Recurrence equations

The same recurrence equation is obtained by solving successive terms of the Smoluchowski equation with 2 S C = Hx equal to //(1 + ft) and 1 for models 1 and 2, respectively. 1... 

By working with orthogonal building blocks [28], it is also possible to go back to only one recurrent equation for the expectation value as for the norm this will be the case in section 3. 

When n = 1/2, the result is Ji/2(x) = (2/jrx)l/2sin x. From the recurrence relations, it can be found that J.1/2 = (2/jtx),/2cos x. The recurrence relation Jn+1 = (2n/x)J - Jn j can the be employed to discover all the other functions of half-integral index. Numerical calculations using recurrence equations are easily impaired by roundoff error, since the error can propagate through successive recurrences. 

The recurrent equation for the total number of extreme units is given by ... 

Fig. 1. General solution of Ibe Sinith-Evrart recurrence equations. N i Ie water phase tertninaiion (adapted Irom UgeUtad et at., 1967).

Lukovits and Janezic182 generated the number of Kekule structures of (1,1) armchair-type carbon nanotubes using the following recurrence equation 184... 

If x —> 0, then p(t = 1 /2, and we have the usual random walk process (i.e., initial effects are ignored). If x 0, then we have a process with memory, i.e. the particle remembers its state at the previous step (the position and direction of motion). Stability embodies the fact that po differs but little from 1. Equations (552) and (553) are now substituted into the recurrence equations (551) yielding... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

In order to solve the hierarchy of recurrence equations [Eqs. (295)-(297)], we introduce a supercolumn vector Cn(s) comprising three subvectors ... 

The subvector c]n m (s) has the dimension n+ 1. The three index recurrence equations [Eqs. (295)—(297)] for h]fk(s) can then be transformed into the matrix three-term differential-recurrence equation... 

The appropriate differential-recurrence equation for the transverse relaxation functions... 

In the transient (step-off) Kerr-effect response, it is also possible to obtain from Eqs. (273)—(275) for l = 2 the system of recurrence equations for the Laplace transforms of the corresponding relaxation functions < " (t) (m = 0,1,2) pertaining to that response, namely,... 

The continued fraction solution of the three-term recurrence equation, Eq. 

One may also readily derive differential-recurrence equations for the statistical moments involving the associated Legendre functions of order 2(1 = 2) pertaining to the dynamic Kerr effect, namely, b (t) [so that = (P2(005 d)) )]. 

Now, e scalar recurrence Eq. (35) can be transformed into the matrix three-term recurrence equation... 

The function 7i,o( ) also can be obtained by using the matrix continued fraction method. The scalar recurrence equation, Eq. (61), can once again be transformed into the matrix three-term recurrence equation... 

In this chapter we shall show how the observed phenomena may be explained by means of elementary catastrophe theory. In principle, the discussion will be confined to examination of non-chemical systems. However, some of the discussed problems, such as a stability of soap films, a phase transition in the liquid-vapour system, diffraction phenomena or even non-linear recurrent equations, are closely related to chemical problems. This topic will be dealt with in some detail in the last section. The discussion of catastrophes (static and dynamic) occurring in chemical systems is postponed to Chapters 5, 6 these will be preceded by Chapter 4, where the elements of chemical kinetics necessary for our purposes will be discussed. 

Sequential models are represented by recurrent equations of the form... 

Many of the considered problems, such as the problem of stability of soap films, the liquid-vapour phase transition, the diffraction phenomena, descriptions of the heartbeat or the nerve impulse transmission, catastrophes described by non-linear recurrent equations have a close relation to chemical problems. 

Schrodinger equation and the corresponding diffraction catastrophes occurring at collisions of molecules are important from the viewpoint of the description of chemical reactions taking place upon contact (collision) of molecules. As shown in Section 1.3, recurrent equations appear from a description of the kinetics of chemical reactions. 

The measurements of x, reveal that the observed phenomena can be modelled by means of the recurrent equation... 

The experimental phenomena observed in the Belousov-Zhabotinskii reaction, such a doubling of the oscillation period, chaotic oscillations or alternate periodical and chaotic oscillations, can be modelled still more exactly by the recurrent equation... 

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