Underlying ordinary differential equation

The presence of such an underlying ordinary differential equation is very useful for conceptual purposes, but in practice it is usually found to be preferable to work directly with the constrained formulation (4.7)-(4.9) for numerical discretization and for describing statistical mechanical properties. 

This is an explicit expression for the solution components X2- Differentiation leads together with Eq. (2.6.5a) to an ODE, the underlying ordinary differential equation (UODE)... 

Turbulence is generally understood to refer to a state of spatiotemporal chaos that is to say, a state in which chaos exists on all spatial and temporal scales. If the reader is unsatisfied with this description, it is perhaps because one of the many important open questions is how to rigorously define such a state. Much of our current understanding actually comes from hints obtained through the study of simpler dynamical systems, such as ordinary differential equations and discrete mappings (see chapter 4), which exhibit only temporal chaosJ The assumption has been that, at least for scenarios in which the velocity field fluctuates chaotically in time but remains relatively smooth in space, the underlying mechanisms for the onset of chaos in the simpler systems and the onset of the temporal turbulence in fluids are fundamentally the same. 

In this chapter we concentrate on dynamic, distributed systems described by partial differential equations. Under certain conditions, some of these systems, particularly those described by linear PDEs, have analytical solutions. If such a solution does exist and the unknown parameters appear in the solution expression, the estimation problem can often be reduced to that for systems described by algebraic equations. However, most of the time, an analytical solution cannot be found and the PDEs have to be solved numerically. This case is of interest here. Our general approach is to convert the partial differential equations (PDEs) to a set of ordinary differential equations (ODEs) and then employ the techniques presented in Chapter 6 taking into consideration the high dimensionality of the problem. 

Our main motivation to develop the specific transient technique of wavefront analysis, presented in detail in (21, 22, 5), was to make feasible the direct separation and direct measurements of individual relaxation steps. As we will show this objective is feasible, because the elements of this technique correspond to integral (therefore amplified) effects of the initial rate, the initial acceleration and the differential accumulative effect. Unfortunately the implication of the space coordinate makes the general mathematical analysis of the transient responses cumbersome, particularly if one has to take into account the axial dispersion effects. But we will show that the mathematical analysis of the fastest wavefront which only will be considered here, is straight forward, because it is limited to ordinary differential equations dispersion effects are important only for large residence times of wavefronts in the system, i.e. for slow waves. We naturally recognize that this technique requires an additional experimental and theoretical effort, but we believe that it is an effective technique for the study of catalysis under technical operating conditions, where the micro- as well as the macrorelaxations above mentioned are equally important. 

Under stationary conditions dc /dt = 0, and an ordinary differential equation results with Eq. (10.5) as boundary conditions, which can be solved explicitly by standard techniques. The resulting expression for the current density is ... 

Simplified mathematical models These models typically begin with the basic conservation equations of the first principle models but make simplifying assumptions (typically related to similarity theory) to reduce the problem to the solution of (simultaneous) ordinary differential equations. In the verification process, such models must also address the relevant physical phenomenon as well as be validated for the application being considered. Such models are typically easily solved on a computer with typically less user interaction than required for the solution of PDEs. Simplified mathematical models may also be used as screening tools to identify the most important release scenarios however, other modeling approaches should be considered only if they address and have been validated for the important aspects of the scenario under consideration. 

Stratonovich SDEs, unlike Ito SDEs, may thus be manipulated using the familiar calculus of differentiable functions, rather than the Ito calculus. This property of a Stratonovich SDE may be shown to follow from the Ito transformation rule for the equivalent Ito SDE. It also follows immediately from the definition of the Stratonovich SDE as the white-noise limit of an ordinary differential equation, since the coefficients in the underlying ODE may be legitimately manipulated by the usual rules of calculus. 

