Partial difference equation

Breakthrough Behavior for Axial Dispersion Breakthrough behavior for adsorption with axial dispersion in a deep bed is not adequately described by the constant pattern profile for this mechanism. Equation (16-128), the partial different equation of the second order Ficldan model, requires two boundaiy conditions for its solution. The constant pattern pertains to a bed of infinite depth—in obtaining the solution we apply the downstream boundaiy condition cf 0 as oo. Breakthrough behavior presumes the existence of... 

Rewriting the diffusion equation into a partial difference equation leads to... 

The observed transients of the crystal size distribution (CSD) of industrial crystallizers are either caused by process disturbances or by instabilities in the crystallization process itself (1 ). Due to the introduction of an on-line CSD measurement technique (2), the control of CSD s in crystallization processes comes into sight. Another requirement to reach this goal is a dynamic model for the CSD in Industrial crystallizers. The dynamic model for a continuous crystallization process consists of a nonlinear partial difference equation coupled to one or two ordinary differential equations (2..iU and is completed by a set of algebraic relations for the growth and nucleatlon kinetics. The kinetic relations are empirical and contain a number of parameters which have to be estimated from the experimental data. Simulation of the experimental data in combination with a nonlinear parameter estimation is a powerful 1 technique to determine the kinetic parameters from the experimental... 

This partial difference equation is of the first order, due to the fact that the coefficients in (6.4) were linear in the y. The characteristics of (6.12) are determined by... 

Courant, R., Friedrichs, K., and Lewy, H. On the partial difference equations of mathematical physics. IBM Journal, 1967. 11(2), 215-234. English translation of the 1928 German original paper. 

Two considerations regarding truncation error that enter into the derivation of the partial difference equations should be pointed out. In some published formulations of these equations, the first radial derivative has been approximated by a forward-difference expression (Kl, S5, Wl). This unsymmetrical formula has no advantage over a symmetrical or central-difference expression, but has a greater—lower order—truncation error. The central-difference approximation... 

Equation (11) represents steady-state conditions within the maternal intervillous channel and is a nonlinear, partial difference equation with two independent variables. Since the dP/dr = 0 when r = R2> a special equation was also required at this position. The same techniques were used as in the fetal capillary equation. 

B. Friedman, The iterative solution of elliptic partial difference equations, AEC Research and Development Report NYO-7698, 1957. 

David Young, Iterative methods for solving partial difference equations of elliptic type, Trans. Amer. Math. Soc. vol. 76 (1954) pp. 92-111. 

The derivation of regular patterns of cellular behavior that can be assigned to certain defined subpopulations (or single cells) from measured data necessarily involves modeling approaches that allow assignment of mechanistic (e.g., metabohc) models to subpopulations with individual parameters or other variations. Further, such models must allow the tracking of individual cell s dynamic behavior even when they fluctuate between different populations. Two principal approaches can be distinguished systems of partial different equations (population balance systems, PBE) and stochastic cell ensemble models (CEMs)... 

Rawlings etal. (1992) analysed the stability of a continuous crystallizer based on the linearization of population and solute balance. Their model did not depend on a lumped approximation of partial difference equations and successfully predicted the occurrence of sustained oscillations. They demonstrated that simple proportional feedback control using moments of CSD as measurements can stabilize the process. It was concluded that the relatively high levels of error in these measurements require robust design for effective control. 

HeBfer and Sjostrand, 1984] B. Helffer and J. Sjostrand. Multiple wells in the semi-classical limit 1, Gommun. Partial Differ. Equations, 9, 337-408, 1984. 

One can use the above equation to discretise a partial difference equation (PDE) and implement a numerical method to solve the PDE. For example, if it is required to calculate numerical solutions of the convection-dispersion-sorption equation (CDSE, see Eq. 3.41b, also shown below) ... 

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