Differentiation-integration method

In Sec. 2.4 it was shown that differentiation enables one to eliminate unwanted background caused by light scattering or noise. But sometimes the undisturbed fundamental signal is called for. In that case, the differentiated spectrum must be retransformed, step by step, into the fundamental spectrum by integration (D I method). 

There are some approximate algorithms for solving determined integrals e. g., the rectangular, trapezoid and tangential formulas, the Simpson and Kepler rules, and integration by polynomials (see special handbooks of mathematics). In many cases the trapezoid formula will suffice for an integration of derivatives see Eq. (2-79) and Fig. 2-34. 

Using Eq. (2-79), the error of integration rises with the rising width of intervals i, and is proportional to if 

Note that the broader the width of intervals is, the smaller the number of data points will be. It is not a sliding data computation (e.g., the Savitzky-Golay polynomial), but a computation, dependent on the width of intervals. In the sliding data manipulation, the interval width can be varied while the number of data points remains constant. To contrast, broader data intervals correspond to a higher influence on the signal shape (deformation), and a reduced amount of information in the spectra. 

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In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

Starting with an initial value of and knowing c t), Eq. (8-4) can be solved for c t + At). Once c t + At) is known, the solution process can be repeated to calciilate c t + 2At), and so on. This approach is called the Euler integration method while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. To achieve accurate solutions with an Eiiler approach, one often needs to take small steps in time. At. A number of more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Bunge Kutta method, which involves the following calculations ... 

Some systems may show stiff properties, especially those for oxidations. Here the system of differential equations to be integrated are not stiff . Even at calculated runaway temperature, ordinary integration methods can be used. The reason is that equilibrium seems to moderate the extent of the runaway temperature for the reversible reaction. 

The principal techniques used to determine reaction rate functions from the experimental data are differential and integral methods. 

Determine the reaction order and rate constant for the reaction by both differential and integral methods of analysis. For orders other than one, C0 will be needed. If so, incorporate this term into the rate constant. 

The generalized physical property approach discussed in Section 3.3.3.2 may be used together with one of the differential or integral methods, which are appropriate for use with reversible reactions. In this case the extent of reaction per unit volume at time t is given in terms of equation 3.3.50 as... 

The first point to be established in any experimental study is that one is dealing with parallel reactions and not with reactions between the products and the original reactants or with one another. One then uses data on the product distribution to determine relative values of the rate constants, employing the relations developed in Section 5.2.1. For simple parallel reactions one then uses either the differential or integral methods developed in Section 3.3 in analysis of the data. 

We further assume that the rate law is of the form ( rA) = kAc c cyc, and that the experiments are conducted at fixed T so that kA is constant. An experimental procedure is used to generate values of cA as a function of t, as shown in Figure 2.2. The values so generated may then be treated by a differential method or by an integral method. 

Use (a) the differential method and (b) the integral method to determine the reaction order, and the value of the rate constant. Comment on the results obtained by the two methods. 

There are two procedures for analyzing kinetic data, the integral and the differential methods. In the integral method of analysis we guess a particular form of rate equation and, after appropriate integration and mathematical manipulation, predict that the plot of a certain concentration function versus time... 

There are advantages and disadvantages to each method. The integral method is easy to use and is recommended when testing specific mechanisms, or relatively simple rate expressions, or when the data are so scattered that we cannot reliably find the derivatives needed in the differential method. The differential method is useful in more complicated situations but requires more accurate or larger amounts of data. The integral method can only test this or that particular mechanism or rate form the differential method can be used to develop or build up a rate equation to fit the data. 

Differentiation of the experimental concentration-time curve would then need interpolation or smoothing, e.g.,by using splines. Parallelization in a typical robotic environment is easy when using the integral method with a few or even only one single well for characterization of one enzyme variant. 

In the sections Associated disorders in western medicine, some disease names are mentioned. However, it should be borne in mind that a disease in western medicine may involve more than one syndrome in traditional Chinese medicine. As such, the diseases mentioned here are only intended to help the reader to understand the syndrome and to have some corresponding orientation in western medicine. The principles, methods and strategies introduced in each chapter of this book are abstracted from a large number of formulas, integrating the knowledge of single herbs, herbal combinations, syndrome differentiation, treatment methods and treatment sequence. They are the essential part of this book. 

M75 Estimation of parameters in differential equations by direct integral method extension of the Himmelblau-Jones-Bischoff method 7S00 8040... 

The rates of liquid-phase reactions can generally be obtained by measuring the time-dependent concentrations of reactants and/or products in a constant-volume batch reactor. From experimental data, the reaction kinetics can be analyzed either by the integration method or by the differential method ... 

Each of these methods has both merits and demerits. For example, the integration method can easily be used to test several well-known specific mechanisms. In more complicated cases the differential method may be useful, but this requires more data points. Analysis by the integration method can generally be summarized as follows. 

The fundamental difficulty in solving DEs explicitly via finite formulas is tied to the fact that antiderivatives are known for only very few functions / M —> M. One can always differentiate (via the product, quotient, or chain rule) an explicitly given function f(x) quite easily, but finding an antiderivative function F with F x) = f(x) is impossible for all except very few functions /. Numerical approximations of antiderivatives can, however, be found in the form of a table of values (rather than a functional expression) numerically by a multitude of integration methods such as collected in the ode... m file suite inside MATLAB. Some of these numerical methods have been used for several centuries, while the algorithms for stiff DEs are just a few decades old. These codes are... 

This is not by itself a kinetic method. It must be combined with either the differential or the integration method and involves keeping all the reactants but one in large excess so that their concentration does not vary through the reaction under these conditions, the observed reaction order is that of the limiting reagent. For example, the simple second-order reaction of Equation 3.18,... 

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