Traditional differential equation model

Soil modeling follows three different mathematical formulation patterns (1) Traditional Differential Equation (TDE) modeling (2) Compartmental modeling and... 

This model reduces to the two-phase model given by Eq. (288) under steady-state conditions. However, for the general case of time-varying inlet conditions this model retains all the qualitative features of the full partial differential equation model and while the traditional two-phase model which does not distinguish between cm and (c) ignores the dispersion effect in the fluid phase. 

Traditional control systems are in general based on mathematical models that describe the control system using one or more differential equations that define the system response to its inputs. In many cases, the mathematical model of the control process may not exist or may be too expensive in terms of computer processing power and memory. In these cases a system based on empirical rules may be more effective. In many cases, fuzzy control can be used to improve existing controller systems by adding an extra layer of intelligence to the current control method. 

In the first chapter several traditional types of physical models were discussed. These models rely on the physical concepts of energies and forces to guide the actions of molecules or other species, and are customarily expressed mathematically in terms of coupled sets of ordinary or partial differential equations. Most traditional models are deterministic in nature— that is, the results of simulations based on these models are completely determined by the force fields employed and the initial conditions of the simulations. In this chapter a very different approach is introduced, one in which the behaviors of the species under investigation are governed not by forces and energies, but by rules. The rules, as we shall see, can be either deterministic or probabilistic, the latter leading to important new insights and possibilities. This new approach relies on the use of cellular automata. 

In the many traditional methods of calculating turbulent flows, these turbulence terms are empirically defined, i.e., turbulence models that are almost entirely empirical are used. Some success has, however, been achieved by using additional differential equations to help in the description of these terms. Empiricism is not entirely eliminated, at present, by the use of these extra equations but the empiricism can be introduced in a more systematic and logical manner than is possible if the turbulence terms in the momentum equation are completely empirically described. One of the most widely used additional equations for this purpose is the turbulence kinetic energy equation and its general derivation will now be discussed. 

One source of nonlinear compartmental models is processes of enzyme-catalyzed reactions that occur in living cells. In such reactions, the reactant combines with an enzyme to form an enzyme-substrate complex, which can then break down to release the product of the reaction and free enzyme or can release the substrate unchanged as well as free enzyme. Traditional compartmental analysis cannot be applied to model enzymatic reactions, but the law of mass-balance allows us to obtain a set of differential equations describing mechanisms implied in such reactions. An important feature of such reactions is that the enzyme... 

The traditional modeling approach in biochemistry is differential equation-based enzyme kinetics. Consequently, the majority of this book so far has been devoted to kinetic modeling. Many examples demonstrate the power and feasibility of kinetic modeling applied to a few enzymatic reactions at a time. It remains to be demonstrated, however, that that approach can be effectively scaled up to an in vivo system of hundreds of reactions and species with thousands of parameters [194], More importantly, it is clear that the kinetic approach is not yet feasible for many large systems simply because the necessary kinetic information is not yet available. 

The mathematical origin of these concentration discontinuities or weak solutions of Eq. 7.1 has been explained by Courant and Friedrichs [24] and by Lax [25]. The mathematical backgroimd has been reviewed in cormection with the discussion of the numerical solution of the equilibrium-dispersive model given by Rou-chon et al. [26]. In the traditional theory of partial differential equations, a solution should be continuous. Lax [27] generalized the concept of solution to include weak solutions, which are not continuously differentiable. A solution of Eq. 7.1 that includes a continuous part, or diffuse bormdary, and a concentration shock is a weak solution of this equation [1,26-28]. A serious problem then arises, since there is no unique weak solution of Eq. 7.1. It is necessary to define the weak solution that is acceptable for the physical problem in order to achieve the determination of the band profile. This solution must make physical sense and prevent the crossing of the characteristics. Oleinik has suggested a selection rule that can... 

Traditionally, dynamical systems are modeled by differential equations. In the case of biochemical networks, Ordinary Differential Equations (ODEs) model the concentration (or activity) of proteins, RNA species, or metabolites by time-dependent... 

Chemical systems are traditionally modeled by reaction-diffusion systems on suitable domains. As was explained above, our main modeling assumption is that the domain is actually unbounded, that is, we consider governing partial differential equations on the entire plane. This assumption may seem unrealistic neither experiments nor numerical simulations can be performed on unbounded domains. In our particular context of spiral waves in the BZ reaction, however, experiments indicate that spiral waves behave much as if there were no boundaries. Therefore until boundary annihilation sets in - typically within only one to two wavelengths from the boundary itself - we consider reaction-diffusion systems ... 

The differential equations are stiff that is, several processes are going on at the same time, but at widely differing rates. This is a common feature of chemical kinetic equations and makes the numerical solution of the differential equations difficult. A steady state is never reached, so the equations cannot be solved analytically. Traditional methods, such as the Euler method and the Runge-Kutta method, use a time step, which must be scaled to fit the fastest process that is occurring. This can lead to large number of iterations even for small time scales. Hence, the use of Stella to model this oscillatory reaction would lead to an impossible situation. 

There are three ways to simulate reaction-diffusion system. The traditional method is to solve partial differential equation directly. Another way is to divide system into cells, which is called cell dynamic scheme (CDS). Typical models are cellular automata (CA)[176] and coupled map lattice (CML)[177]. In cellular automata model, each value of the cell (lattice) is digital. On the other hand, in coupled map lattice model, each value of the lattice (cell) is continuous. CA model is microscopic while CML model is mesoscopic. The advantage of the CML is compatibility with the physical phenomena by smaller number of cells and numerical stability. Therefore, the model based on CML is developed. Each cell has continuum state and the time step is discrete. Generally, each cell is static and not deformable. Deformable cell (lattice) is supposed in order to represent deformation process of the gel. Each cell deforms based on the internal state, which is determined by the reaction between the cell and the environment. 

The problem inherent in population balance modeling of an infinite set of differential equations traditionally has been addressed by the method of moments, as described in the following. 

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