Differential/difference equations

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. 

The unsteady model, originally formulated in terms of a partial differential equation, is thus transformed into N difference differential equations. As a result of the finite-differencing, a solution can be obtained for the variation with respect to time of the water concentration, for every segment, throughout the bed. 

For solution by digital simulation, the depth of filter cake is divided into nine segments, each of which has an equivalent dimensionless thickness Ax. For any element n, the form of the resulting difference differential equation is given by... 

Deactivating catalyst 319 Dead zones 159, 162, 163 Degree of segregation 471 Density influences 492 Desorption of solute 578, 579 Difference differential equation 579 Difference formulae for partial differential equations 268 Differential column 167... 

The controller setting is different depending on which error integral we minimize. Set point and disturbance inputs have different differential equations, and since the optimization calculation depends on the time-domain solution, the result will depend on the type of input. The closed-loop poles are the same, but the zeros, which affect the time-independent coefficients, are not. 

London and Seban (L8) introduced the method of lumped parameters in melting-freezing problems, whereby the partial differential equation is converted into a difference-differential equation by differencing with respect to the space variable. The resulting system of ordinary differential... 

From the Kolmogorov equations (4.41) and (4.42), one obtains the difference-differential equations for the birth-death process. The backward equation is given by... 

Let X(t) be the number of entities which have been replicated in a cell of age T. Of course, X t) is a discrete random variable, which assumes values 0, 1,2, N. If we put PJj) equal to P X x) = a), then it can be shown by a method essentially the same as used for the pure birth process (Section III, A), that P (t) satisfies the difference-differential equation... 

Conventionally used Darcy s law and Richards equation (2 order partial differential equations) are exanq>les of a system with a positive feedback, but witiiout a negative feedback conqmnent. Negative and positive feedback mechanisms are taken into account by using tiie difference-differential equation for soil-moisture balance (75) and the Kuramoto-Sivashinsky equation 74) (see below). 

Laboratory and field experiments show die presence and interplay of such processes as intrafracture film flow along fracture surfaces, coalescence and divergence of multiple flow paths along fracture surteces, and intrafracture water dripping. The nonlinear dynamics of flow and transport processes in unsaturated fractured porous media arise from die dynamic feedback and conpetition between various nonlinear physical processes along widi the complex geometry of flow paths. The apparent randomness of (he flow field does not prohibit the system s determinism and is, in fact, described by deterministic chaotic models using deterministic differential or difference-differential equations. 

On periodic solutions of a functional equation. - In Analytic and Qualitative Methods for Investigation of Differential and Difference-Differential Equations, 121-127, Inst. Matematiki Akad. Nauk Ukrain. SSR, Kiev, 1977. 

Ordinary Difference-Differential Equations. Izdat Inostr. liter., Moscow, 1961. 

Another convenient and effective scheme for the approximate solution of a mathematical description of the polymerization reaction replaces the discrete variable of infinite range, polymer chain length, by a continuous variable. The difference-differential equations become partial differential equations. Barn-ford and coworkers [16,27,28] used this procedure in their analysis of vinyl (radical chain growth) polymerization. Zeman and Amundson [18,19] used it extensively to study batch and continuous polymerizations. Recently, Coyle et al. [4] have applied it to analysis of high conversion free radical polymerizations while Taylor et al. [3] used it in their modelling efforts oriented to control of high conversion polymerization of methyl methacrylate. A rather extensive review of the numerical techniques and approximations has been presented by Amundson and Luss [29] and later by Tirrell et al. [30]. 

Hayes, N. D. 1950. Roots of the Transcendental Equations Associated with a Certain Difference-Differential Equation, J. London Math. Soc. 25, 226-232. 

To my knowledge the point 2, namely the questi on of how to reconstruct trajectories from the return map, is less well studied. In the following I want to show that a whole class of return maps as shown in Fig.10 are caused by topologically equivalent trajectories. In the following I want to construct a typical representative of such a class of trajectories and also derive the differential equation they obey. Clearly the whole flow of trajectories can be deformed so that different differential equations can give rise to the same class of trajectories. But as I want to show, these trajectories are... 

Difference-differential equations (any type of connection of lumped- or distributed-parameter steady- or nonsteady-state subsystems)... 

As in the previous example we assume that the concentrations of activated monomers are buffered and appear implicitely in the rate constants f., i = l,...,n. Under these conditions the system follows different differential equations in the low and high concentration limit. In the first case we find... 

The master equation (5.28) represents 2A + 1 coupled linear difference differential equations forp n t) with n = -N,-N + which are... 

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