Initial value problem, defined

The solution of the initial value problem defined by Eqs. (4.2) is straightforward. For any composition, the right-hand side of Eqs. (4.2) can be computed for the forward and reverse rate coefficients appropriate for the specified temperature. Increments and decrements to the starting concentrations are computed repeatedly by the integration algorithm to generate the solution to the initial value problem (Section 5). 

But the major physical problem remained open Could one prove rigorously that the systems studied before 1979—that is, typically, systems of N interacting particles (with N very large)—are intrinsically stochastic systems In order to go around the major difficulty, Prigogine will take as a starting point another property of dynamical systems integrability. A dynamical system defined as the solution of a system of differential equations (such as the Hamilton equations of classical dynamics) is said to be integrable if the initial value problem of these equations admits a unique analytical solution, weekly sensitive to the initial condition. Such systems are mechanically stable. In order to... 

In both the Ito and Stratonovich formulations, the randomness in a set of SDEs is generated by an auxiliary set of statistically independent Wiener processes [12,16]. The solution of an SDE is defined by a hmiting process (which is different in different interpretations) that yields a unique solution to any stochastic initial value problem for each possible reahzation of this underlying set of Wiener processes. A Wiener process W t) is a Gaussian Markov diffusion process for which the change in value W t) — W(t ) between any two times t and t has a mean and variance... 

The dynamic model consists of the three differential equations (7.104), (7.110), and (7.120). These define an initial value problem with initial conditions at t = 0. The dynamic, unsteady state of this system is described by these highly nonlinear DEs, while the steady states are defined by nonlinear transcendental equations, obtained by setting all derivatives in the system of differential equations (7.104), (7.110), and (7.120) equal to zero. 

Now, the averaged hyperbolic model, Eq. (52), defines a characteristic initial-value problem (Cauchy problem). To complete the model, we need to specify Cm only along the characteristic curves x = 0 and f — 0. Thus, the initial and boundary conditions for the averaged model are obtained by taking the mixing-cup averages of Eqs. (31) and (32) ... 

Equation (E3.13) defines an initial-value problem, with initial values at s = 0,... 

Steady state mass or heat transfer in solids and current distribution in electrochemical systems involve solving elliptic partial differential equations. The method of lines has not been used for elliptic partial differential equations to our knowledge. Schiesser and Silebi (1997)[1] added a time derivative to the steady state elliptic partial differential equation and applied finite differences in both x and y directions and then arrived at the steady state solution by waiting for the process to reach steady state. [2] When finite differences are applied only in the x direction, we arrive at a system of second order ordinary differential equations in y. Unfortunately, this is a coupled system of boundary value problems in y (boundary conditions defined at y = 0 and y = 1) and, hence, initial value problem solvers cannot be used to solve these boundary value problems directly. In this chapter, we introduce two methods to solve this system of boundary value problems. Both linear and nonlinear elliptic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear elliptic partial differential equations and numerical solutions for nonlinear elliptic partial differential equations based on method of lines. 

The model (9.73)—(9.75) was presented as an initial value problem We were interested in the rate at which a system in state 0) decays into the continua L and R and have used the steady-state analysis as a trick. The same approach can be more directly applied to genuine steady state processes such as energy resolved (also referred to as continuous wave ) absorption and scattering. Consider, for example, the absorption lineshape problem defined by Fig. 9.4. We may identify state 0) as the photon-dressed ground state, state 1) as a zero-photon excited state and the continua R and L with the radiative and nonradiative decay channels, respectively. The interactions Fyo and correspond to radiative (e.g. dipole) coupling elements between the zero photon excited state 11 and the ground state (or other lower molecular states) dressed by one photon. The radiative quantum yield is given by the flux ratio Yr = Jq r/(Jq r Jq l) = Tis/(Fijj -F F1/,). 

In these "source models," unlike the usual initial value problem, the wave function is specified for all times at one point of coordinate space. A simple, solvable ID case corresponds to switching-on the source suddenly at T = 0, with a well-defined "carrier frequency" S2o > 0,... 

