An initial-boundary value problem arising in cell population growth modelling : a thesis submitted in partial fulfillment of the requirement for the degree of Master of Science in Mathematics at Massey University, (Manawatu) Institute of Fundamental Sciences, Mathematics
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Date
2012
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Massey University
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Abstract
A partial differential equation modelling cell populations undergoing growth and division
characterised by size is studied. This equation is a special case of the fragmentation
equation studied by Michel et al. [5] with no dispersion term, and the problem is of the
initial-boundary value type.
Eigenfunction solutions are derived for separable solutions and related properties such
as spectrum, uniqueness and unimodality are investigated. We show that the spectrum is
continuous and that the decay of the eigenfunctions is exponential at a critical eigenvalue
and algebraic otherwise.
The existence of a fast decay general solution n(x; t) is then established. The problem
can be solved analytically, and it is shown that the solution is unique and smooth. The
solution properties are illustrated with some numerical simulations.
Finally, the role of exponential decaying eigenfunction solutions is interpreted from the
standpoint of the general solution. The asymptotic behaviour as t ! 1 of the general
solution is examined. Slow decay eigenfunction solutions are briefly discussed, but their
mathematical role remains to be explored.
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Keywords
Boundary value problems, Differential equations, Cell growth, Eigenfunction solutions