Selection differential equations

In this section, formulae for the solution of selected differential equations are presented. The advantage of having the solution expressed in terms of the formula is that the solution is known without having to know the values of the parameters. Sometimes, the solution is given as an implicit nonlinear algebraic equation. In such cases, the advantage of analytical formulae over numerical computation diminishes. 

In the earlier versions of the streamline upwinding scheme the modified weight function was only applied to the convection tenns (i.e. first-order derivatives in the hyperbolic equations) while all other terms were weighted in the usual manner. This is called selective or inconsistent upwinding. Selective upwinding can be interpreted as the introduction of an artificial diffusion in addition to the physical diffusion to the weighted residual statement of the differential equation. This improves the stability of the scheme but the accuracy of the solution declines. 

In the finite-difference appntach, the partial differential equation for the conduction of heat in solids is replaced by a set of algebraic equations of temperature differences between discrete points in the slab. Actually, the wall is divided into a number of individual layers, and for each, the energy conserva-tk>n equation is applied. This leads to a set of linear equations, which are explicitly or implicitly solved. This approach allows the calculation of the time evolution of temperatures in the wall, surface temperatures, and heat fluxes. The temporal and spatial resolution can be selected individually, although the computation time increa.ses linearly for high resolutions. The method easily can be expanded to the two- and three-dimensional cases by dividing the wall into individual elements rather than layers. 

LGs can also serve as powerful alternatives to PDEs themselves in modeling physical systems. The distinction is an important one. It must be remembered, however, that not all PDEs (and perhaps not all physical systems see chapter 12) are amenable to a LG simulation. Moreover, even if a candidate PDE is selected for simulation by a LG. there is no currently known cookbook recipe allowing a researcher to go from the PDE to a LG description (or vice versa). Nonetheless, by their very nature, LGs lend themselves to modeling any partial differential equation (PDE) for which the underlying physical basis for its construction involves a large number of particles with local interactions [wolf86c]. 

As a result, a considerable amount of effort has been expended in designing various methods for providing difference approximations of differential equations. The simplest and, in a certain sense, natural method is connected with selecting a, suitable pattern and imposing on this pattern a difference equation with undetermined coefficients which may depend on nodal points and step. Requirements of solvability and approximation of a certain order cause some limitations on a proper choice of coefficients. However, those constraints are rather mild and we get an infinite set (for instance, a multi-parameter family) of schemes. There is some consensus of opinion that this is acceptable if we wish to get more and more properties of schemes such as homogeneity, conservatism, etc., leaving us with narrower classes of admissible schemes. 

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

The basic principles are described in many textbooks [24, 26]. They are thus only sketchily presented here. In a conventional classical molecular dynamics calculation, a system of particles is placed within a cell of fixed volume, most frequently cubic in size. A set of velocities is also assigned, usually drawn from a Maxwell-Boltzmann distribution appropriate to the temperature of interest and selected in a way so as to make the net linear momentum zero. The subsequent trajectories of the particles are then calculated using the Newton equations of motion. Employing the finite difference method, this set of differential equations is transformed into a set of algebraic equations, which are solved by computer. The particles are assumed to interact through some prescribed force law. The dispersion, dipole-dipole, and polarization forces are typically included whenever possible, they are taken from the literature. 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

Some economies are possible if equilibrium is assumed between selected compartments, an equal fugacity being assignable. This is possible if the time for equilibration is short compared to the time constant for the dominant processes of reaction or advection. For example, the rate of chemical uptake by fish from water can often be ignored (and thus need not be measured or known within limits) if the chemical has a life time of hundreds of days since the uptake time is usually only a few days. This is equivalent to the frequently used "steady state" assumption in chemical kinetics in which the differential equation for a short lived intermediate species is set to zero, thus reducing the equation to algebraic form. When the compartment contains a small amount of chemical or adjusts quickly to its environment, it can be treated algebraically. 

The sequence in which to introduce the range of topics presents a problem. To end up with a theory of chemistry based on relativity and quantum mechanics a minimum background in physical chemistry, mechanics and electromagnetism is essential, which in turn requires a knowledge of vectors, complex numbers and differential equations. The selection of material within the preliminary topics is strictly biased by later needs and presented in the usual style of the parent disciplines. Many readers may like to avoid some tedium by treating the introductory material only for reference, as and when required. 

Equation (xi) must be numerically integrated, using either Ex(t) or E2 t), and the appropriate expressions for cA(t) and cD(t) (see E-Z Solve file ex20-5.msp). Table 20.1 gives the outlet concentration, conversion, yield, and selectivity obtained for each of the two cases, (d) Maximum-mixedness model For the maximum-mixedness model, the rate laws for A and D are substituted into Equation 20.4-6, and the two resulting ordinary differential equations (in dcA/dt and dcD/dt) must be numerically integrated. The respective equations are ... 

Substituting Eq. (35) into Eq. (31), using polar coordinates (i.e., 7 = (r, 0, < )), and selecting a fixed ray at angles and (t> , we obtain the following first order differential equation for the transformation function ... 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

Selecting the same initial step size and threshold as in Example 5.4A, we solve the differential equation (5.58). In order to compare the results,... 

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

Vode, solves stiff systems of ordinary differential equations (ODE) using backward differentiation techniques [49]. It implements rigorous control of local truncation errors by automatic time-step selection. It delivers computational efficiency by automatically varying the integration order. 

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