Functional differential equations arising in the study of a cell growth model : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
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Date
2019
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Massey University
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Abstract
In this thesis we study a class of functional ordinary and partial differential equations that arise in the study of a size structured cell growth model. We study first and second order pantograph equations, which arise as separable solutions, for various constant and non constant coefficients. We discuss several techniques for solving pantograph equations that use the Laplace and the Mellin transforms, including a novel technique based on Mellin convolutions. These techniques are illustrated by applying them to a simple first order equation. The use of the Mellin transform to solve pantograph equation relies upon solving the transform equation, and this can prove formidable. This motivated us to find another avenue to show the existence of a solution. We consider a simple first order pantograph equation and show the uniqueness of the solution. We extend the study to second order pantograph equations and review a few particular second order pantograph equations with constant and non constant coefficients. These equations are solved using established techniques. Among the second order equation, a cell growth model that involves the Hermite operator is a part of research problems. Two interesting features for the Hermite Problem are, the form of the Mellin transform, that is such that the inversion is formidable, and the slow decaying nature of the solution. It is shown that for a range of parameter values α, b and g, there are no pdf solutions to the Hermite Problem; however if we drop the integrability and positivity condition, then there are non trivial solutions.
Although the separable solution is the prime candidate for a steady size distribution, showing analytically that it is this distribution requires more advanced techniques. We thus consider the full problem. In particular, we consider the case where cells do not divide when they are under certain size. This problem differs from earlier ones because the eigenvalue for the separable solution is not known explicitly. In order to show the uniqueness of eigenvalue and to show that there is a steady size distribution solution to this problem, we adapt the analysis of Perthame & Ryzhik, who under certain assumptions on the division rate, established the existence and the uniqueness of the solution to a first order ordinary functional differential equation for a non constant division rate. In addition, we show that we have an alternative technique to find the eigenvalue.
A second order partial differential equation arises when there is stochasticity in the growth rate. The earlier studied techniques are of limited use; however, Efendiev et al. [18] developed a technique that solves these equations for constant coefficients. They proved that the solution to the problem converges to the separable solution as time goes to ∞. We adapt their analysis and solve a second order functional differential equation with linear growth rate. In addition, we show that the solution to this problem does not have an SSD solution.
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Cells, Growth, Mathematical models, Functional differential equations