One Step Methods

In this section we will present one-step methods for DAEs. Though there are successful implementations of extrapolation methods for differential algebraic equations [DHZ87] and mainly for constrained mechanical systems [Lub91] we restrict our discussion of one-step methods to Runge-Kutta methods. 

We consider the equations of motion (5.1.2) in their semi-explicit form 

that the inversion of the regular matrix Ei) y) can be performed in 0 riy) operations by applying the techniques described in Sec. 1.4.1. In order to simplify notation we will assume in the sequel E iy) = /. Furthermore we assume that (5.4.1) has an index of at most 2, i.e. we assume that either condition (5.1.13) or (5.1.14) hold. 

we consider implicit methods and restrict ourselves to those related to collocation polynomials. For this end we have to extend Definition 4.3.1 to semi-explicit DAEs. 

A collocation method then uses this pair of polynomials to determine the numerical solution of the problem at = tn+i, i e. 

Another way of classifying the integration techniques depends on whether or not the method is explicit, semi-implicit, or implicit. The implicit and semi-implicit methods play an important role in the numerical solution of stiff differential equations. To maintain the continuity of the section, we will first describe the explicit integration techniques in the context of one-step and multistep methods. The concept of stiffness and implicit methods are considered in a separate subsection, which also marks the end of this section. 

In single-step methods at each step of integration the solution depends on the prior step only. The generalized form of a single-step method, as derived from equation 2.2, can be written as  

The functional form of j changes according to different methods. The most commonly used methods in this category are Euler s method and Runge-Kutta methods. These methods are described below. 

Euler s method perhaps represents the simplest numerical integration technique, which can be defined as  

Euler s method is derived from a first order Taylor series approximation, where the higher derivative terms starting from the second derivative are ignored. It can be easily proved that the local truncation error, L h), for Euler s method decreases proportionally to the step size, i.e., L h) = 0 h). 

As with most Cl schemes of that period, the construction of the Hamiltonian matrix and its direct diagonalization effectively limited the size of calculations to a few thousand determinants. One possible strategy for extending the capability of this type of calculation is to introduce some sort of selection criterion for the A -particle functions, and to leave out those that do not contribute appreciably. Such methods had been developed within the framework of multireference Cl (MR-CI) calculations, and Hess, Peyerimhoff, and coworkers (Hess et al. 1982) extended this to the case of spin-orbit interactions. Their procedure was based on performing a configuration-selected non-relativistic MR-CI, followed by extrapolation to zero threshold. This technique may be applied in a one-step scheme, where selection criteria are introduced not only for the correlating many-particle states, but also for those that couple to the reference space via spin-orbit interaction. The size of the calculation that has to be performed in the double group may thereby be reduced. The errors introduced by these selection procedures appear to be small. 

The theory of direct Cl was reviewed in section 12.7. The central features of the nonrelativistic theory for the development of spin-orbit Cl are as follows. We write the nonrelativistic Hamiltonian, in the form 

It can be shown that for maximum efficiency this vector should be separated into three parts 

The challenge is to devise an algorithm for the spin-orbit case that incorporates as much as possible of the machinery that makes this nonrelativistic approach so efficient. We begin by writing the spin-orbit operator in second quantization. Looking at the one-electron contribution, we may write this as 

2 2 from which it is obvious that only the off-diagonal spin terms will contribute such that 

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A general one-step method for preparation of primary and secondary nitroparaffins from amines by oxidation with y -chloroperbenzoic acid in 1,2-dichloroethane has been reported (68). This method is particularly useful for laboratory quantities of a wide variety of nitroparaffins because a large number of amines are readily available from ketones by oxime reduction and because the reaction is highly specific for nitroparaffins. 

The phenyl group became a practical protective group for carboxylic acids when Sharpless published a mild, effective one-step method for its conversion to a carboxylic acid. It has recently been used in a synthesis of the amino acid statine, where it served as a masked or carboxylic acid equivalent. ... 

The lead dioxide-alkali method has also been applied successfully by Graebe and Kraft to the three cresols, the three toluic acids, and 2,4-dimethylphenol. For the preparation of 2-hydroxyisophthalic acid, it is the only one-step method that starts from readily obtainable materials. 

