Summing Logarithmic Expansions for Singularly Perturbed Eigenvalue Problems

Ward, Michael J.; Henshaw, William D.; Keller, Joseph B.

doi:10.1137/0153039

Cited by 139 publications

(167 citation statements)

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“…Similar problems in planar domains have been considered in [46] and [27], and on the boundary of the cylinder in [40]. For the case of perfectly absorbing traps of a common radius, we derive a two-term expansion for the principal eigenvalue in powers of −1/ log(εa), where εa 1 is the common trap radius.…”

Section: Daniel Coombs Ronny Straube and Michael Wardmentioning

confidence: 99%

“…For an arbitrary noncircular trap Ω j , (3.32) cannot be solved analytically, but has the far-field behavior (cf. [46]) [46], [33]), and explicit formulae for it are available for different trap shapes such as ellipses, squares, and equilateral triangles (cf. [33]).…”

Section: Partially Absorbing Trapsmentioning

confidence: 99%

“…[5], [27], [40], and [46]). Our goal is to derive an explicit two-term expansion for σ(ε) by deriving simple formulae for the coefficients σ 0 and σ 1 .…”

Section: Partially Absorbing Trapsmentioning

confidence: 99%

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Diffusion on a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points

Coombs¹,

Straube²,

Ward³

2009

SIAM J. Appl. Math.

113

139

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Abstract.A common scenario in cellular signal transduction is that a diffusing surface-bound molecule must arrive at a localized signaling region on the cell membrane before the signaling cascade can be completed. The question then arises of how quickly such signaling molecules can arrive at newly formed signaling regions. Here, we attack this problem by calculating asymptotic results for the mean first passage time for a diffusing particle confined to the surface of a sphere, in the presence of N partially absorbing traps of small radii. The rate at which the small diffusing molecule becomes captured by one of the traps is determined by asymptotically calculating the principal eigenvalue for the Laplace operator on the sphere with small localized traps. The asymptotic analysis relies on the method of matched asymptotic expansions, together with detailed properties of the Green's function for the Laplacian and the Helmholtz operators on the surface of the unit sphere. The asymptotic results compare favorably with full numerical results.

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Section: Daniel Coombs Ronny Straube and Michael Wardmentioning

confidence: 99%

Section: Partially Absorbing Trapsmentioning

confidence: 99%

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Diffusion on a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points

Coombs¹,

Straube²,

Ward³

2009

SIAM J. Appl. Math.

113

139

View full text Add to dashboard Cite

show abstract

“…Matched asymptotics of two-and three-dimensional problems [22], [23], [24], [11] yield the leading term in the expansion of the principal eigenvalue in three dimensions and a full expansion in two dimensions. For the special case of the mixed Neumann problem with a small Dirichlet window in the boundary, the leading term obtained in [17], [18], [19] can be obtained by the application of the matched asymptotics expansion to this problem.…”

Section: Deep Well-amentioning

confidence: 99%

“…Here we consider the narrow escape problem for a Brownian motion in a field of force. The closely related problem of computing the principal eigenvalue of the Laplace operator for mixed boundary conditions on large and small pieces of the boundary was considered in [22], [23], [24], [11] (see section 6 for discussion).…”

Section: Introduction Kramers' Theorymentioning

confidence: 99%