Oscillatory integrals

NIRVANA Non-oscillatory, Integrally Restricted, Volume-Averaged Numerical Advection... 

In many cases the integral (3.24) may be expressed in terms of oscillatory integrals of a simpler type... 

The solution to the Airy equation may be written in terms of the oscillatory integral (see Section 3.4.5)... 

Let us recall that in the case of computing the oscillatory integrals (3.25) in which the function F has a degenerate critical point, the stationary phase method, described in Section 3.4.2, fails. There are two basic methods of computation of such integrals which will be exemplified by the Airy function. The information provided below is brief and intended to facilitate the reader an access to suitable references (see bibliographical remarks at the end of this chapter). 

The second method consists in writing an oscillatory integral in the form... 

Control parameters are the positions of atomic nuclei R, a function of state is the wave function satisfying the time-dependent Schrodinger equation — an equation of state. The problem of determining the dependence of the integral (6.172) on parameters is related to the problem associated with the investigation of oscillatory integrals occurring in optics and in diffraction problems, see Section 3.4. 

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

On Some Difficulties in Integrating Highly Oscillatory Hamiltonian Systems... 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

This latter modified midpoint method does work well, however, for the long time integration of Hamiltonian systems which are not highly oscillatory. Note that conservation of any other first integral can be enforced in a similar manner. To our knowledge, this method has not been considered in the literature before in the context of Hamiltonian systems, although it is standard among methods for incompressible Navier-Stokes (where its time-reversibility is not an issue, however). 

L. Mandelstam and N. Papalexi performed an interesting experiment of this kind with an electrical oscillatory circuit. If one of the parameters (C or L) is made to oscillate with frequency 2/, the system becomes self-excited with frequency/ this is due to the fact that there are always small residual charges in the condenser, which are sufficient to produce the cumulative phenomenon of self-excitation. It was found that in the case of a linear oscillatory circuit the voltage builds up beyond any limit until the insulation is ultimately punctured if, however, the system is nonlinear, the amplitude reaches a stable stationary value and oscillation acquires a periodic character. In Section 6.23 these two cases are represented by the differential equations (6-126) and (6-127) and the explanation is given in terms of their integration by the stroboscopic method. 

The technique of photoemission electron spectroscopy (PEEM) is a particularly attractive and important one for spatially resolved work function measurements, as both the Kelvin probe technique and UPS are integral methods with very poor ( mm) spatial resolution. The PEEM technique, pioneered in the area of catalysis by Ertl,72-74 Block75 76 and Imbihl,28 has been used successfully to study catalytic oscillatory phenomena on noble metal surfaces.74,75... 

Datura stramonium rheogniometer oscillatory membrane integrity non-growth [31]... 

Integral action is brought in with high Xi values. These are reduced by factors of 2 until the response is oscillatory, and Tj is set at 2 times this value. 

Proportional gain, integral and derivative time constants to PI and PID controllers. Different settings for load and set point changes. Different settings for different definitions of the error integral. The minimum ITAE criterion provides the least oscillatory response. 

The two constants of integration, C and D, are determined as before by the initial conditions. This solution is oscillatory, although the amplitude of the oscillations decreases exponentially in time, as shown in Fig. 2. 

---