Green’s function boundary conditions

First-Principles Green s Function Boundary Condition Method... 

FP-GFBC First-principle Green s function boundary condition... 

A, 77,231 (1998). Green s Function Boundary Conditions in Two-Dimensional and Three-Dimensional Atomistic Simulations of Dislocations. 

Fig. 2. Schematic representation of the domain decomposition scheme used to implement flexible Green s function boundary conditions in our GFBC/MGPT atomistic simulation code for dislocation calculations, (a) The three main computational regions separated into a layered

By using the method of the dyadic Green s function [4] and the adequate boundary conditions [5], the expressions of the electric field in the zone where the transducer is placed can be written as [2]... 

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... 

As we already know a determination of the function G q, p) satisfying all these conditions represents a solution of the boundary value problem and in accordance with the theorem of uniqueness these conditions uniquely define the function G q, p). In general, a solution of this problem is a complicated task, but there are exceptions, including the important case of the plane surface Sq, when it is very simple to find the Green s function. Let us introduce the point s, which is the mirror reflection of the point p with respect to the plane of the earth s surface, Fig. 1.10, and consider the function G (p, q,. s) equal to... 

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

Finally, for completeness, the Green s function corresponding to a pair of reactants initially formed with separation r0 and subjected to the partially reflecting boundary condition, is quoted (Pagistas and Kapral [37], Naqvi et al. [38]. 

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

So far, the Lagrangian density for a homogenous problem (no sink or source term in the diffusion equation) has been considered, subject to the requirement that the approximate trial function, ip, can be forced to satisfy the boundary conditions. In this sub-section, these limitations are removed and the Lagrangian density for the Green s function developed. The Green s functions for the forward and backward time process satisfy the equations... 

The terms in delta functions will be integrated to get the invariant 5 , which is still space- and time-independent. Hence, these two Green s functions can be combined as they can have the same value at the moment of particle creation (t = 0) and at this time only these terms are non-zero. This expression needs a little further manipulation. Following Lebedev et al. [506] on the variational principle, drop the term V ZJe- 17 VGG and include a new term of the form V (Ae fJUG G — e u4>G — e u4tG )r where f is a unit radial vector and

scalar functions of r which are related directly to the boundary conditions which both G and G satisfy. Multiply the delta function term by 2. 

This is a rather large expression However, because 5G and 5G can be made arbitrarily small, all the first few expressions in the square brackets are zero. The first two brackets are just eqns. (260b) and (260a), respectively. The second two are the boundary conditions on G and G (identical) with A effectively kuct/4nR2 if reaction is to be represented by a boundary condition and 0 = 0. The outer boundary condition is obtained by letting 0 = 0 or for the Green s function or homogenous problem, respectively, with A = 0. Finally, the last term contains (G 6G — G<5G )" , which is zero, since G or G, SG or 5G are zero, respectively at + co and... 

The equation should be compared with eqn, (318). When the boundary and initial conditions which the Green s function has to satisfy are chosen to be the same as those satisfied by e p, the problem reduces to finding the Green s function and evaluating the third term on the right-hand side of eqn. (320). 

The difference between eqn. (330) and eqn. (338) is that y andy2 in eqn. (338) are forced to satisfy the boundary conditions prior to using these independent solutions, while in eqn. (330), the independent solutions do not satisfy the boundary conditions. The form of eqns. s33) and (337) are very similar and shows that the Green s function method and the variation of parameters method of solution are equivalent. 

G(r, — tlrls — fI) = 0, since — t +> < — tx]. This first term is zero. The second is also zero because both the Green s functions satisfy the same boundary conditions. Hence, the reciprocity relationship (339) is proved. 

Summations over point-, line-, or planar-source solutions are useful examples of the general method of Green s functions. 6 For instance, the boundary and initial conditions for a triangular source are... 

Such purely mathematical problems as the existence and uniqueness of solutions of parabolic partial differential equations subject to free boundary conditions will not be discussed. These questions have been fully answered in recent years by the contributions of Evans (E2), Friedman (Fo, F6, F7), Kyner (K8, K9), Miranker (M8), Miranker and Keller (M9), Rubinstein (R7, R8, R9), Sestini (S5), and others, principally by application of fixed-point theorems and Green s function techniques. Readers concerned with these aspects should consult these authors for further references. 

The Usadel equation is a nonlinear equation for the 4x4 matrix Green s function [15]. The Usadel equation is complemented by the boundary conditions at the S/F interface [12]... 

E.G. Karpov et al A Green s function approach to deriving non-reflecting boundary conditions in molecular dynamics simulations. Int. J. Num. Meth. Eng 62, 1250-1262 (2005)... 

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