In the traditional interpretation of the Fangevin equation for a constrained system, the overall drift velocity is insensitive to the presence or absence of hard components of the random forces, since these components are instantaneously canceled in the underlying ODF by constraint forces. This insensitivity to the presence of hard forces is obtained, however, only if both the projected divergence of the mobility and the force bias are retained in the expression for the drift velocity. The drift velocity for a kinetic interpretation of a constrained Langevin equation does not contain a force bias, and does depend on statistical properties of the hard random force components. Both Fixman and Hinch nominally considered the traditional interpretation of the Langevin equation for the Cartesian bead coordinates as a limit of an ordinary differential equation. Both authors, however, neglected the possible existence of a bias in the Cartesian random forces. As a result, both obtained a drift velocity that (after correcting the error in Fixman s expression for the pseudoforce) is actually the appropriate expression for a kinetic interpretation. 

The mathematical difficulty increases from homogeneous reactions, to mass transfer, and to heterogeneous reactions. To quantify the kinetics of homogeneous reactions, ordinary differential equations must be solved. To quantify diffusion, the diffusion equation (a partial differential equation) must be solved. To quantify mass transport including both convection and diffusion, the combined equation of flow and diffusion (a more complicated partial differential equation than the simple diffusion equation) must be solved. To understand kinetics of heterogeneous reactions, the equations for mass or heat transfer must be solved under other constraints (such as interface equilibrium or reaction), often with very complicated boundary conditions because of many particles. 

The original model regarding surface intermediates is a system of ordinary differential equations. It corresponds to the detailed mechanism under an assumption that the surface diffusion factor can be neglected. Physico-chemical status of the QSSA is based on the presence of the small parameter, i.e. the total amount of the surface active sites is small in comparison with the total amount of gas molecules. Mathematically, the QSSA is a zero-order approximation of the original (singularly perturbed) system of differential equations by the system of the algebraic equations (see in detail Yablonskii et al., 1991). Then, in our analysis... 

Under conditions of constant shear, dy /dt = 0, Eq. (5.67) becomes an ordinary differential equation, which can be solved by separation of variables and integration using the boundary conditions r = tq at t = 0 and r = r at t = t to give the following relation for the shear stress, r, as a function of time... 

Now we turn to the problem of constructing conformally invariant ansatzes that reduce systems of partial differential equations invariant under the group C(1,3) to systems of ordinary differential equations. 

Consequently, to describe all the ansatzes of the form (53),(54) reducing the Yang-Mills equations to a system of ordinary differential equations, one has to construct the general solution of the overdetermined system of partial differential equations (54),(86). Let us emphasize that system (54),(86) is compatible since the ansatzes for the Yang-Mills field ( ) invariant under the three-parameter subgroups of the Poincare group satisfy equations (54),(86) with some specific choice of the functions F, F2, , 7Mv, [35]. 

Then i) is invariant under the scaling corresponding to Eq. 4.19 and c becomes a function of the single variable, r). The diffusion equation becomes an ordinary differential equation (i.e., d — d). 

Chemical processes in condensed media often cannot be reduced to simple mono- and bimolecular reactions simply because chains of reaction take place. Therefore their kinetics is described by a set of ordinary differential equations (2.1.1) which are generally nonlinear due to bimolecular stages. Independent variables n ( ), i = 1,..., s (intermediate reactions products) define a number of equations under study. 

Equation (6.2.21) is an ordinary differential equation having large parameter h r). Its solution under certain standard conditions could be obtained by means of the steepest descent (called also VKB) method. 

Here E and F are initial reactants whereas P and Q are final products, A, B and C are intermediate compounds HBr02, Br and Ce4+. Concentrations nE and nF of the initial reactants are assumed to be constant in an open system under study due to stationary matter source. Under well-stirring condition, the kinetic law of mass action leads to a set of the ordinary differential equations... 

