First-order equation derivation

Time derivatives in expansion (2.113) can now be substituted using the differential equation (2.112) (Donea, 1984 The first order time derivative in expansion (2.113) is substituted using Equation (2.112) as... 

The selection of a time increment dependent on parameter a (i.e. carrying out Taylor series expansion at a level between successive time steps of n and n+Y) enhances the flexibility of the temporal discretizations by allowing the introduction of various amounts of smoothing in different problems. The first-order time derivatives are found from the governing equations as... 

Effective computer codes for the optimization of plants using process simulators require accurate values for first-order partial derivatives. In equation-based codes, getting analytical derivatives is straightforward, but may be complicated and subject to error. Analytic differentiation ameliorates error but yields results that may involve excessive computation time. Finite-difference substitutes for analytical derivatives are simple for the user to implement, but also can involve excessive computation time. 

Wolbert et al. in 1991 proposed a method of obtaining accurate analytical first-order partial derivatives for use in modular-based optimization. Wolbert (1994) showed how to implement the method. They represented a module by a set of algebraic equations comprising the mass balances, energy balance, and phase relations ... 

Quantitative measurements of simple and enzyme-catalyzed reaction rates were under way by the 1850s. In that year Wilhelmy derived first order equations for acid-catalyzed hydrolysis of sucrose which he could follow by the inversion of rotation of plane polarized light. Berthellot (1862) derived second-order equations for the rates of ester formation and, shortly after, Harcourt observed that rates of reaction doubled for each 10 °C rise in temperature. Guldberg and Waage (1864-67) demonstrated that the equilibrium of the reaction was affected by the concentration ) of the reacting substance(s). By 1877 Arrhenius had derived the definition of the equilbrium constant for a reaction from the rate constants of the forward and backward reactions. Ostwald in 1884 showed that sucrose and ester hydrolyses were affected by H+ concentration (pH). 

A general systematic technique applicable to second-order differential equations, of which (11.31) is a particular example, is that of phase plane analysis. We have seen this approach before (chapter 3) in the context of systems with two first-order equations. These two cases are, however, equivalent. We can replace eqn (11.31) by two first-order equations by introducing a new variable g, which is simply the derivative of the concentration with respect to z. Thus... 

The direction of differencing for the two first-order equations, Eqs. 7.60 and 7.61, is opposite that is, the pressure derivative involves information at points j + 1 and j, while the radial-coordinate equation involves information at points j and j — 1. This differencing is essential to the implicit-boundary condition specification that sets the radial coordinate at both boundaries. The opposite sense of the differencing permits information to be propagated from both boundaries into the interior of the domain. 

The continuity equation at the inlet boundary can be viewed as a constraint equation. Referring to the difference stencil (Fig. 17.14), it is seen that this first-order equation itself is evaluated at the boundary and no explicit boundary condition is needed. Moreover, since the inlet temperature, pressure, and composition are specified, the density is fixed and thus dp/dt = 0. Therefore, at the boundary, the continuity equation (Eq. 17.15) has no time derivative it is an algebraic constraint. There is no explicit boundary condition for A. At the inlet boundary, the value of A must be determined in such a way that all the other boundary conditions are satisfied. Being an eigenvalue, A s effect is felt through its influence on the V velocity in the radial momentum equation, and subsequently by V s influence on u through the continuity equation. 

A one-dimensional Fokker-Planck equation was used by Smoluchowski [19], and the bivariate Fokker-Planck equation in phase space was investigated by Klein [21] and Kramers [22], Note that, in essence, the Rayleigh equation [23] is a monovariate Fokker-Planck equation in velocity space. Physically, the Fokker-Planck equation describes the temporal change of the pdf of a particle subjected to diffusive motion and an external drift, manifest in the second- and first-order spatial derivatives, respectively. Mathematically, it is a linear second-order parabolic partial differential equation, and it is also referred to as a forward Kolmogorov equation. The most comprehensive reference for Fokker-Planck equations is probably Risken s monograph [14]. 

When accurate data can be obtained over a range of both concentrations and temperatures, it is possible from the Michaelis-Menton model to obtain data on the first-order rate constant kz and the constant Km = kz + kz)/ki and their apparent activation energies Ez and Unfortunately, most of the values quoted in the literature for the activation energies of enzyme-catalyzed reactions are derived from the use of overly simple first-order equations to describe the reaction. Consequently these values are a composite of Kmj kzy and the other constants in the Michaelis-Menton equation and cannot be used for interpretive purposes. Where the constants have been separated it is found that the values of Ez are low and of order of magnitude of 5 to 15 Kcal/mole. It is of interest to note that enzyme preparations from different biological sources, which may show different specific activity for a given reaction, have very nearly the same temperature coefficient for their specific rate constants. ... 

By the definition of the steady-state condition, the first terms on the right-hand side are zero and provided the first-order partial derivative terms do not all vanish, we can ignore the additional terms which are second-order in the perturbations. Thus, we obtain a pair of linear equations for the evolution of these perturbations in the vicinity of the steady-state point... 

