Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-11T00:32:57.838Z Has data issue: false hasContentIssue false
  • English
  • Français

A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model

Published online by Cambridge University Press:  05 January 2011

B. van BRUNT  and
G. C. WAKE
B. van BRUNT
Affiliation:
Institute of Fundamental Sciences, Massey University Manawatu, Private Bag 11-222, Palmerston North 4442, New Zealand
G. C. WAKE
Affiliation:
Centre for Mathematics-in-Industry, Institute of Information and Mathematical Sciences, Massey University Auckland, Private Bag 102-904, NSMC, Auckland 0745, New Zealand email: g.c.wake@massey.ac.nz
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the probability density function solutions to a second-order pantograph equation with a linear dispersion term. The functional equation comes from a cell growth model based on the Fokker–Planck equation. We show that the equation has a unique solution for constant positive growth and splitting rates and construct the solution using the Mellin transform.

Keywords

Functional differential equations, existence and uniqueness, exact solutionssteady-size distributions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 2 , April 2011 , pp. 151 - 168
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ambartsumyan, V. A. (1944) On the fluctuation of the brightness of the Milky Way. Dokl. Akad. Nauk SSSR 44, 223–226.Google Scholar
[2]Basse, B., Baguley, B., Marshall, E., Joseph, W., van Brunt, B., Wake, G. & Wall, D. (2004) Modelling cell death in human tumour cell lines exposed to the anticancer drug paclitaxel. J. Math. Bio. 49 (4), 329–357.CrossRefGoogle Scholar
[3]Basse, B., Wake, G., Wall, D. & van Brunt, B. (2004) On a cell-growth model for plankton. IMA J. Math. Med. Bio. 21, 49–61.CrossRefGoogle ScholarPubMed
[4]Bogachev, L., Derfel, G., Molchanov, S. & Ockendon, J. (2008) On bounded solutions of the balanced generalized pantograph equation. In: Chow, P. L., Mordukhovich, B. & Yin, G. (editors), Topics in Stochastic Analysis and Nonparametric Estimation, IMA, Vol. 145, Springer, New Mexico, pp. 29–49.Google Scholar
[5]Derfel, G. (1989) Probabilistic method for a class of functional differential equations. Ukrain. Mat. Zh. 41, 1322–1327. (English translation: (1990) Ukrainian Math. J. 41, 1137–1141.CrossRefGoogle Scholar
[6]Derfel, G. & Iserles, A. (1997) The pantograph equation in the complex plane. J. Math. Anal. Appl. 213, 117–132.CrossRefGoogle Scholar
[7]Epstein, B. (1948) Some applications of the Mellin transform in statistics. Ann. Math. Statist. 19, 370–379.CrossRefGoogle Scholar
[8]Fox, L., Mayers, D. F., Ockendon, J. R. & Tayler, A. B. (1971) On a function differential equation. J. Inst. Math. App. 8, 271–307.CrossRefGoogle Scholar
[9]Gardiner, C. W. (2004) Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[10]Gaver, D. P. (1964) An absorption probablility problem. J. Math. Anal. Appl. 9 (N3), 384–393.CrossRefGoogle Scholar
[11]Hall, A. J., Wake, G. C. & Gandar, P. W. (1991) Steady size distributions for cells in one dimensional plant tissues. J. Math. Bio. 30 (2), 101–123.CrossRefGoogle Scholar
[12]Hall, A. J. & Wake, G. C. (1989) A functional differential equation arising in the modelling of cell-growth. J. Aust. Math. Soc. Ser. B 30, 424–435.CrossRefGoogle Scholar
[13]Hall, A. J. & Wake, G. C. (1990) A functional differential equation determining steady size distributions for populations of cells growing exponentially. J. Aust. Math. Soc. Ser. B, 31, 344–353.CrossRefGoogle Scholar
[14]Iserles, A. (1993) On the generalized pantograph functional differential equation. Eu. J. Appl. Math. 4, 1–38.CrossRefGoogle Scholar
[15]Jakeman, E. (1999) K-distributed noise. J. Opt. A: Pure Appl. Opt. 1, 784–789.CrossRefGoogle Scholar
[16]Kato, T. & McLeod, J. B. (1971) The functional-differential equation y′(x) = ay(λx) + by(x). Bull. Am. Math. Soc. 77, 891–937.Google Scholar
[17]Kim, H. K. (1998) Advanced Second Order Functional Differential Equations, Ph.D. thesis, Massey University, New Zealand.Google Scholar
[18]Marshall, J., van-Brunt, B. & Wake, G. (2002) Natural boundaries for solutions to a certain class of functional differential equations. J. Math. Anal. Appl. 268, 157–170.CrossRefGoogle Scholar
[19]Morgan, D. (2000) A remarkable sequence derived from Euler products. J. Math. Phys. 41 (10), 7109–7121.CrossRefGoogle Scholar
[20]Ockendon, J. & Tayler, A. (1971) The dynamics of a current collection system for an electric locomotive. Proc. R. Soc. Lond. Ser. A 322, 447–468.Google Scholar
[21]Springer, M. & Thompson, W. (1966) The distribution of products of independent random variables. SIAM J. Appl. Math. 14 (3), 511–526.CrossRefGoogle Scholar
[22]Tough, R. (1987) A Fokker-Planck description of K-distributed noise. J. Phys. A: Math. Gen. 20, 551–567.CrossRefGoogle Scholar
[23]van-Brunt, B., Wake, G. C. & Kim, H. K. (2001) On a singular Sturm–Liouville problem involving an advanced functional differential equation. Euro. J. Appl. Math. 12, 625–644.CrossRefGoogle Scholar
[24]Wake, G. C., Cooper, S., Kim, H. K. & van-Brunt, B. (2000) Functional differential equations for cell-growth models with dispersion. Commun. Appl. Anal. 4, 561–574.Google Scholar