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A functional partial differential equation arising in a cell growth model with dispersion
Author(s) -
Efendiev Messoud,
van Brunt Bruce,
Wake Graeme C.,
Zaidi Ali Ashher
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4684
Subject(s) - mathematics , exponential growth , term (time) , partial differential equation , convergence (economics) , boundary value problem , dispersion (optics) , parabolic partial differential equation , rate of convergence , mathematical analysis , exponential function , order (exchange) , key (lock) , ecology , physics , finance , quantum mechanics , optics , economics , biology , economic growth
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.