Poisson equations transformation

The above procedure guaranties that all coefficients are smooth functions of the wave-vectors. Since the field B satisfies an anisotropic Poisson equation, transformed to Fourier space, its long-range character is evident. In nematics its effect turns out to enhance transverse modulations in distinct contrast to isotropic fluids in most cases. As will become clear below, the mean flow contributions can be neglected in the immediate vicinity of threshold. 

If a gaussian function is chosen for the charge spread function, and the Poisson equation is solved by Fourier transformation (valid for periodic... 

The Poisson equation can also be solved using the discrete sine transform (Zenger and Bader 2004). The discrete sine transform of a two-dimensional function f(x, y) defined on... 

Thus, the Poisson equation may be solved by applying the discrete sine transform, multiplying the result by... 

Let us consider the nonlocal Poisson equation V(sW) = -4irp in the uniform space. The singular boundary condition on the surface of the solute cavity is neglected. Note that this condition furnishes the mechanisms of the excluded volume effect. The solute is charged and spherical, i.e. p(r) = p(R). The solution T (A ) is obtained by using Fourier transform [6,16] it is valid outside the cavity (R > a),... 

Evaluate the Poisson equation for the potential of Eq. (5.37) to determine the charge density implied. Hints relationships developed in Arfken (1985) are helpful. Additionally, the theta-function transformation (Ziman, 1972)... 

C. Fast Fourier Transform Methods for Poisson Equations... 

Special linear systems arise from the Poisson equation, d uldx + d uldy = f x, y) on a rectangle, 0 Laplace equation of Section II.A is a special case where fix, y) = 0.] If finite differences with N points per variable replace the partial derivatives, the resulting linear system has equations. Such systems can be solved in 0(N log N) flops with small overhead by special methods using fast Fourier transform (FFT) versus an order of AC flops, which would be required by Gaussian elimination for that special system. Storage space also decreases from 2N to units. Similar saving of time and space from O(N ) flops, 2N space units to 0(N log N) flops and space units is due to the application of FFT to the solution of Poisson equations on a three-dimensional box. 

The interface position is tracked by introducing a curvilinear interface-fitted non-orthogonal coordinate system. By means of a coordinate transformation, the physical domain is converted to a computational domain with known boundaries that are coordinate isoUnes. A boundary-fitted grid is generated around the deformed interface at each iteration by a pair of Poisson equations associated with spacing control functions. 

The Coulomb contribution to this Fourier transform is as obtained from the Poisson equation. If one introduces the effects of the... 

From the form of equation (2) one might expect similar difficulties in calculating potentials numerically however, the integral can be transformed into a Poisson equation, ... 

Sadygov R G and Yarkony D R 1998 On the adiabatic to diabatic states transformation in the presence of a conical intersection a most diabatic basis from the solution to a Poisson s equation. I J. Chem. Rhys. 109 20... 

From Poisson s equation (265), we get for the Fourier transform of the potential ... 

Poisson s equation (6.9) can be solved directly by writing the potential, K(r), in terms of its Fourier transform, K(q), that is... 

Solve the Poisson-Boltzmann equation for a spherically symmetric double layer surrounding a particle of radius Rs to obtain Equation (38) for the potential distribution in the double layer. Note that the required boundary conditions in this case are at r = Rs, and p - 0 as r -> oo. (Hint Transform p(r) to a new function y(r) = r J/(r) before solving the LPB equation.)... 

The previous section used the constant strain three-noded element to solve Poisson s equation with steady-state as well as transient terms. The same problems, as well as any field problems such as stress-strain and the flow momentum balance, can be formulated using isoparametric elements. With this type of element, the same (as the name suggests) shape functions used to represent the field variables are used to interpolate between the nodal coordinates and to transform from the xy coordinate system to a local element coordinate system. The first step is to discretize the domain presented in Fig. 9.12 using the isoparametric quadrilateral elements as shown in Fig. 9.15. 

The macroscopic system under consideration is regarded as being composed of np 1023 particles. The state variables describing it are (26). Their time evolution is governed by (72) extended in an obvious way from one to np particles. If we pass from (26) to the state variable (27), the time evolution (72) transforms into the Liouville equation corresponding to (72), i.e., to Equation (74) with n replaced by (27) and the Poisson bracket that extends (75) in an obvious way to np particles (i.e.,... 

Batch Polymerization. The case of stepwise polymerization without termination was originally treated by Dostal and Mark (16) by using the Eigenzeit transformation to linearize the equations, as discussed earlier. Gee and Melville (21) extended this by the same technique to a case where the propagation rate constant varied with molecular size, contrary to the usual assumption. In the case of stepwise polymerization without termination, batch reactions can give a very narrow (Poisson) distribution. Abraham (2) and Kilkson (35) both showed that the use of the Z-transform simplified the handling of this type of mechanism. 

This transformed the nonlinear Poisson-Boltzmann equation into the linear Helmholtz-type equation... 

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