Diffusion equation 366 index

It was shown in Chap. 7 that the three-point second spatial derivative on an unequally spaced grid, leading to (8.1) with the coefficients defined in (8.3), can be improved with relatively small effort to an asymmetric four-point, formula, spanning the indices i — 1, i, % f 1, % I 2, with the second derivative referred to the point at index i. The diffusion equation is then semi-discretised to... 

The solution to the fractional diffusion equation is clearly dependent on fluctuations that have occurred in the remote past note the time lag k in the index on the fluctuations and the fact that it can be arbitrarily large. The extent of the influence of these distant fluctuations on the system response is determined by the relative size of the coefficients in the series. Using Stirling s approximation on the gamma functions determines the size of the coefficients in Eq. (25) as the fluctuations recede into the past, that is, as k — oo we obtain... 

Finally, the fractional calculus was used to construct fractional diffusion equations. One such equation, in particular, models the evolution of the Levy nestable probability density describing Levy diffusion, another mechanism for generating anomalous diffusion. It was shown that this probability density satisfies the scaling relation [Eq. (35)] with the Levy index a such that 8 = 1 /a. The dynamics of a Levy diffusion process, using a Langevin equation, were also considered. The probability density for a simple dissipative process being driven by Levy noise is also Levy but with a change in parameters. This is a possible alternate model of the fluctuations in the interbeat intervals for the human heart shown to be Levy stable over a decade ago [25],... 

When this is substituted into the diffusion equations and boundary conditions for bj, the result is obtained that both are the same for all values of the index j, so that, necessarily ... 

If the number of tube segments is large enough Z S> 1, its discrete character can be negleaed. In this case, eqn [59] can be transformed into the diffusion equation for y/ s,t), where 5 = 0... L is a continuous variable measuring the distance along the tube, which replaces the index k ... 

For unequal time intervals, we look at the example transformed diffusion equation 5.85. This is to be discretised at intervals of SO and H. We use index j to count 0 intervals and i to count X intervals (of H). 

At any point with index i, that is at X = iH, the diffusion equation (1.1) is discretised on the left-hand side in the Euler manner (Sect. 4.4, or in other words the forward difference formula (3.1)) and on the right-hand side with the central three-point approximation (3.41), giving for the iteration going from time T to the... 

First, it is worthwhile detailing the implementation of BDF itself, ignoring startup for the moment. We choose three-point BDF. Based on (4.28) given on page 68 for ode%, the diffusion equation (8.8) is discretised at index i as... 

The refractive index profile of Gl POFs prepared by the extrusion method can also be predicted prior to the actual fabrication. The diffusion phenomena of dopants in polymers while passing through the diffusion section can be described by the advection-diffusion equation [20]... 

In previous studies, it was presumed that a GI POP fabricated by this process would have a refractive index distribution with a tail at the core/cladding boundary, not as steep as that prepared by the conventional batch process [50]. Therefore, the bandwidth of the GI POF obtained by dopant diffusion coextrusion would be inferior to that obtained by the batch process. However, contrary to this expectation, a GI POF without any tail prepared by the process was reported in 2007 [57]. If low-molecular-weight molecules diffuse in a radial direction, their diffusion can be expressed by Fick s diffusion equation... 

Here, the propagator q(T, s) (where 0 < s < 1 is the index denoting the position along the diblock chain) and its counterpart qf(r, s) are given by the modified diffusion equations ... 

The spectrophotometer measures the transmission and, if an absorption measurement is carried out, converts the transmission into absorbance using these equations. This conversion works fine for samples where there is no reflection, either specular or diffuse, as is the case for nonturbid solutions. However, for films there is invariably some reflection, which is often quite large, particularly for films of high dielectric constant (or refractive index) materials, such as PbS and PbSe. Additionally, if the films are not completely transparent, then scattering introduces an extra element of reflection. Therefore, to measure the real absorption of a film, a reflection measurement must also be carried out and correction for this reflection made. The correction will be approximate and depends on the nature of the film itself. However, that most commonly used is... 

We wish to see what the overall conversion of a continuous mixture will be, but, first, we have to ask which parameters will depend on jc, the index variable of the continuous mixture. Clearly k the rate constant in the Damkohler number will be a function of jc, and, if monotonic, can be put equal to Da.x. The parameter /3 is clearly hydrodynamic and so, for the most part, are the terms in the Davidson number. The only term in the equation 6.21 of Davidson and Harrison that might depend on x is the gas phase diffusivity,... 

In Equation (9.6), x is the direction of flux, nt [mol m-3 s 1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m-2 s 1] is the mole flux due to molecular diffusion, D k [m2 s 1] is the effective Knudsen diffusion coefficient, D [m2 s 1] is the effective bimolecular diffusion coefficient (D = Aye/r), e is the porosity of the electrode, r is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983 Warren, 1969). This model modifies Equation (9.6) as ... 

The results of estimation of coefficient of self-diffusion due to simulation for macromolecules with different lengths are shown in Fig. 12. The introduction of local anisotropy practically does not affect the coefficient of diffusion below the transition point M, the position of which depends on the coefficient of local anisotropy. For strongly entangled systems (M > M ), the value of the index —2 in the reptation law is connected only with the fact of confinement of macromolecule, and does not depend on the value of the coefficient of local anisotropy. At the particular value ae = 0.3, the simulation reproduces the results of the conventional reptation-tube model (see equation (5.21)) and corresponds to the typical empirical situation (M = 10Me). 

The multi-mode model for a tubular reactor, even in its simplest form (steady state, Pet 1), is an index-infinity differential algebraic system. The local equation of the multi-mode model, which captures the reaction-diffusion phenomena at the local scale, is algebraic in nature, and produces multiple solutions in the presence of autocatalysis, which, in turn, generates multiplicity in the solution of the global evolution equation. We illustrate this feature of the multi-mode models by considering the example of an adiabatic (a = 0) tubular reactor under steady-state operation. We consider the simple case of a non-isothermal first order reaction... 

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