Ordinary differential equations homogeneous equations

When solving the problem on controlling the chemical reaction, the spatially homogeneous chemical system under isothermal conditions is described by a set of ordinary differential equations. Unlike equation (3.1), in the following system of kinetic equations the control parameters u(t) are separated... 

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

For fast reactions Da becomes large. Based on that assumption and standard correlations for mass transfer inside the micro channels, both the model for the micro-channel reactor and the model for the fixed bed can be reformulated in terms of pseudo-homogeneous reaction kinetics. Finally, the concentration profile along the axial direction can be obtained as the solution of an ordinary differential equation. 

The reactor model adopted for describing the lab-scale experimental setup is an isothermal homogeneous plug-flow model. It is composed of 2NP + 2 ordinary differential equations of the type of Equation 16.11 with the initial condition of Equation 16.12, NP + 3 algebraic equations of the type of Equation 16.13, and the catalytic sites balance (Equation 16.14) ... 

An ordinary differential equation has only one variable. Those with more variables are partial differential equations. In most applications to be considered here the differential equations are of the homogeneous type. This means that if yi(x) and 92(2) are two solutions of the equation... 

DNS of homogeneous turbulence thus involves the solution of a large system of ordinary differential equations (ODEs see (4.3)) that are coupled through the convective and pressure terms (i.e., the terms involving T). 

CONP Kee, R. J., Rupley, F. and Miller, J. A. Sandia National Laboratories, Livermore, CA. A Fortran program (conp.f) that solves the time-dependent kinetics of a homogeneous, constant pressure, adiabatic system. The program runs in conjunction with CHEMKIN and a stiff ordinary differential equation solver such as LSODE (lsode.f, Hindmarsh, A. C. LSODE and LSODI, Two Initial Value Differential Equation Solvers, ACM SIGNUM Newsletter, 15, 4, (1980)). The simplicity of the code is particularly valuable for those not familiar with CHEMKIN. 

Consider the system described by the linear, homogeneous ordinary differential equations... 

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. 

Ordinary differential equations are suitable only for describing homogeneous systems, and we need partial differential equations if the variables depend also on spatial coordinates. The solution of such equations is beyond the scope of this book. 

Many chemical-kinetics problems, such as the homogeneous mass-action kinetics problems discussed in Section 16.1, are easily posed as a system of standard-form ordinary differential equations (ODE),... 

One of the simplest practical examples is the homogeneous nonisothermal and adiabatic continuous stirred tank reactor (CSTR), whose steady state is described by nonlinear transcendental equations and whose unsteady state is described by nonlinear ordinary differential equations. 

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. 

A non-linear mathematical model, which is a set of ordinary differential equations, for the process in the SPBER was developed.19 The model accounts for the heterogeneous electrochemical reaction and homogeneous reaction in the bulk solution. The lateral distributions of potential, current density and concentration in the reactor were studied. The potential distribution in the lateral dimension, x, of the packed bed was described by a one dimensional Poisson equation as ... 

Let us first consider a homogeneous situation in which the transport terms become zero. Then, the temporal evolution of the system is described by a set of ordinary differential equations of the general form... 

We also use a restricted form of Equation 19 for the kinetics studies described previously. Smog chamber analyses uses just the first and last terms so that they depend on ordinary differential equations. These are solutions which describe the time-dependent behavior of a homogeneous gas mixture. We used standard Runge-Kutta techniques to solve them at the outset of the work, but as will be shown here, adaptations of Fade approximants have been used to improve computational efficiency. 

The rate of change of concentration in a homogeneous reaction system can be described by the following system of ordinary differential equations (odes) ... 

Substitution of Eqs. (9) into Eqs. (8) and subsequent differentiation with respect to lead to the equilibrium equations in terms of microstresses and microstrains (i.e. strains averaged across the layer thickness). To exclude the latter, constitutive equations for the damaged layer and the outer sublaminate, equations of the global equilibrium of the laminate as well as generalised plane strain conditions are employed. Finally, a system of coupled second order non-homogeneous ordinary differential equations is obtained... 

To locate the marginal state, expressions such as Eq. (11) are substituted into Eqs. (8-10). The real part of q is then set equal to zero, and the resulting set of linear homogeneous ordinary differential equations is solved, subject to the appropriate boundary conditions which are also generally homogeneous. 

Each of the two enzymes thus behaves as phosphofructokinase in the model considered for glycolytic oscillations (chapter 2). To limit the study to temporal organization phenomena, the system is considered here as spatially homogeneous, as in the case of experiments on glycolytic oscillations (Hess et ai, 1969). In the case where the kinetics of the two enzymes obeys the concerted allosteric model (Monod et al, 1965), the time evolution of the model is governed by the kinetic equations (4.1), which take the form of three nonlinear, ordinary differential equations ... 

The three variables in that model are intracellular ATP (a), intracellular cAMP (j8), and extracellular cAMP (y). We assume that the complex obeys the allosteric model of Monod et al. (1965), i.e. each regulatory or catalytic subunit can exist in two states, one of which has a larger affinity towards its ligand or, in the case of the catalytic subunit, is more active than the other the transition between these two states is concerted. When we etssume, moreover, that the system remains spatially homogeneous (which corresponds to the conditions in continuously stirred cell suspensions), the time evolution of the system is governed by the three ordinary differential equations ... 

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