An Eigenvalue Problem Involving a Functional Differential Equation Arising in a Cell Growth Model

Brunt, Bruce van; Vlieg-Hulstman, M.

doi:10.1017/s1446181110000866

Cited by 7 publications

(5 citation statements)

References 9 publications

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“…. These values correspond to the eigenvalues determined by van Brunt and Vlieg-Hulstman [19]. Here, we simply note that D(0, λ) > 0 for all λ > bα.…”

Section: A Class Of Dirichlet Seriessupporting

confidence: 55%

“…The focus in most of these studies was on particular values of λ. The interpretation of λ as an eigenvalue parameter, aside from its rôle in cell division models as the value that produces a pdf solution, was made for a related problem in [19,20]. We also note that an eigenvalue problem for a second order version was studied in [22].…”

Section: A Class Of Dirichlet Seriesmentioning

confidence: 97%

See 1 more Smart Citation

A Cell Growth Model Adapted for the Minimum Cell Size Division

Brunt¹,

Gul²,

Wake³

2015

ANZIAM J.

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We study a cell growth model with a division function that models cells which divide only after they have reached a certain minimum size. In contrast to the cases studied in the literature, the determination of the steady size distribution entails an eigenvalue that is not known explicitly, but is defined through a continuity condition. We show that there is a steady size distribution solution to this problem.

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“…. These values correspond to the eigenvalues determined by van Brunt and Vlieg-Hulstman [19]. Here, we simply note that D(0, λ) > 0 for all λ > bα.…”

Section: A Class Of Dirichlet Seriessupporting

confidence: 55%

Section: A Class Of Dirichlet Seriesmentioning

confidence: 97%

A Cell Growth Model Adapted for the Minimum Cell Size Division

Brunt¹,

Gul²,

Wake³

2015

ANZIAM J.

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“…Finally, we note that the Dirichlet series defined by the V k correspond to the eigenfunctions derived by van-Brunt and Vlieg-Hulstman [30,31] for the pantograph equation.…”

Section: The Limiting Solution and Asymptotics As T → ∞mentioning

confidence: 65%

Solutions to an advanced functional partial differential equation of the pantograph type

2015

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A model for cells structured by size undergoing growth and division leads to an initial boundary value problem that involves a first-order linear partial differential equation with a functional term. Here, size can be interpreted as DNA content or mass. It has been observed experimentally and shown analytically that solutions for arbitrary initial cell distributions are asymptotic as time goes to infinity to a certain solution called the steady size distribution. The full solution to the problem for arbitrary initial distributions, however, is elusive owing to the presence of the functional term and the paucity of solution techniques for such problems. In this paper, we derive a solution to the problem for arbitrary initial cell distributions. The method employed exploits the hyperbolic character of the underlying differential operator, and the advanced nature of the functional argument to reduce the problem to a sequence of simple Cauchy problems. The existence of solutions for arbitrary initial distributions is established along with uniqueness. The asymptotic relationship with the steady size distribution is established, and because the solution is known explicitly, higher-order terms in the asymptotics can be readily obtained.

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“…It has been studied in the complex plane [11]. Connections have been made with complex dynamics [25,38], second order singular Sturm Liouville problems [37], and eigenfunction expansions even in the first order case [41,42]. In terms of probability theory, the pantograph equation has been interpreted as a special case of a much broader class of equations [8].…”

Section: Some Generalizationsmentioning

confidence: 99%

Cell Division And The Pantograph Equation

2018

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Simple models for size structured cell populations undergoing growth and division produce a class of functional ordinary differential equations, called pantograph equations, that describe the long time asymptotics of the cell number density. Pantograph equations arise in a number of applications outside this model and, as a result, have been studied heavily over the last five decades. In this paper we review and survey the rôle of the pantograph equation in the context of cell division. In addition, for a simple case we present a method of solution based on the Mellin transform and establish uniqueness directly from the transform equation.Article published online by EDP Sciences and available at https://www.esaim-proc.org or https://doi.

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An Eigenvalue Problem Involving a Functional Differential Equation Arising in a Cell Growth Model

Cited by 7 publications

References 9 publications

A Cell Growth Model Adapted for the Minimum Cell Size Division

A Cell Growth Model Adapted for the Minimum Cell Size Division

Solutions to an advanced functional partial differential equation of the pantograph type

Cell Division And The Pantograph Equation

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