This is a functional equation for the boundary position X and the unknown constant parameter n. Upon substituting Eq. (256) into Eq. (251) an ordinary differential equation is obtained for X(t, n), and a family of curves in the phase plane (X, X) can be obtained. For n sufficiently close to unity two functions in the phase plane can be determined which serve as upper and lower bounds for the trajectories. The choice is guided by reference to the exact solution for the limiting case of constant surface temperature. It is shown that the upper and lower bounds are quite close to the one-parameter phase plane solution, although no comparison is made with a direct numerical solution. The one-parameter solution also agrees well with experiments on the solidification of aluminum under conditions of low surface heat transfer coefficient (hi = 0.02 cm.-1). 

In order to complete the solution of the problem of gas motion under the action of a short impulse, we must not only find the exponents and dimensionless functions, which is accomplished by integrating the ordinary differential equations. We must also determine the numerical coefficients A and B in the formulas. 

The group structure under a change of A forces Y(A) to change according to a first order ordinary differential equation. To show this we write... 

There are a variety of special methods used to solve ordinary differential equations. It was Sophus Lie (1842-1899) in the nineteenth century who showed that all the methods are special cases of integration procedures which are based on the invariance of a differential equation under a continuous group of symmetries. These groups became known as Lie groups.2 A symmetry group... 

The problem under consideration is shown schematically in Figure 9.9. The governing equation for this problem was given previously in Equation (9.8), where the species fluxes, are determined from Equation (9.7). Equation (9.7), for i = 1 to J — 1, provides a linear set of independent ordinary differential equations. To solve for all J components, an additional equation is needed. For this we use the conservation of species ... 

To analyze this phenomenon further, 2D numerical simulations of (49) and (50) were performed using a central finite difference approximation of the spatial derivatives and a fourth order Runge-Kutta integration of the resulting ordinary differential equations in time. Details of the simulation technique can be found in [114, 119]. The material parameters of the polymer blend PDMS/PEMS were used and the spatial scale = (K/ b )ll2 and time scale r = 2/D were established from the experimental measurements of the structure factor evolution under a homogeneous temperature quench. 

The first term in the evolution operator has the form of a Poisson bracket and evolution under this part of the operator can be expressed in terms of characteristics. The corresponding set of ordinary differential equations is... 

The ordinary differential equations describing a steady-state adiabatic PFR can be written with axial length z as the independent variable. Alternatively the weight of catalyst w can be used as the independent variable. There are three equations a component balance on the product C, an energy balance, and a pressure drop equation based on the Ergun equation. These equations describe how the molar flowrate of component C, temperature T, and the pressure P change down the length of the reactor. Under steady-state conditions, the temperature of the gas and the solid catalyst are equal. This may or may not be true dynamically ... 

Effect Of Number Of Lumps If the number of plotting points in Aspen Plus is set at 10 (the default), the resulting exit temperature from the reactors under steady-state conditions in Aspen Dynamics is 578 K. Remember that it should be 583 K from the rigorous integration of the ordinary differential equations describing the steady-state tubular reactor that are used in Aspen Plus. Changing the number of points to 20 produces an exit temperature of 580 K. Changing the number of points to 50 produces an exit temperature of 582 K, which is very close to the correct value. Therefore a 50-lump model should be used. 

Formulating appropriate rate laws for CO adsorption, OH adsorption and the reaction between these two surface species, a set of four coupled ordinary differential equations is obtained, whereby the dependent variables are the average coverages of CO and OH, the concentration of CO in the reaction plane and the electrode potential. In accordance with the experiments, the model describes the S-shaped I/U curve and thus also bistability under potentiostatic control. However, neither oscillatory behavior is found for realistic parameter values (see the discussion above) nor can the nearly current-independent, fluctuating potential be reproduced, which is observed for slow galvanodynamic sweeps (c.f. Fig. 30b). As we shall discuss in Section 4.2.2, this feature might again be the result of a spatial instability. 

All modem olefins plants now under design and construction use indirectly fired tubular pyrolysis reactors. Although this type of reactor is presently undergoing several challenges (1,2), it should continue to hold a dominant position for many years to come. A tubular reactor may be simulated by a set of ordinary differential equations (3). Reaction... 

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