The LDA radial Schrodinger equation is solved by matching the outward numerical finite-difference solution to sin inward-going solution (which vanishes at infinity) of the same energy, near the classical turning point. Continuity of P t(r) = rAn/(r) and its derivative determines the eigenvalue /. The second order differential equation is actually solved as a pair of simultaneous first-order equations, so that the nonrelativistic and relativistic (Dirac equation) procedures appear similar. 

This is a second-order differential equation, since the second derivative 0 is the highest one that appears. We are not yet equipped to analyze second-order equations, so we would like to find some conditions under which we can safely neglect the mr term. Then (1) reduces to a first-order equation, and we can apply our machinery to it. 

Fd(E/N) is the fractional power transferred to dissociative electronic excitation at a given E/N, as derived from Fig. 41, W is the electric power density transferred to the discharge (W cm-3) and 6.7 x 105 J mol-1 is the 7 eV threshold energy of process XLIV, N°o2 is the initial number density of C02. The experimentally determined degree of C02 dissociation zd at the exit of the discharge zone of length 5 has been fitted by a first order equation with volume variation... 

Strictly speaking, any order between one and zero might be applied, unless the experimenter takes into account where the test solution fits in Figure 3.12. In an experiment using a higher concentration of drug, it would be rare to follow the reaction to the extent (>50% conversion) necessary to decide clearly which kinetic order is applicable. In most cases, the data have been treated according to the first-order equations, which may not be a correct interpretation, but is acceptable because the relative or apparent rate constants derived are not absolute and apply only to the apparatus used. 

The derivative (D) being approximated by the finite-difference operator (FD) to within a truncation error (TE) (or, discretization error). The foregoing mathematical consideration provides an estimate of the accuracy of the discretization of the difference operators. It shows that TE is of the order of (Ax)2 for the central difference, but only O(Ax) for the forward and backward difference operators of first order. Equations (4.41) and (4.42) involve 2 or 3 nodes around node i at x , leading to 2- and 3-point difference operators. Considering additional Taylor series expansions extending to nodes i + 2 and i - 2 etc., located at x + 2Ax and x. — 2Ax, etc., respectively, one may derive 4- and 5-point difference formulas with associated truncation errors. Results summarized in Table 4,8 show that a TE of O(Ax)4 can be achieved in this manner. The penalty for this increased accuracy is the increased complexity of the coefficient matrix of the resulting system of equations. 

In studying the kinetics of benzene oxidation to maleic anhydride and in order to eliminate diffusion hindrance, Ioffe and Lyubarskii (151) used the flow-circulating method. The rate of this reaction was found to be proportional to benzene concentration to the power of 0.78, and that of high conversion to the power of 0.71. The rate of maleic anhydride oxidation followed a first order equation. Kinetic equations were derived from experimental results ... 

Simplified surface-area based rate model. A simple isotope exchange rate model derived by Northrop and Clayton (1966) was modified by Cole et al. (1983) to account for the surface area of the solid in experimental mineral-fluid systems. As a first approximation, this model assumes that the rate-limiting step involves the addition and removal of atoms (O, H, C) from the surface of the solid. The overall rate of reaction, R, can be expressed in the following pseudo-first order equation with the inclusion of a factor. As, representing the total surface area (m ) of the mineral... 

The Newton-Gauss method consists of linearizing the model equation using a Taylor series expansion around a set of initial parameter values bo, also called preliminary estimates, whereby only the first-order partial derivatives are consider... 

Often, an nth-order differential equation is placed in state-variable form. This is a set of n first-order equations. When deriving equations via material and energy balances, this state-variable form arises naturally. By developing the equations in state-variable form, we must solve the simultaneous sets of linear first-order differential equations. The state variable form is... 

In general, the mutarotations which cannot be expressed by the first-order equation conform to equations derived on the assumption of three components in the equilibrium mixture. The equilibrium involved may be ... 

Non-monotonic density profiles are unstable. Since, however, the influence of the wall decays exponentially with the distance, the dynamics is practically frozen whenever the interphase boundary is separated from the wall by a layer thick compared to the characteristic width of the diffuse interface. A static solution with a fixed h exists only at a certain fixed value of i, which can be determined using a solvability condition of the first-order equation as in Section 1.3. In a wider context, an appropriate solvability condition serves to obtain an evolution equation for the nominal position h of the interphase boundary. The technique of derivation of solvability conditions for a problem involving a semi-infinite region and exponentially decaying interactions is non-standard and therefore deserves some attention. 

We here concentrate on the CSF representation in somewhat greater detail. The first order nuclear derivative CSF coupling elements are expressed in terms of the following equation... 

The coefficients a, b, etc., may be functions of x. The order of the equation is the order of the highest derivative that occurs in it, so eqn 6.12a is a first-order equation and the expression above is a second-order equation. Solving a differential equation is the process of determining the function, in this casey(x), that satisfies it. 

Here, the XC-kemel [p] has been defined as the seeond functional derivative with respect to p, and it has been tacitly assumed that for all functionals in use fliis yields a factor (x - x ), which reduces the double integral to a single integral. Taken together, this results in the first-order equations... 

Exercise 11.5 Derive the first-order equation Eq. (11.41) from the Ehrenfest theorem... 

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