It was found that a high-impact strength is obtained in PP-EPDM blends by slow curing with sulfur. Thiuram disulfide N-(cyclohexylthio)phthalimide was used as an inhibitor of curing, and its effect on the impact strength of dynamically cured PP-EPDM blends was studied (Table 6). It was also found that the one-step method of blend preparation also has a favorable effect on the impact strength of the resultant blend system. 

Occasionally, chemists need to invert the stereochemistry of an alkene—that is, to convert a cis alkene to a trans alkene, or vice versa. There is no one-step method fordoing an alkene inversion, but the transformation can be carried out by combining several reactions in the proper sequence. How would you carry out the following reactions ... 

This synthetic process is applicable to the preparation of other ketene acetal derivatives of /3-alkoxy alcohols. Examples include the ketene acetal derivatives of tetrahydrofurfuryl alcohol and l-methoxy-2-propanol.3 There are a number of advantages in its use, including a simple, time-saving procedure, readily available and inexpensive reagents, and good yields of ketene acetal obtained by a one-step method. 

The present procedure is a convenient, one-step method of preparing optically active /raw.s -pin ocarveol. Although lower in yield than the lithium diethylamide procedure, it is more readily adaptable to large-scale work. Moreover, the two methods are complimentary in the conditions required (neutral vs. basic) and in the overall transformation accomplished ... 

This one-step method of preparing iodoxybenzene is preferable to earlier methods because it is simpler and the yield is substantially higher. The procedure seems general for iodoxyarenes,... 

This one step method has been used for the syntheses of rodlike polyimides, for example with 3,3,4,4/-biphenyltetracarboxylic dianhydride (BPDA) and 2,2 -bisBrinuoromcthyll- T-diaminobiphcnyl (PFMB).109... 

The use of tetraoctylammonium salt as phase transfer reagent has been introduced by Brust [199] for the preparation of gold colloids in the size domain of 1-3 nm. This one-step method consists of a two-phase reduction coupled with ion extraction and self-assembly using mono-layers of alkane thiols. The two-phase redox reaction controls the growth of the metallic nuclei via the simultaneous attachment of self-assembled thiol monolayers on the growing clusters. The overall reaction is summarized in Equation (5). 

Musso C, Silva-Santos M, Pereira F. Cotton block method one-step method of cell block preparation after fine needle aspiration. Acta Cytol. 2005 49 22-26. 

For one-step methods, the use of Ellman s reagent (previous section) to yield a similar reversible sulfhydryl blocking group is probably a better choice with protein molecules. 

Mesoionic compounds (Section 5.07.1.3) are fully aromatic and usually have an exocyclic heteroatom bearing a charge attached to the ring. A new one-step method for converting the exocyclic oxygen of 3-phenyl-l,2,3-thiadiazolium-5-olate 52 into the exocyclic sulfur of 3-phenyl-l,2,3-thiadiazolium-5-thiolate 53 makes use of Lawesson s reagent (Equation 14) <1988BCJ2977>. 

Electron Transfer Type of Dehydrogenase Sensors To fabricate an enzyme sensor for fructose, we found that a molecular interface of polypyrrole was not sufficient to realize high sensitivity and stability. We thus incorporated mediators (ferricyanide and ferrocene) in the enzyme-interface for the effective and the most sensitive detection of fructose in two different ways (l) two step method first, a monolayer FDH was electrochemically adsorbed on the electrode surface by electrostatic interaction, then entrapment of mediator and electro-polymerization of pyrrole in thin membrane was simultaneously performed in a separate solution containing mediator and pyrrole, (2) one-step method co-immobilization of mediator and enzyme and polymerization of pyrrole was simultaneously done in a solution containing enzyme enzyme, mediator and pyrrole as illustrated in Fig.22. 

Radiation grafting is an extremely valuable one-step method for directly modifying the properties of polymers (y,2). The technique has been used with a wide variety of naturally occurring macromolecules such as wool (3) and cellulose (4) and also with many synthetic polymers, particularly the polyolefins (2, 5, 6) ... 

