Summing logarithmic expansions for singularly perturbed eigenvalue problems
01 Jun 1993-Siam Journal on Applied Mathematics (Society for Industrial and Applied Mathematics)-Vol. 53, Iss: 3, pp 799-828
TL;DR: In each case, it is shown that the entire infinite series is contained in the solution of a single related problem that does not involve the size or shape of the hole.
Abstract: Strong localized perturbations of linear and nonlinear eigenvalue problems in a bounded two-dimensional domain D are considered. The effects on an eigenvalue $\lambda _0 $ of the Lapla-cian, and on the fold point $\lambda _{c0} $ of a nonlinear eigenvalue problem, of removing a small subdomain $D_\epsilon $, of “radius” $\epsilon $, from D and imposing a condition on the boundary of the resulting hole, are determined. Using the method of matched asymptotic expansions, it is shown that the expansions of the eigenvalues and fold points for these perturbed problems start with infinite series in powers of $( - 1/\log [ \epsilon d( \kappa ) ] )$. Here $d( \kappa )$ is a constant that depends on the shape of $D_\epsilon $ and on the precise form of the boundary condition on the hole. In each case, it is shown that the entire infinite series is contained in the solution of a single related problem that does not involve the size or shape of the hole. This related problem is not stiff and can be solved numerically...
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TL;DR: A wide range of analytical methods and models of intracellular transport is presented, including Brownian ratchets, random walk models, exclusion processes, random intermittent search processes, quasi-steady-state reduction methods, and mean-field approximations for active transport.
Abstract: The interior of a living cell is a crowded, heterogenuous, fluctuating environment. Hence, a major challenge in modeling intracellular transport is to analyze stochastic processes within complex environments. Broadly speaking, there are two basic mechanisms for intracellular transport: passive diffusion and motor-driven active transport. Diffusive transport can be formulated in terms of the motion of an overdamped Brownian particle. On the other hand, active transport requires chemical energy, usually in the form of adenosine triphosphate hydrolysis, and can be direction specific, allowing biomolecules to be transported long distances; this is particularly important in neurons due to their complex geometry. In this review a wide range of analytical methods and models of intracellular transport is presented. In the case of diffusive transport, narrow escape problems, diffusion to a small target, confined and single-file diffusion, homogenization theory, and fractional diffusion are considered. In the case of active transport, Brownian ratchets, random walk models, exclusion processes, random intermittent search processes, quasi-steady-state reduction methods, and mean-field approximations are considered. Applications include receptor trafficking, axonal transport, membrane diffusion, nuclear transport, protein-DNA interactions, virus trafficking, and the self-organization of subcellular structures.
583 citations
Additional excerpts
...(2.73) It can also be shown that the principal eigenvalue has an infinite logarithmic expansion (Ward et al., 1993): λ0 = νΛ1 + ν 2Λ2 + . . . , ν ≡ − 1 log ε ....
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TL;DR: In this article, the authors use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a hyperasymptotic approximation.
Abstract: Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter e which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.
261 citations
IBM1
TL;DR: In this article, a method is described to solve the time-dependent incompressible Navier-Stokes equations with finite differences on curvilinear overlapping grids in two or three space dimensions.
255 citations
TL;DR: This paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded settings.
Abstract: This paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded o...
211 citations
Cites methods from "Summing logarithmic expansions for ..."
...A more general procedure to sum logarithmic expansions resulting from singularly perturbed eigenvalue problems is given in [12]....
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TL;DR: Recent developments in the non-standard asymptotics of the narrow escape problem are reviewed, which are based on several ingredients: a better resolution of the singularity of Neumann's function,resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components.
Abstract: The narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain. The problem is equivalent to solving the mixed Dirichlet--Neumann boundary value problem for the Poisson equation with small Dirichlet and large Neumann parts. The mixed boundary value problem, which goes back to Lord Rayleigh, originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the Neumann problem in domains with bottlenecks. We review here recent developments in the non-standard asymptotics of the problem, which are based on several ingredients: a better resolution of the singularity of Neumann's function, resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components. The new methodology applies to two- and higher-dimensional problems. Selected applications are r...
175 citations
Cites background or methods from "Summing logarithmic expansions for ..."
...In the case of sufficiently smooth boundaries near the absorbing window considered in Ward and Keller (1993); Ward, Henshaw, and Keller (1993); Ward and Van De Velde (1992); Kolokolnikov, Titcombe, and Ward (2005); Cheviakov, Ward, and Straube (2010); Coombs, Straube, and Ward (2009); and Bénichou…...