In order to be able to easily compare the properties of different methods in a unified way, we focus in this chapter primarily on a particular class of schemes, generalized one-step methods. Suppose that the system under study has a well defined flow map , defined on the phase space (which is assumed to exclude any singular points of the potential energy function). The solution of the initial value problem, z = /(z), z(0) = may be written z(f,<) (with z(0,( ) = <), and the flow-map satisfies = z(f, 5). A one-step method, starting from a given point, approximates a point on the solution trajectory at a given time h units later. Such a method defines a map % of the phase space as illustrated in Fig. 2.1. 

We saw before that we could think of the simple SDE initial value problem (6.8) as having a solution defined by a certain random process. In the same way we would like to obtain, for the Ornstein-Uhlenbeck SDE (6.19), an explicit stochastic process which is in some sense equivalent to solving the SDE. Multiply both sides of (6.19) by the integrating factor exp(yf), and observe that... 

Chapter 7 Numerical Solution Methods (Initial Value Problems) To account for the second order derivative, we define... 

When this information is available, one has a well-defined stochastic initial-value problem, which one might, in principle, hope to solve by the transport equation. Since the neutron population is here assumed to have spherical symmetry, this is an integro-differential equation in four independent variables. However, a stepwise numerical integration of it is a formidable task, even for modem computing machines. 

The above analyses of species concentrations and net reaction rates clearly indicate which reactions and which chemical species are most important in this reaction mechanism, under the particular conditions considered. However, for purposes of refining a reaction mechanism by eliminating unimportant reactions and species and by improving rate parameter estimates and thermochemical property estimates for the most important reactions and species, it would be helpful to have a quantitative measure of how important each reaction is in determining the concentration of each species. This measure is obtained by sensitivity analysis. In this approach, we define sensitivity coefficients as the partial derivative of each of the concentrations with respect to each of the rate parameters. We can write an initial value problem like that given by equation (35) in the general form... 

This method is very efficient when it is applicable, and is one of the many integral transformations that are useful for solving initial value problems. Integral transforms are certain functions that are defined by an integral, such as... 

An appropriate question is when is this approach mathematically legal The answer is whenever the initial value problem has analytic coefficients and forcing functions. The term analytic can be defined as any function having a Taylor series representation in some open interval about a given point. 

Subsequent II,(a , t) (i > 1) are defined as solutions of the linear initial value problems... 

Initial value problems with ordinary differential equations (ODE) have well-defined conditions (based on Lipschitz continuity of the time derivatives) that guarantee unique solutions. Conditions for unique solutions of DAEs (Equations 14.2 and 14.3) are less well defined. One way to guarantee existence and uniqueness of DAE solutions is to confirm that the DAE can be converted (at least implicitly) to an initial value ODE. A general analysis of these DAE properties can be found in [5] and is beyond the scope of this chapter. On the other hand, for a workable analysis, one needs to ensure a regularity condition on the DAE characterized by its index. 

Another important point, often overlooked in the literature, is that a time-dependent problem in quantum mechanics is mathematically defined as an initial value problem. This stems from the fact that the time-dependent Schrodinger equation is a first-order differential equation in the time coordinate. The wave-function (or the density) thus depends on the initial state, which implies that the Runge-Gross theorem can only hold for a fixed initial state (and that the xc potential depends on that state). In contrast, the static Schrodinger equation is a second order differential equation in the space coordinates, and is the typical example of a boundary value problem. 

To be clear, let us first define the correct solution to the initial-value problem asj (x). The approximative solution using a single-step numerical method, e.g. the Euler method, with an increment function /(x , y ) and step size h is hf Xn,yn)r. and it... 

Reduced systems modelling based on the initial value problem in Eq. (7.88) requires the application of three functions. Function d=g(a) defines the time derivative of a, function Y = h(a) calculates the concentrations from the parameters of the manifold (mapping whilst function a = h(Y) (mapping... 

Stiff initial value problems were first encountered in the study of the motion of springs of varying stiffness, from which the problem derives its name. For linear ordinary differential equations, the stiffness of the system can be defined in terms of the stiffness ratio SR (Finlayson, 1980), given by ... 

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