Explicit integration

Fig. 7. Writhe distributions for closed circular DNA as obtained by LI (see Section 4.1) versus explicit integration of the Langevin equations. Data are from [36].

The solution Xh(t) of the linearized equations of motion can be solved by standard NM techniques or, alternatively, by explicit integration. We have experimented with both and found the second approach to be far more efficient and to work equally well. Its handling of the random force discretization is also more straightforward (see below). For completeness, we describe both approaches here. 

Replace the holdup derivatives in Eqs. (13-149) to (13-151) by total-stage material-balance equations (e.g., dMj/dt = Vj + i + Ej- — Vj — Lj) and solve the resulting equations one at a time by the predictor step of an explicit integration method for a time increment that is determined by stability and truncation considerations. If the mole fraclions for a particular stage do not sum to 1, normalize them. 

From Table 13-31, a total of 394 time increments were necessary to distill all hut 22.08 Ih-mol of the initial charge of 99.74 Ih-mol following the establishment of total-reflux conditions. If this problem had to he solved by an explicit integrator, approximately 25,000 time increments would have been necessary. 

Through an explicit integration by parts, we can show that... 

By contrast, a numerical computer program for solving such integration problems would depend on approximating the mathematical expression by a series of algebraic equations over explicit integration limits. 

To date, there has been no explicit solution for this problem for p > 3, since the surface concentrations of electroactive species O and R are time dependent and therefore the Superposition Principle cannot be applied (see also Sect. 4.3) [1,5]. In these conditions, a non-explicit integral solution has been deduced using the Laplace transform method (see Appendix H). 

Finally, by introducing the expressions for and in the Nemst condition (H.37), a non-explicit integral expression for the CV current can be deduced. 

Let us start with some simple considerations and three explicitly integrated examples. We can immediately give two simple mechanisms for the generation of slow relaxations. It is a delay of the motion near the unstable fixed point and (for the systems that are dependent of the parameter) a delay of the motion in the region, where at small variations of the parameter there appears a fixed point. 

Graph the wavefunction T = A= (i//1 + r/r3), where rjr 1 and 3 are the first and third stationary states for the particle in a box (Equation 6.20), and verify that it satisfies Equation 6.4 (the definite integral needed to verify this can be found in Appendix B). Without doing any explicit integrals, determine (x) and (p). 

This method often requires very small integration step sizes to obtain a desired level of accuracy. Runge-Kutta integration has a higher level of accuracy than Euler. It is also an explicit integration technique, since the state values at the next time step are only a function of the previous time step. Implicit methods have state variable values that are a function of both the beginning and end of the current... 

In the case of a weak coupling, a better way to go is to construct an effective theory which, on the one hand, correctly describes the physics at non-relativistic scales, and on the other hand does not have relativistic degrees of freedom [12]. The price to pay for explicitly integrating out relativistic degrees of freedom is that the new Hamiltonian is an infinite sum of operators and is not renormaliz-able. These are, however, minor problems compensated by significant technical and conceptual advantages. 

For the deactivation process studied,the time step At=30 min utes has been found to be satisfactory for both the one-and two-dimensional models. The axial step size Az=0.02m has been used for both models and 10 grid points in the radial direction have been adapted for the two-dimensional descriptions. To improve the order of approximation of the explicit integration process,the nonlinear reaction rate term has been evaluated at the (i+1/2)level. The values of concentration and temperature at (i+1/2) have been determined by extrapolation from the profiles i-1 and i. 

Explicit integration of Eqn. (40) is intractable due to the form of the limit state function which defines the integration domain ([Pg.380]

Higinio Ramos, A non-standard explicit integration scheme for initial-value problems. Applied Mathematics and Computation, 2007, 189, 710-718. 

A. Askar and A.S. Cakmak, Explicit integration method for the time-dependent Schrodinger equation for collision problems, J. Chem. Phys., 68 (1978) 2794. 

From the last formula we find at once the explicit integral expression for anomalous square slowness ... 

Integration into the business. Explicit integration of EHS standards into existing business practices, including new product development, facility, procurement, key suppliers, property transactions, etc. 

Molecular dynamics examines the temporal evolution of a collection of atoms on the basis of an explicit integration of the equations of motion. From the point of view of diffusion, this poses grave problems. The time step demanded in the consideration of atomic motions in solids is dictated by the periods associated with lattice vibrations. Recall our analysis from chap. 5 in which we found that a typical period for such vibrations is smaller than a picosecond. Hence, without recourse to clever acceleration schemes, explicit integration of the equations of motion demands time steps yet smaller than these vibrational periods. 

The impact parameter can be estimated simply from the following consideration the transverse electrostatic pulse contains frequencies of the order of v/b, where v is the velocity of the fast electron. Thus a core electron bound with an energy AE can only be promoted to empty states above the Fermi level in the conduction band if rfundamental property of the radiation can also be seen by explicitly integrating the resultant electric field at an atomic site a distance b away (as given in eq. (3.12)), over the momentum space d q giving ... 

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