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...The NET calculated in section 2 was calculated for small absorbing windows in a smooth reflecting boundary in Ward and Keller (1993); Ward, Henshaw, and Keller (1993); Ward and Van De Velde (1992); Kolokolnikov, Titcombe, and Ward (2005); Cheviakov, Ward, and Straube (2010); Coombs, Straube, and…...
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...…for the solution of the mixed Neumann–Dirichlet problem for the Poisson equation in geometries in which the methodologies of Ward and Keller (1993); Ward, Henshaw, and Keller (1993); Ward and Van De Velde (1992); Kolokolnikov, Titcombe, and Ward (2005); Cheviakov, Ward, and Straube (2010);…...
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...The method was developed in Ward and Keller (1993); Ward, Henshaw, and Keller (1993); Ward and Van De Velde (1992); Kolokolnikov, Titcombe, and Ward (2005); and Cheviakov, Ward, and Straube (2010)....
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...Further refinements of (3.13) are given in Ward and Keller (1993); Ward, Henshaw, and Keller (1993); Ward and Van De Velde (1992); Kolokolnikov, Titcombe, and Ward (2005); Cheviakov, Ward, and Straube (2010); Singer, Schuss, and Holcman (2006a); and Schuss, Singer, and Holcman (2007)....
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References
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Book•
01 Dec 1964
TL;DR: The method of power series Equations of the first order Classification of partial differential equations Cauchy's problem for equations with two independent variables The Dirichlet and Neumann problems as mentioned in this paper.
Abstract: The method of power series Equations of the first order Classification of partial differential equations Cauchy's problem for equations with two independent variables The fundamental solution Cauchy's problem in space of higher dimension The Dirichlet and Neumann problems Dirichlet's principle Existence theorems of potential theory Integral equations Eigenvalue problems Tricomi's problem formulation of well posed problems Finite differences Fluid dynamics Free boundary problems Partial differential equations in the complex domain Bibliography Index.
1,025 citations
IBM1
TL;DR: The generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them are discussed and some techniques for the solution of elliptic and time-dependent PDEs on composite meshes are described.
577 citations
TL;DR: Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed and the resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.
Abstract: Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed. The aspects of this method considered include error estimation, adaptive mesh selection, B-spline basis function evaluation, linear system solution and nonlinear problem solution. The resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.
514 citations
TL;DR: This paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded settings.
Abstract: This paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded o...
211 citations
TL;DR: In this paper, critical values for the Biot number and for the dimensionless central temperature excess have been evaluated for the whole range of Biot numbers from the uniform case (Semenov extreme, Bi $\rightarrow$ 0) to the Frank-Kamenetskii extreme (Bi $rightarrow\infty$).
Abstract: The conductive theory of thermal explosion in its original form (Frank-Kamenetskii, Acta phys.-chim. URSS (1939)) expresses the balance between heat generation and heat conduction in terms of the dimensionless parameter $\delta = [QEAa^2\_0 c^n\_0 \exp (-E/RT\_a)]/\kappa RT^2\_a$. Stability is lost when $\delta$ exceeds a critical value which, in this approximation, depends only on the geometry of the system. Matters are usually more complicated than this. First, heat transfer is often impeded both in the interior (by conduction) and at the surface; the relative importance of these impedances is expressed by the Biot number Bi = $\chi a\_0/\kappa$. Second, the temperature-dependence of reaction rate may not be well enough represented by the \`exponential approximation' (which simply implies a doubling of rate every so many degrees). The natural and convenient dimensionless measure here is the parameter $\epsilon = RT\_a/E$. In the present paper, critical values for the parameter $\delta$ and for the dimensionless central-temperature excess $\theta\_0$ have been evaluated for the whole range of Biot number from the uniform case (Semenov extreme, Bi $\rightarrow$ 0) to the Frank-Kamenetskii extreme (Bi $\rightarrow\infty$). The procedures can handle any temperature-dependence of rate and are illustrated here for the Arrhenius and \`bimolecular' forms for which, k$\propto \exp$ (-E/RT) and k$\propto$ T$^1/2 \exp$ (-E/RT) respectively. When E/RT$\_a$ is not large, criticality is lost at $\epsilon = \epsilon\_{tr} \leqslant \frac{1}{4}$. Such transitional values for the reduced ambient temperature $\epsilon$, for the critical value of $\delta$, and for the dimensionless central temperature excess $\theta\_0$ have also been obtained. They are represented both graphically and numerically. The present results are also compared with earlier work.
39 citations