Time-dependent equation initial conditions

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

The transient heat equation (Eq. 3.285) often serves as the model for parabolic equations. Here the solution depends on initial conditions, meaning a complete description of T(0, x) for the entire spatial domain at t =0. Furthermore the solution T(t,x) at any spatial position x and time t depends on boundary conditions up to the time t. The shading in Fig. 3.14 indicates the domain of influence for the solution at a point (indicated by the dot). 

The equilibrium initial conditions are determined by another variational principle on the free energy functional, and lead to corresponding appropriate equations which are not presented here. They serve as the given initial conditions for the time-dependent equations given above. The various equations have the same form as above except that all the B s are replaced by their free... 

Equation 11.1-2 is an ordinary first-order differential equation. Before it can be solved to yield an expression for M t), a boundary condition must be provided—a specified value of the dependent variable (At) at some value of the independent variable (/), Frequently, the value of Af at time / = 0 (an initial condition ) is specified. The complete balance equation would be... 

One way of quantifying the sensitive dependence on initial conditions in a nonlinear dynamics model is via Lyapunov exponents. Usually this is done by introducing the variational equations which describe the time-dependent variation of perturbations of a solution of a dynamical system. Besides giving us a means to verify the chaotic nature of given system, the variational equations prove useful in describing the symplectic structure (taken up in the next chapter) which is essential to the design of effective numerical methods for molecular dynamics. 

Vibrational motion is thus an important primary step in a general reaction mechanism and detailed investigation of this motion is of utmost relevance for our understanding of the dynamics of chemical reactions. In classical mechanics, vibrational motion is described by the time evolution and l t) of general internal position and momentum coordinates. These time dependent fiinctions are solutions of the classical equations of motion, e.g. Newton s equations for given initial conditions and I Iq) = Pq. 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

This is a second-order ODE with independent variable z and dependent variable k C t,z), which is a function of z and of the transform parameter k. The term C(t, 0) is the initial condition and is zero for an initially relaxed system. There are two spatial boundary conditions. These are the Danckwerts conditions of Section 9.3.1. The form appropriate to the inlet of an unsteady system is a generalization of Equation (9.16) to include time dependency ... 

Time dependent integration of the evolution equations (8) with an initial condition given by... 

Equation (4.a) states that the wave function must obey the time-dependent Schrodinger equation with initial condition /(t = 0) = < ),. Equation (4.b) states that the undetermined Lagrange multiplier, x t), must obey the time-dependent Schrodinger equation with the boundary condition that x(T) = ( /(T))<1> at the end of the pulse, that is at f = T. As this boundary condition is given at the end of the pulse, we must integrate the Schrodinger equation backward in time to find X(f). The final of the three equations, Eq. (4.c), is really an equation for the time-dependent electric field, e(f). 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

The time dependence of the various species concentrations will depend on the relative magnitudes of the four rate constants. In some cases the curves will involve a simple exponential rise to an asymptote, as is the case for irreversible reactions. In other cases the possibility of overshoot (exists, as indicated in Figure 5.2. Whether or not this phenomenon will occur depends on the relative magnitudes of the rate constants and the initial conditions. However, the fact that both roots of equation 5.2.34 must be real requires that there be only one maximum in the curve for R(t) or S(t). 

In general, full time-dependent analytical solutions to differential equation-based models of the above mechanisms have not been obtained for nonlinear isotherms. Only for reaction kinetics with the constant separation factor isotherm has a full solution been found [Thomas, J. Amer. Chem. Soc., 66, 1664 (1944)]. Referred to as the Thomas solution, it has been extensively studied [Amundson, J. Phys. Colloid Chem., 54, 812 (1950) Hiester and Vermeulen, Chem. Eng. Progress, 48,505 (1952) Gilliland and Baddour, Ind. Eng. Chem., 45, 330 (1953) Vermeulen, Adv. in Chem. Eng., 2,147 (1958)]. The solution to Eq. (16-130) for item 4C in Table 16-12 for the same initial and boundary conditions as Eq. (16-146) is... 

The decoupled equations can be integrated to any given order. Equation (7) simply confirms that the cj are the time-dependent initial conditions of the problem. The system is assumed to be in an initial well-defined stationary... 

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