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- Title
A functional partial differential equation arising in a cell growth model with dispersion.
- Authors
Efendiev, Messoud; van Brunt, Bruce; Wake, Graeme C.; Zaidi, Ali Ashher
- Abstract
In this paper we solve an initial‐boundary value problem that involves a pde with a nonlocal term. The problem comes from a cell division model where the growth is assumed to be stochastic. The deterministic version of this problem yields a first‐order pde; the stochastic version yields a second‐order parabolic pde. There are no general methods for solving such problems even for the simplest cases owing to the nonlocal term. Although a solution method was devised for the simplest version of the first‐order case, the analysis does not readily extend to the second‐order case. We develop a method for solving the second‐order case and obtain the exact solution in a form that allows us to study the long time asymptotic behaviour of solutions and the impact of the dispersion term. We establish the existence of a large time attracting solution towards which solutions converge exponentially in time. The dispersion term does not appear in the exponential rate of convergence.
- Subjects
FUNCTIONAL analysis; PARTIAL differential equations; DISPERSION (Chemistry); BOUNDARY value problems; STOCHASTIC processes
- Publication
Mathematical Methods in the Applied Sciences, 2018, Vol 41, Issue 4, p1541
- ISSN
0170-4214
- Publication type
Article
- DOI
10.1002/mma.4684