Milne approach

An alternative to the WKBJ approach is the so-called Milne approach. It is not so evident and has been recently rediscovered by Killingbeck (1980). The Milne approach is also applied to the generalization of the... 

WKBJ method (Hecht and Mayer, 1957) and to the study of not only bound-state energies but also quasi-bound states at resonance (complexvalued) energies (Korsch and Laurent, 1981 Korsch et al., 1982). The Milne approach provides in a numerical sense a rapid second-order convergence (Korsch and Laurent, 1981). In the Milne approach the quantization condition is the same as in the WKBJ method. 

The Milne approach was of course proposed by Milne (1930) and independently by Wilson (1930) and Young (1931), and is based on the following nonlinear differential equation (Milne, 1930) ... 

The Milne approach is a powerful method for study of the one-dimensional Schrddinger equations involving the class of diatomic potential functions. 

Hopwood D, Milne G. Fixation, in Electron Microscopy in Biology. A Practical Approach (Harris JR, ed.), IRL Press, Oxford, UK, 1991, pp. 1-15. 

Typically, these methods arrive at the same finite difference representation for a given problem. However, we feel that Taylor-series expansions are easy to illustrate and we will therefore use them here in the derivation of finite difference equations. We encourage the student of polymer processing to look up the other techniques in the literature, for instance, integral methods and polynomial fitting from Tannehill, Anderson and Pletcher [26] or from Milne [16] and finite volume approach from Patankar [18], Versteeg and Malalasekera [27] or from Roache [20]. 

D. Price, K. Pyrah, T.R. Huh, G.J. Milnes, J.R. Ebdon, BJ. Hunt, P. Joseph, and C.S. Konkel, Flame retarding poly(methyl methacrylate) with phosphorus-containing compounds Comparison of an additive with a reactive approach. Polym. Degrad. Stab., 74,441 447 (2001). 

The predictor calls for four previous values in Adams-Moulton and Milne s algorithms. We obtain these by the fourth-order Runge-Kutta method. Also, we can reduce the step size to improve the accuracy of these methods. Milne s method is unstable in certain cases because the errors do not approach zero as we reduce the step size, h. Because of this instability, the method of Adams-Moulton is more widely used. 

Stuart M. Tumhull and Frank Milne, A Simple Approach to Interest-Rate Option Pricing, The Review of Financial Studies, 4, no. 1 (1991), pp. 87-120. 

Somerville C. (2006). Cellulose synthesis in higher plants. Annu Rev Cell Dev Biol, 22,53—78. Somerville C, Bauer S, Brininstool G, Facette M, Hamann T, Milne J, Osborne E, Paredez A, Persson S, Raab T. (2004). Toward a systems approach to understanding plant cell walls. 

Processed data sets based on multiple omics analyses are valuable for further system biology approaches, that is, genome scale or metabolic network models, for better understanding of the clostridial metabolic switch and its mechanism of butanol production (Lee et al., 2008 Senger and Papoutsakis, 2008a,b Haus et al., 2011 Milne et al., 2011 McAnulty et al., 2012). 

Somerville, C., Bauer, S., Brininstool, G., Facette, M., Hamann, T., Milne, J., Osborne, E., Paredez, A., Persson, S., Raab, T., Vorwerk, S., and Youngs, H. (2004). Toward a systems approach to understanding plant-cell walls. Science 306, 2206-2211. 

Basing on their experimental studies, Borgwardt and Bruce recommended the value of m as 0.62. This approach gives a reaction order of less than 1. Experimental results of Yang et al (1978) indicated a reaction order of 0.59. This approach was extended by Milne et al (1990) considering the finite vapor pressure of SO2 above the solid CaS04. Instead of Eq.2.11 they advise the use of following equilibrium relation ... 

In 1979, Cramer and Milne made a first attempt to compare molecules by aligning them in space and by mapping their molecular fields to a 3D grid. In the following years, this approach was further developed as the DYLOMMS (dynamic lattice-oriented molecular modelling system) method but was not very well accepted by the scientific community. Several important facts had to work together to allow a broader application of this approach ... 

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