Differential-recurrence equation

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]. 

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... 

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)) )]. 

We also remark that Eq. (5.44) may be decomposed into separate sets of equations for the odd and even ap(t) which are decoupled from each other. Essentially similar differential recurrence relations for a variety of relaxation problems may be derived as described in Refs. 4, 36, and 73-76, where the frequency response and correlation times were determined exactly using scalar or matrix continued fraction methods. Our purpose now is to demonstrate how such differential recurrence relations may be used to calculate mean first passage times by referring to the particular case of Eq. (5.44). 

The numerical model-Simulator NV-Simulator V. At this point, we must find the more suitable variant for passing from the differential or partly differential model equations to the numerical state. For the case of the monodimensional model, we can select the simplest numerical method - the Euler method. In order to have a stable integration, an acceptable value of the integration time increment is recommended. In a general case, a differential equations system given by relations (3.55)-(3.56) accepts a simple numerical integration expressed by the recurrent relations (3.57) ... 

The result given here may also be extended to rotation in space. As far as the cage motion is concerned, the complex susceptibility will still be governed by the Rocard equation because the equations of motion factorize. However, the solution for the dipole correlation function is much more complicated because of the difficulty of handling differential recurrence relations pertaining to rotation in space in the presence of a potential. 

The coefficients of these series will be found from the system of recurrence equations that are derived in the standard way. To this end, the series (4.1) are substituted into Eqs. (4.0) and the expressions for like powers of e are equated. Ordinary differential equations and corresponding initial conditions are obtained. 

Using methods parallel to those of the previous section, the recurrence equations and the corresponding roots for the modified Euler, Adams, and Adams-Moulton methods can be derived [9]. For the differential equation (5.170), these are ... 

The partial differential equations used to model the dynamic behavior of physicochemical processes often exhibit complicated, non-recurrent dynamic behavior. Simple simulation is often not capable of correlating and interpreting such results. We present two illustrative cases in which the computation of unstable, saddle-type solutions and their stable and unstable manifolds is critical to the understanding of the system dynamics. Implementation characteristics of algorithms that perform such computations are also discussed. 

We next derive some recurrence relations for the Hermite polynomials. If we differentiate equation (D.l) with respect to s, we obtain... 

To find the differential equation that is satisfied by the Hermite polynomials, we first differentiate the second recurrence relation (D.6) and then substitute (D.6) with n replaeed by n — 1 to eliminate the first derivative of i ( )... 

A third recurrence relation may he obtained by differentiating equation (E.4) to give... 

Recurrence Relations for the Legendre Polynomials. If wc differentiate both sides of equation (13.2) with respect to k we Slave... 

Recurrence Relations for the Bessel coefficients. If we differentiate the generating equation (25.0) with respect to x obtain the relation I / 1 f. / 1... 

Recurrence formulae for the Hermite polynomials follow directly from the defining relation (38.1). If we differentiate both sides of that equation with respect to we obtain the relation... 

We now ask the system to solve (D2) for Y as a function of X using the ODE command The general solution with the two integration constants, %K1 and %K2 is given in (D3) in about two CPU seconds The program can also find powerseries solutions for some differential equations when it can solve the recurrence relation ... 

We therefore advise that the reader should consult a recent series of papers published by Galvez et al. [171, 172] encompassing all the mechanisms mentioned in Sect. 7.1, elaborated for both d.c. and pulse polarography. The principles of the Galvez method are clearly outlined in the first part of the series [171]. It is similar to the dimensionless parameter method of Koutecky [161], which enables the series solutions for the auxiliary concentration functions cP and cQ exp (kt) and

combined directly with the partial differential equations of the type of eqn. (203). In some of the treatments, the sphericity of the DME is also accounted for. The results are usually visualized by means of predicted polarograms, some examples of which are reproduced in Fig. 38. Naturally, the numerical description of the surface concentrations at fixed potential are also immediately available, in terms of the postulated power series, and the recurrent relationships obtained for the coefficients of these series. 

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

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