Functional differential equations arising in cell‐growth

Wake, G. C.; Begg, Ronald

doi:10.1002/pamm.200700950

Cited by 16 publications

(25 citation statements)

References 3 publications

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“…Both papers are devoted to the classes of uniqueness for Cauchy problem to PDE with linearly transformed arguments. The cell division problem has been generalized to include dispersion [17,18] and this led to the study of second-order pantograph equations [19]. The problem has also been studied for certain non-constant coefficients [20,21], and a multi-compartment model has been developed for an application to the treatment of cancer [22].…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

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

2015

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“…Wake et al [10] proved the existence and uniqueness of SSDs for nonzero values of ε, along with the fact that as ε → 0 + the Dirichlet series expression for those SSDs reduces to that of the SSD from [9]. Theorem 16 provides an error estimate for the difference of the cumulative SSDs for nonzero ε from the cumulative SSD for ε = 0.…”

Section: An Examplementioning

confidence: 96%

Existence theorems for a class of nonlocal differential equations

Begg

2006

Journal of Mathematical Analysis and Applications

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A class of nonlocal second-order ordinary differential equations of the formfor continuous f and λ is studied. The equation is supplemented with none, one, or two Robin boundary conditions depending on whether the interval of interest I is finite, semi-infinite or infinite. The only other restriction on the function λ is that it maps I into itself. Sufficient conditions for the existence of a solution are found, which include the assumption of the existence of 'upper' and 'lower' solutions. The upper and lower solutions provide bounds for the solution on I .

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“…This is an important challenge for numerical analysts since functional differential equations with advanced proportional arguments arise in practical applications: second-order pantograph-type DDEs with q > 1 arise for example in the mathematical modelling of the growth of a population of cells where the individual cells grow, such that each mother cell divides into q > 1 daughter cells of the same size. See [33,34,66] for the details; the theory of such functional differential equations is discussed in [59].…”

Section: Pantograph-type Vfides With Advanced Argumentsmentioning

confidence: 99%

Recent advances in the numerical analysis of Volterra functional differential equations with variable delays

Brunner

2009

Journal of Computational and Applied Mathematics

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MSC: 65R20 34K06 34K28Keywords: Volterra functional integro-differential equations Vanishing delays Proportional delays Pantograph equation Collocation solutions Optimal order of superconvergence a b s t r a c tThe numerical analysis of Volterra functional integro-differential equations with vanishing delays has to overcome a number of challenges that are not encountered when solving 'classical' delay differential equations with non-vanishing delays. In this paper I shall describe recent results in the analysis of optimal (global and local) superconvergence orders in collocation methods for such evolutionary problems. Following a brief survey of results for equations containing Volterra integral operators with non-vanishing delays, the discussion will focus on pantograph-type Volterra integro-differential equations with (linear and nonlinear) vanishing delays. The paper concludes with a section on open problems; these include the asymptotic stability of collocation solutions u h on uniform meshes for pantograph-type functional equations, and the analysis of collocation methods for pantograph-type functional equations with advanced arguments.

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Functional differential equations arising in cell‐growth

Cited by 16 publications

References 3 publications

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

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

Existence theorems for a class of nonlocal differential equations

Recent advances in the numerical analysis of Volterra functional differential equations with variable delays

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