<i>Theory of Functions of a Complex Variable</i>

Markushevich, A. I.; Silverman, Richard A.; Gillis, James T.

doi:10.1063/1.3048359

Cited by 409 publications

(110 citation statements)

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“…By the Riemann mapping theorem [4], there exists a conformal mapping ξ which maps the complement of a closed disk with center at the origin and radius ρ onto the complement of D, satisfying the normalization condition, ξ(w)/w → 1 as |w| → ∞. Then its Laurent expansion at ∞ is given by…”

Section: Faber Polynomial Propagation Schemementioning

confidence: 99%

“…By definition, f ∞ = max z∈D |f (z)|. The maximum principle for analytic functions states [4] that if f is analytic in D and continuous in the closure of D, then |f | cannot attain its maximum at interior points of D. According to (4.2) and the maximum principle for functions analytic in D bounded by Γ, the accuracy ǫ n (Γ) of a polynomial approximation of an analytic function f of a matrix H (in our case, f (z) = exp(−itz)) is…”

Section: Accuracy and Efficiency Assessmentmentioning

confidence: 99%

“…For a sufficiently small time step ∆t, typically, ∆t ∼ H −1 , where H is the (matrix) norm of H, the action of the infinitesimal evolution operator exp(−i∆tH) on the state vector Ψ can be approximated by various means that do not require any direct diagonalization of H. Section 3 is devoted to an algorithm that involves neither a direct diagonalization of H nor many time steps. It is based on the well known approximation of an analytical function by the Faber polynomial series [3] (see also the textbooks [4]). The Faber approximation method has been applied to quantum scattering problems [5] to compute the causal Green's function for the Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

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Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media

Borisov¹,

Shabanov²

2006

Journal of Computational Physics

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Maxwell's equations for propagation of electromagnetic waves in dispersive and absorptive (passive) media are represented in the form of the Schrödinger equation i∂Ψ/∂t = HΨ, where H is a linear differential operator (Hamiltonian) acting on a multi-dimensional vector Ψ composed of the electromagnetic fields and auxiliary matter fields describing the medium response. In this representation, the initial value problem is solved by applying the fundamental solution exp(−itH) to the initial field configuration. The Faber polynomial approximation of the fundamental solution is used to develop a numerical algorithm for propagation of broad band wave packets in passive media. The action of the Hamiltonian on the wave function Ψ is approximated by the Fourier grid pseudospectral method. The algorithm is global in time, meaning that the entire propagation can be carried out in just a few time steps. A typical time step is much larger than that in finite differencing schemes, ∆t F ≫ H −1 . The accuracy and stability of the algorithm is analyzed. The Faber propagation method is compared with the Lanczos-Arnoldi propagation method with an example of scattering of broad band laser pulses on a periodic grating made of a dielectric whose dispersive properties are described by the Rocard-Powels-Debye model. The Faber algorithm is shown to be more efficient. The Courant limit for time stepping, ∆t C ∼ H −1 , is exceeded at least in 3000 times in the Faber propagation scheme.

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Section: Faber Polynomial Propagation Schemementioning

confidence: 99%

Section: Accuracy and Efficiency Assessmentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media

Borisov¹,

Shabanov²

2006

Journal of Computational Physics

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“…In this paper we present a thorough analysis of the narrow escape problem for the circular disk and note that our calculations apply in a straightforward manner to any simply connected domain in the plane that can be mapped conformally onto the disk. According to Riemann's mapping theorem [12], this covers all simply connected planar domains whose boundary contains at least one point. The same conclusion holds for the narrow escape problem on two-dimensional Riemannian manifolds that are conformally equivalent to a circular disk.…”

Section: Introductionmentioning

confidence: 99%

Narrow Escape, Part II: The Circular Disk

2006

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We consider Brownian motion in a circular disk Ω, whose boundary ∂Ω is reflecting, except for a small arc, ∂Ω a , which is absorbing. As ε = |∂Ω a |/|∂Ω| decreases to zero the mean time to absorption in ∂Ω a , denoted Eτ , becomes infinite. The narrow escape problem is to find an asymptotic expansion of Eτ for ε ≪ 1. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously derived expansion for a general smooth domain, Eτ = |Ω| Dπ log 1 ε + O(1) , (D is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is E[τ | x(0) = 0] = R 2 D log 1 ε + log 2 + 1 4 + O(ε) . The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because log 1 ε is not necessarily large, even when ε is small. We also find the singular behavior of the probability flux profile into ∂Ω a at the endpoints of ∂Ω a , and find the value of the flux near the center of the window.

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“…The following specialised version of Bernstein's theorem [11] will be needed. PROOF: In what follows M is a generic constant whose value may change from occurrence to occurrence.…”

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confidence: 99%