Poisson’s integral

As was shown in Chapter 1, these conditions uniquely define the function T. For determination of the disturbing potential we will make use of Poisson s integral, described in the Chapter 1, which allows one to find the harmonic function E outside the spherical surface of the radius R, Fig. 2.9b, if this function, E p), is known at points of this surface ... 

The final step is Gegenbauer s generalization of Poisson s integral (Watson,... 

The SSW form an ideal expansion set as their shape is determined by the crystal structure. Hence only a few are required. This expansion can be formulated in both real and reciprocal space, which should make the method applicable to non periodic systems. When formulated in real space all the matrix multiplications and inversions become 0(N). This makes the method comparably fast for cells large than the localisation length of the SSW. In addition once the expansion is made, Poisson s equation can be solved exactly, and the integrals over the intersitital region can be calculated exactly. 

Substituting the expression for the potential If, derived above, we see that it is a solution of Poisson s equation. In other words, the integral... 

The convenience of Eq. (6) is realizable only in the rather unrealistic situation where the charge distribution exhibits cylindrical or spherical symmetry. For storage silos, blenders, fluidized bed reactors, and other real vessel geometries, integral solutions are usually not possible, necessitating an alternate problem formulation. Poisson s equation serves this need, relating the volume charge distribution to the electrostatic potential. 

The lower limit of the integration is determined by the condition (13) and it secures that a self-consistent field can be sought from Poisson s equation,... 

The water thickness measurement uncertainty due to neutron counting statistics can be calculated from Poisson counting statistics. For a random process, the standard deviation, Ah in the observed counts I is A/ = y7. The number of neutrons in the incident, or open beam, I0, is the product of the neutron fluence rate (cur2 s 1), integration time T (s), integration area A (cm2), and neutron detection efficiency, ip... 

Substituting these velocities into the integrated continuity equation results in Poisson s equation for the pressure as the governing equation... 

In engineering applications it is often convenient to obtain integral representations which directly involve the field and its fluxes, rather than equations for single- or double-layer densities. This methodology is commonly called the direct method. For Poisson s equation this can be done using the Green s identities for scalar fields. As we already know, Poisson s equation is widely used in transport phenomena and polymer processing, and it is defined as,... 

We can now generate an equivalent integral formulation for Poisson s equation... 

The integral formulation for Poisson s equation is found the same way as for Laplace s equation (using Green s second identity, Theorem (10.1.3)), except that now the second volume integral is kept in Green s second identity. For a point xq V the integral formulation... 

In terms of heat transfer this can be physically interpreted that, if the infinite heat source is at the boundary (the infinite character is given by the delta function), then for a smooth surface only half of the delta function must be included. The general integral representation of Poisson s equation becomes,... 

Morgan, J. van W. (1977). Integration of Poisson s equation for a complex system with arbitrary geometry, J. Phys. C10,1181-1202. 

In continuum notation, this relation would constitute one form of Poisson s equation of electrostatics. The continuum forms of E(x) and V (x) are valid if the charge density planes are so close together that over small regions of space the charge density can be viewed as a continuous function p(x) of position x. [The local space charge density p(x) has units of Coulombs m-3]. In such cases, the sums in eqns. (37) and (40) for E (x) and V (x) can be approximated by integrals to give... 

This latter relation constitutes the integral form of Poisson s equation of electrostatics. 

Note that the potential ( I ) decays from its surface value to a value of zero at some point in the bulk. Inserting Eqs. (4) and (7) into Poisson s equation and applying the appropriate boundary conditions ( I = I o al v = 0 and 1 = 0 at x = oo), the resulting equation can be integrated twice to yield... 

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