Linked differential equations

Both the heat flow analogy and the NMR experiment can be represented by a pair of linked differential equations ... 

Using equations (4) through (9), the unknown concentration Cr can be eliminated and a system of linked differential equations for the fluid concentration Cq and the core radius rc is obtained. 

In this section, we will discuss in detail the linked differential equations for mass and heat transfer which describe the drying of a spherical green body. This same analysis can also be used for plate and cylinder green bodies with corrections for the geometry. The equations for cylinder and plate drying are presented in Tables 14.3 and 14.4. 

Thus, the Lotka-Volterra model consists of these two linked differential equations that cannot be separated from each other and that cannot be solved in closed form. The numerical solution, which comes close to the real dynamics of the changes in the abundance of the lynx and snowshoe hare (Figure 4.10.30) is shown in Figure 4.10.31. For the arbitrarily chosen value of the maximum hare population of = 100 (a.u.), the parameters are (unit a ) A = 0.8, 6 = 0.05, C = 0.6,... 

What is a mathematical model The group of unknown physical quantities which interest us and the group of available data are closely interconnected. This link may be embodied in algebraic or differential equations. A proper choice of the mathematical model facilitates solving these equations and providing the subsidiary information on the coefficients of equations as well as on the initial and boundary data. 

In chemical kinetics, one finds linked sets of differential equations expressing the rates of change of the interacting species. Overall, mathematical models have been exceedingly successfiil in depicting the broad outlines of an enormously diverse variety of phenomena in nature. Some scientists have even commented in surprise at how well mathematics works in describing nature. So successful have these mathematical models been that their use has spread from the hard sciences to areas as diverse as economics and the analysis of athletic performance [3]. 

At the same time it is worth to notice that in modern numerical methods of a solution of boundary value problems, based on replacement of differential equations by finite difference, these steps are performed simultaneously. In accordance with the theorem of uniqueness, the field inside the volume V is defined by a distribution of masses inside this volume and boundary conditions, and correspondingly it is natural to derive an equation establishing this link. With this purpose in mind we will again proceed from Gauss s theorem,... 

The principle of the perfectly-mixed stirred tank has been discussed previously in Sec. 1.2.2, and this provides essential building block for modelling applications. In this section, the concept is applied to tank type reactor systems and stagewise mass transfer applications, such that the resulting model equations often appear in the form of linked sets of first-order difference differential equations. Solution by digital simulation works well for small problems, in which the number of equations are relatively small and where the problem is not compounded by stiffness or by the need for iterative procedures. For these reasons, the dynamic modelling of the continuous distillation columns in this section is intended only as a demonstration of method, rather than as a realistic attempt at solution. For the solution of complex distillation problems, the reader is referred to commercial dynamic simulation packages. 

Writing unsteady-state component balances for each liquid phase results in the following pair of partial differential equations which are linked by the mass transfer rate and equilibrium relationships... 

Strictly speaking, finite difference or finite element solutions to differential equations are simply multiplying the number of comparments many times, but the mathematical rules for linking cells in difference calculations are rigorously set by the form of the equations. 

Choosing the concentration of ADP as the dependent variable, the network is described by a set of four linearly independent differential equations. The link matrix L is... 

Although the partial differential equation Eq. 25-10 is linear and looks rather simple, explicit analytical solutions can be derived only for special cases. They are characterized by the size of certain nondimensional numbers that completely determine the shape of the solutions in space and time. A reference distance x0 and a reference time f0 are chosen that are linked by ... 

Scalar equations, as studied earlier, depend on one or several real variables x, y, t, T,. Differential equations instead link various derivatives of one or more functions /(...), y(...),..., each with any number of variables. These mathematical functions describe state variables in engineering parlance. Differential equations are equations in one or more variables and in one or more functions of the variables and in their derivatives. They involve independent variables such as space and time, and dependent, so called state variables or functions and their derivatives. Many physico-chemical processes are governed by differential equations or systems thereof, that involve unknown functions/, g,. .. in various variables and various of their derivatives/, g1, g",. .. 

The second group, i.e. at a slightly lower level, is the one where all factors that are part of an observed phenomenon are known, but we know or are only partly aware of their interrelationships, i.e. influences. This is usually the case when we are faced with a complex phenomenon consisting of numerous factors. Sometimes we can link these factors as a system of simultaneous differential equations but with no solutions to them. As an example we can cite the Navier-Stokes simultaneous system of differential equations, used to define the flow of an ideal fluid ... 

Thus, the determinant equation represents a connecting link between differential equations (3.31) promoting their uniting in a system. To determine the determinant with sufficient... 

The two results can be written together as a system of two linked (coupled) first-order differential equations. This means that the return of Mz to equilibrium depends on how far Mz is from equilibrium, and vice versa. 

The initial idea is to use the differential equations of a probabilistic transfer model with hazard rates varying with the age of the molecules, i.e., to enlarge the limiting hypothesis (9.2). The objective is to find nonexponential families of survival distributions that are mathematically tractable and yet sufficiently flexible to fit the observed data. In the simplest case, the differential equation (9.7) links hazard rates and survival distributions. Nevertheless, this relation was at the origin of an erroneous use of the hazard function. In fact, substituting in this relation the age a by the exogenous time t, we obtain... 

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. 

This provides a pair of coupled, non-linear (through the Arrhenius temperature dependence) ordinary differential equation for the two variables a and T. If the temperature increases, the reaction rate increases through the increase in k. The consequent increase in T will lead to increases in the heat transfer rates and also to a decrease in the concentration of A, which in turn tends to decrease the reaction rate term ka. To quantify this effect, we can examine the adiabatic case a = 0- In this situation, the temperature rise above the inflow is uniquely linked to the extent of reaction = ao a)/ao through the condition... 

We have developed a two-step procedure for the in silico screening of compound libraries based on biopharmaceutical property estimation linked to a mechanistic simulation of GI absorption. The first step involves biopharmaceutical property estimation by application of machine learning procedures to empirical data modeled with a set of molecular descriptors derived from 2D and 3D molecular structures. In silico methods were used to estimate such biopharmaceutical properties as effective human jejunal permeability, cell culture permeability, aqueous solubility, and molecular diffusivity. In the second step, differential equations for the advanced compartmental absorption and transit model were numerically integrated to determine the rate, extent, and approximate GI location of drug liberation (for controlled release), dissolution, and absorption. Figure 17.3 shows the schematic diagram of the ACAT model in which each one of the arrows represents an ordinary differential equation (ODE). 

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