In this part, we prepared and studied the Ag/Si02 catalyst by one-step and two-step sol-gel methods. The results show that the Ag/Si02 catalyst prepared here is one kind of bulk material which has a high surface area. The Ag/Si02 catalyst is made up with functional component of Ag or silver oxide in 20 to 30 nm and carrier Si02. Moreover, we found that the different preparation methods have great effect on crystal structure of the samples. The structure of the sample prepared by the one-step method is always a single crystal structure. And the structure of the sample prepared by the two-step method is always a mixed crystal structure. 

FIGURE 6.9. Scheme of the preparation of Ag/Si02 composite nanocatalyst by (a) one-step method, (b) two-step method. 

The TEM images (Fig. 6.14a) show that Ag nanoparticles about 5 nm are dispersed uniformly in Si02 carrier before the Ag/Si02 composite prepared by the one-step method is calcined. With the increase of the calcination temperature, Ag nanoparticles grow out from the Si02 carrier. Ag nanoparticles are about 10 nm and disperse on the surface of the Si02 carrier when the composite is calcined at 450°C (Fig. 6.14b). 

FIGURE 6.15. TEM diffraction images of Ag/Si02 sample preparedby different methods (a)one-step method (2.0 h), (b) one-step method (2.5 h), (c) two-step method. 

Ag/Si02 composites prepared by the one-step method and two-step methods were characterized by TGA/DTA, FTIR, XRD, and TEM. The results showed that Ag or silver oxide nanoparticles in 20 to30 nm were dispersed on the surface of the carrier Si02. The calcination temperature and prepared method affect the crystal phase of the functional component. 

OpticaUy active iV-tosylsulfoximides produced in the copper-catalyzed reaction of chiral sulfoxides with tosyl azide may be hydrolyzed with strong acid (H2SO4) to optically active free sulfoximides. However, this procedure often fails and/or results in decomposition. It is interesting to note in this connection that a simple one-step method for the preparation of optically active unsubstituted sulfoximides has been reported recently by Johnson and co-workers (180). It involves the reaction between optically active sulfoxides and 0-mesi-tylsulfonylhydroxylamine and results in sulfoximides 60 of high optical purity. As expected, this imidation process occurs with retention of configuration at sulfur. 

Another one-step method of similarly limited value for the synthesis of multiply substituted phenazines 84 is the Beirut reaction [80]. It involves the transformation of the easily accessible benzofuroxans 85 together with, in... 

Quantitative entrapment of vaccines into small (up to about 200 nm diameter) liposomes in the absence of microfluidization (which can damage DNA and other labile materials when extensive) can be carried out by a novel one-step method (7) as follows SUVs (e.g., cationic) prepared as in section Preparation of Small Unilamellar Vesicles are mixed with sucrose to give a range of sucrose-to-lipid weight/weight ratio of 1.0 to 5.0 and the appropriate amount of plasmid DNA (e.g., 10-500 pg) and/or protein (e.g., up to 1 mg). The mixture is then rapidly frozen and subjected to dehydration by freeze-drying, followed by rehydration as in section Preparation of Vaccine-Containing Dehydration-Rehydration Vesicles. ... 

Much more efficient method of the synthesis of metal hydrides is a one-step method of reactive ball milling of elemental metal directly under hydrogen gas. The method of reactive mechanical alloying/milling of either elemental metals or preaUoyed... 

In a very similar way, hydroxy functionalized ATRP initiators such as 2,2,2-tribromoethanol can be used for the simultaneous polymerization of eCL and MMA (Scheme 25) [83]. Purposely, the ROP of eCL is promoted by Al(OfPr)3 added in catalytic amount so that the rapid alcohol-alkoxide exchange reaction (see Sect. 2.4) activates all the hydroxyl functions. In order to avoid initiation by the isopropoxy groups of Al(0/Pr)3. The in-situ formed zPrOH is removed by distillation of the zPrOH/toluene azeotrope. On the other hand, the ATRP of MMA is catalyzed by NiBr2(PPh3)3. The two aforementioned one-step methods provide block copolymers with controlled composition and molecular weights, but with a slightly broad MWD (PDI=1.5-2). 

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