case of three solutions, is open-loop unstable due to the slope condition discussed in Chap. 4. This was verified by Douglas et al. (1962) in a control study of a reactor heat exchange system. 

Douglas, J., Jr., DuPoint, T., Galerkin Methods for Parabolic Equations, ... 

The first use of FCS in chemical relaxation was demonstrated by Elliott Elson, Douglas Magde, and Watt Webb [4], [6]. The system analyzed was the fluctuation due to association and dissociation between an intercalating fluorescent dye and double stranded DNA. Since the diffusion terms constitute own eigenvalues of the diffusion reaction equation, differences in the diffusion of free and bound dye constitute an obstacle in the analysis of the relaxation terms by FCS. We have for this reason focused our interest on cases with little or no change in translational diffusion when using the new FCS in its confocal form. 

The discretisation of the heat conduction equation can also be undertaken for three-dimensional temperature fields, and this is left to the reader to attempt. The stability condition (2.304) is tightened for the explicit difference formula which means time steps even smaller than those for planar problems. The system of equations of the implicit difference method cannot be solved by applying the ADIP-method, because it is unstable in three dimensions. Instead a similar method introduced by J. Douglas and H.H. Rachford [2.71], [2.72], is used, that is stable and still leads to tridiagonal systems. Unfortunately the discretisation error using this method is greater than that from ADIP, see also [2.53]. 

For a moru exact significance of AH" in equation (33.28), even when the heat content varies with temperature, see Douglas and Crockford, J. Am. Chem Soc., 57, 97 (1935) Walde, J. Fhys. Chem., 45, 431 (1939). 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

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