Numerical differentiation INDEX

Use small increments(< 0.1) in pH were used in setting up the spreadsheet(s) in Problem 8-1, then perform numerical differentiation by calculating (pH - pH )) /(V(3) - V(2)) (subscripts refer to row numbers) over the entire pH range of the titrations. Plot this ApH/AV vs V to get the sharpness index (S.I.) over the entire titration range. Also plot the reciprocal S.I. V to get the variation of buffo- index with titration volume (compare to equations 8-20 to 8-23). 

A characteristic of DAEs besides their form is their differentiation index [32]. For a definition and an example see Appendix C. It is an indicator for the problems to be encountered with the numerical solution of a set of DAEs. Systems of index > 1 are usually called higher index DAEs and the higher the index the more severe numerical difficulties can be. As the mathematical description of problems in various disciplines often leads to DAE system, they have been a research subject for more than two decades. A large body of publications and a number software programs for their numerical solution have emerged. DAE systems of index 1 can be safely numerically computed by means of the backward differentiation formula (BDF) [33, 34] implemented in solvers such as the well known and widely used DASSL code [35]. 

Brenan, K. E., and Petzold, L. R., The numerical solution of higher index differential/algebraic equations by implicit Runge-Kutta methods," UCRL-95905, preprint, Lawrence Livermore National Laboratories, Livermore, California (1987). 

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

Thus for numerical solution, the equations are the (at) equations n. C. 9., the (x-at) equations n. C. 10., n. C. 11. andH. C. 12. for the p + 2 variables T, P, and pX1. With all quantities known at some starting point z = 0, a computing machine can be programmed to calculate the derivatives in equations n. C. 10-12. Various machine integration routines are then available to solve simultaneous, first order differential equations. Such routines should have a variable step-wise feature for automatically doubling or halving the internal to satisfy a chosen precision index. 

For these computational rules to be valid, all the differential quotients appearing on the left and right sides must make sense. This means that the quantities in the numerator must really be differentiable functions of the variables appearing in the denominator and index. 

Of the four main quantities —p, V, T, S mentioned above, —p and T are most easily controlled (—p orp because one often works in the laboratory with containers which are open to the atmosphere) so they are the preferred independent variables. These preferred quantities appear in the denominator or index of the differential quotients that, possibly multiplied by certain factors, can be found in Tables. The three coefficients y, mentioned above, are of this type. Consequently, we will attempt to convert a given coefficient so that only preferred quantities (in this case, p and T) appear in the denominator or index of the differential quotient, but never in the numerator. If we abbreviate all the other quantities with a, a, . .. and the preferred ones with b, b, . .. (here we have only two of these, but the method remains the same even when there are more), the given differential quotients are to be replaced by some of the type da/db)y. ... 

If the differential equations are infinitely differentiable, we may take the truncation index k as large as we like, but the constants appearing in the above theorem will depend on the truncation index in a complicated way. It is possible to prove (see discussions in [164, 227] for more detail), that for many standard classes of numerical methods, there are real, positive constants C, D such that... 

Index reduction entails another well known problem, namely that of numerical drift. Original constraint equations get lost by differentiation and cannot be taken into account in the numerical solution of the reduced index system. Hence, the numerical solution of the reduced index problem can only approximate the original constraints. This suggests to keep the original constraints and differentiated equations in the numerical solution of the reduced index problem resulting in more equations than unknowns in the reduced index problem. The dummy derivative method addresses this problem by considering the derivatives of some variables as new independent algebraic variables called dummy derivatives so that the number of unknowns matches the number of equations. This approach, however, requires to decide which variables are selected as dummy derivatives and which ones as states. 

The numerical value of the glass-transition temperature depends on the rate of measurement (see Section 10.1.2). The techniques are therefore subdivided into static and dynamic measurements. The static methods include determinations of heat capacities (including differential thermal analysis), volume change, and, as a consequence of the Lorentz-Lorenz volume-refractive index relationship, the change in refractive index as a function of temperature. Dynamic methods are represented by techniques such as broad-line nuclear magnetic resonance, mechanical loss, and dielectric-loss measurements. Static and dynamic glass transition temperatures can be interconverted. The probability p of segmental mobility increases as the free volume fraction / Lp increases (see also Section 5.5.1). For /wlf = of necessity, p = 0. For / Lp oo, it follows that p = 1. The functionality is consequently... 

Fig. 27.3a,b. The volumetric shape index and curvedness, a The shape index maps the topological shape of a surface patch into a numerical value. The figure shows five sample shapes with their shape index values cup, rut, saddle, ridge, and cap. Color coding of these different types of shapes can differentiate among the structures on the colonic wall (see Fig. 27.4). b The curvedness represents the size and scale of the shape. For example, a gentle cap-like shape and a sharp cap-like shape have the same shape index value, but have a different curvedness values... 

The occupational environment can be neutral, cold or hot. A combined action between the four environmental parameters (temperature, relative humidity, velocity and radiant heat) and the two individual parameters (clothing worn by the occupants and their activity) can lead to a thermal comfort, discomfort, or to a thermal stress situation (Parsons, 2013). The integration of these parameters can be done in a thermal index in a way that will provide a single value that is related to the effects on the occupants. Three types of indices can be identified empirical, rational and derived. According to Parsons (2000), rational indices are derived from mathematical models that describe the behavior of the human body in thermal environments. The analysis of these situations can be achieved using diverse techniques and comfort models, such as Computation Fluid Dynamics (CFD) and other numerical simulations (Murakami et ah, 2000). The human thermal software (Teixeira et al., 2010) is based on differential... 

Abstract. A class (m,k)-methods is discussed for the numerical solution of the initial value problems for impHcit systems of ordinary differential equations. The order conditions and convergence of the numerical solution in the case of implementation of the scheme with the time-lagging of matrices derivatives for systems of index 1 are obtained. At A < 4 the order conditions are studied and schemes optimal computing costs are obtained. 

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