Differential equation describing time

SOLUTION OF DIFFERENTIAL EQUATION DESCRIBING TIME DEPENDENCE OF PRODUCT DETECTION... 

Zeeman has given a system of differential equations, describing time evolution of the variables x, a, b, for which a slow surface is the cusp catastrophe surface (3.78). The system of Zeeman equations has an attracting and stable stationary point E. The process of nerve impulse transmission represented on the catastrophe surface M3 is shown in Fig. 57. 

Differential equation, describing time-expense of initial reagent, is completely congruent in its form with equation of first-order simple reaction, therefore... 

Now,wecansubstitutethisexpressionfor Cb intermsof Ca into the differential equation describing the change in Ca, make it autonomous, and derive an expression for the time dependenceofCa ... 

Sometimes the time variable is eliminated from the set of differential equations describing the kinetics of the coupled system, e.g. by dividing... 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

The differential equation describing the temperature distribution as a function of time and space is subject to several constraints that control the final temperature function. Heat loss from the exterior of the barrel was by natural convection, so a heat transfer coefficient correlation (2) was used for convection from horizontal cylinders. The ends of the cylinder were assumed to be insulated. The equations describing these conditions are ... 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

To solve the kinetics for the most general case, in which, for example, we allow partial pressures to vary with time, we need the full set of differential equations describing the coverage of all species participating in the reaction ... 

The procedure for the solution of unsteady-state balances is to set up balances over a small increment of time, which will give a series of differential equations describing the process. For simple problems these equations can be solved analytically. For more complex problems computer methods would be used. 

A phase space is established for a typical particle, whose coordinates specify the location of the particle as well as its quality. Then, ordinary differential equations describe how these phase coordinates evolve in time. In other words, the state of a particle in a processing system is specified by the values of a number of phase coordinates z. The only requirement on z is that they describe the state of the particle fully enough to permit one to write a set of first order ode s of the form ... 

LES/FDF-approach. An In situ Adaptive Tabulation (ISAT) technique (due to Pope) was used to greatly reduce (by a factor of 5) the CPU time needed to solve the set of stiff differential equations describing the fast LDPE kinetics. Fig. 17 shows some of the results of interest the occurrence of hot spots in the tubular LDPE reactor provided with some feed pipe through which the initiator (peroxide) is supplied. The 2004-simulations were carried out on 34 CPU s (3 GHz) with 34 GB shared memory, but still required 34 h per macroflow time scale they served as a demo of the method. The 2006-simulations then demonstrated the impact of installing mixing promoters and of varying the inlet temperature of the initiator added. 

In the 1960 s the meteorologist Edward Lorenz worked on systems of differential equations describing weather patters, and found something utterly different. The smallest modification in the initial conditions can have a dramatic effect, resulting in a completely different outcome after a certain time. Such behaviour is called chaotic. The sets of differential equations initially were rather complex but later he developed a simpler set which shows the same effect. 

Easterby proposed a generalized theory of the transition time for sequential enzyme reactions where the steady-state production of product is preceded by a lag period or transition time during which the intermediates of the sequence are accumulating. He found that if a steady state is eventually reached, the magnitude of this lag may be calculated, even when the differentiation equations describing the process have no analytical solution. The calculation may be made for simple systems in which the enzymes obey Michaehs-Menten kinetics or for more complex pathways in which intermediates act as modifiers of the enzymes. The transition time associated with each intermediate in the sequence is given by the ratio of the appropriate steady-state intermediate concentration to the steady-state flux. The theory is also applicable to the transition between steady states produced by flux changes. Apphcation of the theory to coupled enzyme assays makes it possible to define the minimum requirements for successful operation of a coupled assay. The theory can be extended to deal with sequences in which the enzyme concentration exceeds substrate concentration. 

We now resort to the crucial approximation that a it) varies slower than either e(t) or 0(t). This approximation is justifiable in the weak-coupling regime (to second order in Hj) as discussed below. Under this approximation, Eq. (4.41) is transformed into a differential equation describing relaxation at a time-dependent rate ... 

Molecules throughout a gas have a distribution of velocities and density depending on the temperature, external forces, concentration gradients, chemical reactions, and so on. The properties of a dilute gas are known completely if the velocity distribution function /(r, p, 1) can be found. The Boltzmann equation [38], is an integro-differential equation describing the time evolution of /. The physical derivation of the Boltzmann equation is easy to state, and is presented next. However, its solution is extremely difficult, and relies on varying degrees of approximation. 

A most convenient way to solve the differential equations describing a mass transport problem is the Laplace transform method. Applications of this method to many different cases can be found in several modern and classical textbooks [21—23, 53, 73]. In addition, the fact that electrochemical relationships in the so-called Laplace domain are much simpler than in the original time domain has been employed as an expedient for the analysis of experimental data or even as the basic principle for a new technique. The latter aspect, especially, will be explained in the present section. 

In contrast to noncompartmental analysis, in compartmental analysis a decision on the number of compartments must be made. For mAbs, the standard compartment model is illustrated in Fig. 3.11. It comprises two compartments, the central and peripheral compartment, with volumes VI and V2, respectively. Both compartments exchange antibody molecules with specific first-order rate constants. The input into (if IV infusion) and elimination from the central compartment are zero-order and first-order processes, respectively. Hence, this disposition model characterizes linear pharmacokinetics. For each compartment a differential equation describing the change in antibody amount per time can be established. For... 

The description of the intermediate dynamics in Equation (5.21) involves only the flow rates us. However, it was demonstrated that these flow rates do not affect the total holdup of the impurity in the recycle loop, since the total holdup of I is influenced only by the inflow of impurity, by its transfer rate in the separator, and by the purge stream, which, as can be seen from Equation (5.21), have no influence on the dynamics in this intermediate time scale. Consequently, one of the differential equations describing the intermediate dynamics is redundant, and Equations (5.21) are not independent. Correspondingly, the steady-state conditions... 

Accurately, a RCM is obtained by solving the differential equation describing the evolution in time of the liquid composition in a batch distillation still ... 

Let us consider some drug administration practicalities. Up to now, the administered amounts were considered as initial units introduced simultaneously into several compartments at the beginning of the experiment. These amounts were considered as initial conditions to the differential equations describing the studied processes. Nevertheless, this concept seems to have limited applications in pharmacokinetics. In this section, we develop the probabilistic transfer and retention-time models associated with an extravascular or intravascular route of administration. 

When one looks into the basic functions of the link and indirect response models, it is clear that one of the differences resides in the input functions to the effect and the receptor protein site, respectively. For the link model a linear input operates in contrast to the indirect model, where a nonlinear function operates. For the link model the time is not directly present and the pharmacological time course is exclusively dictated by the pharmacokinetic time, whereas the indirect model has its own time expressed by the differential equation describing the dynamics of the integrated response. 

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