Derivation of Green’s function using addition theorem

Chen, Jeng‐Tzong; Chou, K. H.; Kao, Shing‐Kai

doi:10.1016/j.mechrescom.2008.10.001

Cited by 15 publications

(15 citation statements)

References 10 publications

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“…To our knowledge, the GMSV has not been applied to compute Green functions for Laplacian boundary value problems in three-dimensional domains with disconnected spherical boundaries (however, see [66] for the planar case). The Green function is harmonic everywhere in a given domain except for a fixed singularity point, and satisfies imposed homogeneous boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

Grebenkov

Traytak

2019

Journal of Computational Physics

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We apply the generalized method of separation of variables (GMSV) to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (i.e., an arbitrary configuration of non-overlapping partially reactive spherical sinks or obstacles). We consider both exterior and interior problems and all most common boundary conditions: Dirichlet, Neumann, Robin, and conjugate one. Using the translational addition theorems for solid harmonics to switch between the local spherical coordinates, we obtain a semi-analytical expression of the Green function as a linear combination of partial solutions whose coefficients are fixed by boundary conditions. Although the numerical computation of the coefficients involves series truncation and matrix inversion, the use of the solid harmonics as basis functions naturally adapted to the intrinsic symmetries of the problem makes the GMSV particularly efficient, especially for exterior problems. The obtained Green function is the key ingredient to solve boundary value problems and to determine various characteristics of stationary diffusion such as reaction rate, escape probability, harmonic measure, residence time, and mean first passage time, to name but a few. The relevant aspects of the numerical implementation and potential applications in chemical physics, heat transfer, electrostatics, and hydrodynamics are discussed.

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Section: Introductionmentioning

confidence: 99%

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

Grebenkov

Traytak

2019

Journal of Computational Physics

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“…In turn, as the particle starts the search at the jth stage on the surface of the (j − 1)th target, the hindering effect might be notable at short times. While exact solutions are not available in such geometric configurations, asymptotic methods [27,63,82] and semi-analytical approaches [83][84][85][86][87] can be applied.…”

Section: Discussionmentioning

confidence: 99%

A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes

Grebenkov,

Metzler,

Oshanin

2021

Preprint

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We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g., a signal transduction proceeds via a series of consecutive "messengers": the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two-and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.

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“…Then, all the unknown boundary data can be determined and the potential is obtained by substituting the boundary data into Eq. (15) or (19). Based on the null-field integral equation approach, successful applications to Laplace [10,20], Helmholtz [12,13], biharmonic [21], and bi-Helmholtz [22] problems have been done.…”

Section: ∂U(s) ∂Nsmentioning

confidence: 99%

“…Not only perfect but also imperfect interface problems were addressed. Chen et al [19] have also proposed a logical approach to construct the Green's function of Laplace operator by using the addition theorem and the superposition technique. The null-field integral formulation has been successfully used to solve the Laplace [10,20], Helmholtz [11][12][13], biharmonic [21], and bi-Helmholtz [22] problems.…”

Section: Introductionmentioning

confidence: 99%

Scattering of sound from point sources by multiple circular cylinders using addition theorem and superposition technique

Chen

Lee

Lin

et al. 2011

Numer. Methods Partial Differential Eq.

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In this study, we use the addition theorem and superposition technique to solve the scattering problem with multiple circular cylinders arising from point sound sources. Using the superposition technique, the problem can be decomposed into two individual parts. One is the free-space fundamental solution. The other is a typical boundary value problem (BVP) with specified boundary conditions derived from the addition theorem by translating the fundamental solution. Following the success of null-field boundary integral formulation to solve the typical BVP of the Helmholtz equation with Fourier densities, the second-part solution is easily obtained after collocating the observation point exactly on the real boundary and matching the boundary condition. The total solution is obtained by superimposing the two parts which are the fundamental solution and the semianalytical solution of the Helmholtz problem. An example was demonstrated to validate the present approach. The parameter study of size and spacing between cylinders are addressed. The results are well compared with the available theoretical solutions and experimental data.

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Derivation of Green’s function using addition theorem

Cited by 15 publications

References 10 publications

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes

Scattering of sound from point sources by multiple circular cylinders using addition theorem and superposition technique

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