Absorbing boundary conditions

Here we are interested in escape out of the domain L specified by a single cycle of the potential that is out of a domain of length n that is the domain of the well. Because the bistable potential of Eq. (5.42) has a maximum at x = n/2 and minima at x = 0, x = 7t, it will be convenient to take our domain as the interval —7t/2 < x < n/2. Thus we will impose absorbing boundaries at x = —n/2, x = n/2. Next we shall impose a second condition that all particles are initially located at the bottom of the potential well so that x0 = 0. The first boundary condition (absorbing barriers at —n/2, n/2) implies that only odd terms in p in the Fourier series will contribute to Y (x). While the second ensures that only the cosine terms in the series will contribute because there is a null set of initial values for the sine terms. Hence... 

Neuhauser D and Baer M 1989 The time dependent Schrodinger equation application of absorbing boundary conditions J. Chem. Phys. 90 4351... 

Mandelshtam V A and Taylor H S 1995 A simple recursion polynomial expansion of the Green s function with absorbing boundary conditions. Application to the reactive scattering J. Chem. Phys. 102... 

Seideman T and Miller W H 1992 Quantum mechanical reaction probabilities via a discrete variable representation-absorbing boundary condition Green function J. Chem. Phys. 97 2499... 

B. Engquist and A. Majda, Absorbing Boundary Conditions for the Numerical Simulation of Waves, Math. Comput. 31, No. 139 (1977). 

Let us discuss the four main types of boundary conditions reflecting, absorbing, periodic, and the so-called natural boundary conditions that are much more widely used than others, especially for computer simulations. 

Absorbing Boundary. The absorbing boundary may be represented as an infinitely deep potential well just behind the boundary. Mathematically, the absorbing boundary condition is written as... 

Natural Boundary Conditions. If the Markov process is considered in infinite interval, then boundary conditions at oo are called natural. There are two possible situations. If the considered potential at +oo or —oo tends to —oo (infinitely deep potential well), then the absorbing boundary should be supposed at Too or —oo, respectively. If, however, the considered potential at Too or — oo tends to Too, then it is natural to suppose the reflecting boundary at Too or —oo, respectively. 

The Transition Probability. Suppose we have a Brownian particle located at an initial instant of time at the point xo, which corresponds to initial delta-shaped probability distribution. It is necessary to find the probability Qc,d(t,xo) = Q(t,xo) of transition of the Brownian particle from the point c 0 Q(t,xo) = W(x, t) dx + Jrf+ X W(x, t) dx. The considered transition probability Q(t,xo) is different from the well-known probability to pass an absorbing boundary. Here we suppose that c and d are arbitrary chosen points of an arbitrary potential profile (x), and boundary conditions at these points may be arbitrary W(c, t) > 0, W(d, t) > 0. 

The Eqs. (10) and (11) functionally connect thep parameter, the initial electron s velocity Vo and velocity s constituents Vqx and Voy in plane-parallel structure with distributed potential with the coordinate of its entry s point xo, on the electrode with distributed potential. Let us analyze Eq. (10) Under pxo < B the electron will lose the initial kinetic energy completely with generation of electromagnetic radiation (the kinetic energy is absorbed completely). Under these conditions the electron doesn t leave the structure. There is the partial selection of energy under px0 > B and the electron comes beyond the limits of structure. If electron enters the structure normally (Voy = Vo, Vox = Vo) the boundary condition after that electron leaves the structure can be written as ... 

Cumulative Reaction Probability via a Discrete Variable Representation with Absorbing Boundary Conditions. 

Recursion Polynomial Expansion of the Green s Function with Absorbing Boundary Conditions. Application to the Reactive Scattering. 

Reaction Probabilities via a Discrete Variable Representation- Absorbing Boundary Condition Green Function. 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

In this model there is no stomach compartment but possible dissolution in the stomach can be partly accounted for by defining the boundary conditions (C and r ) in the beginning of the intestine. If fast dissolution is expected in the stomach, the boundary conditions for C can be set to 1, that is, the luminal concentration when entering the intestine equals the saturation solubility in the intestine. If no dissolution is expected in the stomach, the starting value for C will be 0. C can then have all values between 0 and 1 as the boundary condition and this results in the restriction that super-saturation in the intestine due to fast dissolution and high solubility in the stomach can not be accounted for. Assuming that the difference between the mass into and out of the intestine is equal to the mass absorbed at steady state, the Tabs can be calculated from Eq. 17 [40]... 

This simple form for the burning rate expression is possible because the equations are developed for the conditions in the gas phase and the mass burning rate arises explicitly in the boundary condition to the problem. Since the assumption is made that no radiation is absorbed by the gases, the radiation term appears only in the boundary condition to the problem. 

W.C. Chew, J.M. Jin and E. Michelsen, Complex coordinate stretching as a generalized absorbing boundary condition, Microwave and Opt. Technol. Lett. 15, 363-369 (1997). 

The diffusive random walk of the Helfand moment is mled by a diffusion equation. If the phase-space region is defined by requiring Ga(t) < x/2, the escape rate can be computed as the leading eigenvalue of the diffusion equation with these absorbing boundary conditions for the Helfand moment [37, 39] ... 

A realistic boundary condition must account for the solubility of the gas in the mucus layer. Because ambient and most experimental concentrations of pollutant gases are very low, Henry s law (y Hx) can be used to relate the gas- and liquid-phase concentrations of the pollutant gas at equilibrium. Here y is the partial pressure of the pollutant in the gas phase expressed as a mole fraction at a total pressure of 1 atm x is the mole fraction of absorbed gas in the liquid and H is the Henry s law constant. Gases with high solubilities have low H value. When experimental data for solubility in lung fluid are unavailable, the Henry s law constant for the gas in water at 37 C can be used (see Table 7-1). Gas-absorption experiments in airway models lined with water-saturated filter paper gave results for the general sites of uptake of sulfur dioxide... 

Table III shows that the experimental and predicted evaporation rates are in good agreement at all beam intensities. There is some inconsistency at the highest power levels. It was difficult to maintain the droplet in the center of the laser beam at the highest power level, and the measured evaporation rate is somewhat low as a result of that problem. Additional computations demonstrate that the predicted evaporation rate is quite sensitive to the choice of the imaginary component of N, so the results suggest that this evaporation method is suitable for the determination of the complex refractive index of weakly absorbing liquids. For strong absorbers, the linearizations of the Clausius-Clapeyron equation and of the radiation energy loss term in the interfacial boundary condition may not be valid. In this event, a numerical solution of the governing equations is required. The structure of the source function, however, makes this a rather tedious task.

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

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