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ESTIMATES FOR APPROXIMATE SOLUTIONS TO A FUNCTIONAL DIFFERENTIAL EQUATION MODEL OF CELL DIVISION
- Part of
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- Journal:
- The ANZIAM Journal / Volume 62 / Issue 4 / October 2020
- Published online by Cambridge University Press:
- 12 March 2021, pp. 469-488
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- Article
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The functional partial differential equation (FPDE) for cell division,
$$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t), \end{align*} $$is not amenable to analytical solution techniques, despite being closely related to the first-order partial differential equation (PDE)
$$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$which, with known
$F(x,t)$, can be solved by the method of characteristics. The difficulty is due to the advanced functional terms 
$n(\alpha x,t)$ and 
$n(\beta x,t)$, where 
$\beta \ge 2 \ge \alpha \ge 1$, which arise because cells of size x are created when cells of size 
$\alpha x$ and 
$\beta x$ divide.The nonnegative function,
$n(x,t)$, denotes the density of cells at time t with respect to cell size x. The functions 
$g(x,t)$, 
$b(x,t)$ and 
$\mu (x,t)$ are, respectively, the growth rate, splitting rate and death rate of cells of size x. The total number of cells, 
$\int _{0}^{\infty }n(x,t)\,dx$, coincides with the 
$L^1$ norm of n. The goal of this paper is to find estimates in 
$L^1$ (and, with some restrictions, 
$L^p$ for 
$p>1$) for a sequence of approximate solutions to the FPDE that are generated by solving the first-order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.
