Special Differential Equations

Eq. (44) can be used to compute the influence of the external mass transfer parameter Hi using r calculated according Eq. (43). When reaction orders are different, mass transfer limitation of A will favor the reaction of lower order (see also [33]). In case of porous particles, also intraparticle diffusion could be incorporated into the analysis by solving the relevant differential equations. Special situations can arise here, when Ai, A2 and A3 are present in comparable concentration inside the catalyst particle. 

In Chapter 3, we covered the mathematical modeling of lumped systems as well as some preliminary examples of distributed systems using a systematic, generalized approach. The examples for distributed systems were preliminary and they were not sufficiently generalized. In this chapter, we introduce sufficient generalization for distributed systems and give more fundamentally and practically important examples, such as the axial dispersion model resulting in two-point boundary-value differential equations. These types of model equations are much more difficult to solve than models described by initial-value differential equations, specially for nonlinear cases, which are solved numerically and iteratively. Also, examples of diffusion (with and without chemical reaction) in porous structures of different shapes will be presented, explained, and solved for both linear and nonlinear cases in Chapter 6. 

The fimdamental kinetic master equations for collisional energy redistribution follow the rules of the kinetic equations for all elementary reactions. Indeed an energy transfer process by inelastic collision, equation (A3.13.5). can be considered as a somewhat special reaction . The kinetic differential equations for these processes have been discussed in the general context of chapter A3.4 on gas kmetics. We discuss here some special aspects related to collisional energy transfer in reactive systems. The general master equation for relaxation and reaction is of the type [H, 12 and 13, 15, 25, 40, 4T ] ... 

Mathematical and Computational Implementation. Solution of the complex systems of partial differential equations governing both the evolution of pollutant concentrations and meteorological variables, eg, winds, requires specialized mathematical techniques. Comparing the two sets of equations governing pollutant dynamics (eq. 5) and meteorology (eqs. 12—14) shows that in both cases they can be put in the form ... 

There are special numerical analysis techniques for solving such differential equations. New issues related to the stabiUty and convergence of a set of differential equations must be addressed. The differential equation models of unsteady-state process dynamics and a number of computer programs model such unsteady-state operations. They are of paramount importance in the design and analysis of process control systems (see Process control). 

Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fliiid mechanics texts (e.g., Slatteiy [ibid.] Denu Whitaker and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are... 

These equations form a fourth-order system of differential equations which cannot be solved analytically in almost all interesting nonseparable cases. Further, according to these equations, the particle slides from the hump of the upside-down potential — V(Q) (see fig. 24), and, unless the initial conditions are specially chosen, it exercises an infinite aperiodic motion. In other words, the instanton trajectory with the required periodic boundary conditions,... 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

The population balance is a partial integro-differential equation that is normally solved by numerical methods, except for special simplified cases. Numerical solution of the population balance for the general case is not, therefore, entirely straightforward. Ramkrishna (1985) provides a comprehensive review. 

A specially orthotropic laminate has either a single layer of a specially orthotropic material or multiple specially orthotropic layers that are symmetrically arranged about the laminate middle surface. In both cases, the laminate stiffnesses consist solely of A, A 2> 22> 66> 11> D 2, D22, and Dgg. That is, neither shear-extension or bend-twist coupling nor bending-extension coupling exists. Thus, for plate problems, the transverse deflections are described by only one differential equation of equilibrium ... 

Symmetric angle-ply laminates were described in Section 4.3.2 and found to be characterized by a full matrix of extensional stiffnesses as well as bending stiffnesses (but of course no bending-extension coupling stiffnesses because of middle-surface symmetry). The new facet of this type of laminate as opposed to specially orthotropic laminates is the appearance of the bend-twist coupling stiffnesses D. g and D2g (the shear-extension coupling stiffnesses A. g and A2g do not affect the transverse deflection w when the laminate is symmetric). The governing differential equation of equilibrium is... 

The solution to the governing differential equation, Equation (5.32), is not as simple as for specially orthotropic laminated plates because of the presence of D. g and D2g. The Fourier expansion of the deflection w. Equation (5.29), is an example of separation of variables. However, because of the terms involving D.,g and D2g, the expansion does not satisfy the governing differential equation because the variables are not separable. Moreover, the deflection expansion also does not satisfy the boundary conditions. Equation (5.33), again because of the terms involving D. g and D2g. 

Antisymmetric cross-ply laminates were found in Section 4.3.3 to have extensional stiffnesses A i, A., 2, A22 = Aj, and Aeei bending-extension coupling stiffnesses and 822 =-Bn and bending stiffnesses Di2. D22 = Dn. and Dge. The new terms here in comparison to a specially orthotropic laminate are and 822- Because of this bending-extension coupling, the three buckling differential equations are coupled ... 

It is a first-order differential equation in time, but second-order in the spatial variables. Space and time do not enter on an equal footing, as required by the special theory of relativity. 

If for certain values of a parameter A in the differential equation, the qualitative aspect of the solution (i.e., the phase portrait ) of the differential equation remains the same (in other words the changes are only quantitative) such values of A are called ordinary values. If however, for a certain value A = A0 this qualitative aspect changes, such a special value is called a critical or bifurcation value. 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

In the previous discussion of the one- and two-compartment models we have loaded the system with a single-dose D at time zero, and subsequently we observed its transient response until a steady state was reached. It has been shown that an analysis of the response in the central plasma compartment allows to estimate the transfer constants of the system. Once the transfer constants have been established, it is possible to study the behaviour of the model with different types of input functions. The case when the input is delivered at a constant rate during a certain time interval is of special importance. It applies when a drug is delivered by continuous intravenous infusion. We assume that an amount Z) of a drug is delivered during the time of infusion x at a constant rate (Fig. 39.10). The first part of the mass balance differential equation for this one-compartment open system, for times t between 0 and x, is given by ... 

Dynamic models expressed in terms of transform functions can be solved by digital simulation by transposing the transfer function into an equivalent set of differential equations, as shown by Ord-Smith and Stephenson (1975) and by Matko et al. (1992). Also some languages include special transfer function subroutines. 

Let us consider the special class of problems where all state variables are measured and the parameters enter in a linear fashion into the governing differential equations. As usual, we assume that x is the n-dimemional vector of state variables and k is the p-dimemional vector of unknown parameters. The structure of the ODE mode is of the form... 

The Hermite polynomials introduced above represent an example of special functions which arise as solutions to various second-order differential equations. After a summary of some of the properties of these polynomials, a brief description of a few others will be presented. The choice is based on their importance in certain problems in physics and chemistry. 

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