Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T22:34:36.818Z Has data issue: false hasContentIssue false
  • English
  • Français

EIGENFUNCTIONS ARISING FROM A FIRST-ORDER FUNCTIONAL DIFFERENTIAL EQUATION IN A CELL GROWTH MODEL

Part of: Functional-differential and differential-difference equations

Published online by Cambridge University Press:  19 April 2011

BRUCE VAN BRUNT  and
M. VLIEG-HULSTMAN
BRUCE VAN BRUNT*
Affiliation:
Institute of Fundamental Sciences, Mathematics, Massey University, Palmerston North, New Zealand (email: [email protected])
M. VLIEG-HULSTMAN
Affiliation:
Institute of Fundamental Sciences, Mathematics, Massey University, Palmerston North, New Zealand (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A boundary-value problem for cell growth leads to an eigenvalue problem. In this paper some properties of the eigenfunctions are studied. The first eigenfunction is a probability density function and is of importance in the cell growth model. We sharpen an earlier uniqueness result and show that the distribution is unimodal. We then show that the higher eigenfunctions have nested zeros. We show that the eigenfunctions are not mutually orthogonal, but that there are certain orthogonality relations that effectively partition the set of eigenfunctions into two sets.

Keywords

pantograph equationcell growth modelnonlocal eigenvalue problem

MSC classification

Secondary: 34K06: Linear functional-differential equations
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 1 , July 2010 , pp. 46 - 58
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Adams, C. R., “Linear q-difference equations”, Bull. Amer. Math. Soc. 37 (1931) 361–400.CrossRefGoogle Scholar
[2]Ambartsumian, V. A., “On the fluctuation of brightness of the Milky Way”, Dokl. Akad. Nauk USSR 44 (1944) 223–226.Google Scholar
[3]Basse, B., Baguley, B., Marshall, E., Joseph, W., van Brunt, B., Wake, G. C. and Wall, D. J. N., “Modelling cell death in human tumour cell lines exposed to the anticancer drug Paclitaxel”, J. Math. Biol. 49 (2004) 329–357.CrossRefGoogle Scholar
[4]Basse, B., Wake, G. C., Wall, D. J. N. and van Brunt, B., “On a cell-growth model for plankton”, IMA J. Math. Med. Biol. 21 (2004) 49–61.CrossRefGoogle ScholarPubMed
[5]Begg, R., “Existence theorems for a class of nonlocal differential equations”, J. Math. Anal. Appl. 322 (2006) 1168–1187.CrossRefGoogle Scholar
[6]Begg, R., Wall, D. J. N. and Wake, G. C., “On a multicompartment age-distribution model of cell growth”, IMA J. Appl. Math. 75 (2010) 905–931.CrossRefGoogle Scholar
[7]Bogachev, L., Derfel, G., Molchanov, S. and Ockendon, J. R., “On bounded solutions of the balanced generalized pantograph equation”, in: Topics in stochastic analysis and nonparametric estimation, Volume 145 of IMA Volumes in Mathematics and its Applications (eds Chow, P., Mordukhovich, B. and Yin, G.), (Springer, New York, 2008) 29–49.CrossRefGoogle Scholar
[8]Carr, J. and Dyson, J., “The matrix functional differential equation y ′(x)=Ay(λx)+By(x)”, Proc. Roy. Soc. Edinburgh A 75 (1975) 5–22.CrossRefGoogle Scholar
[9]Čermák, J., “On matrix differential equations with several unbounded delays”, European J. Appl. Math. 17 (2006) 417–433.CrossRefGoogle Scholar
[10]Chang, Y. H. and Jau, G. C., “The behaviour of the solution for a mathematical model for analysis of the cell cycle”, Commun. Pure Appl. Anal. 5 (2006) 779–792.CrossRefGoogle Scholar
[11]da Costa, F. P., Grinfeld, M. and McLeod, J. B., “Unimodality of steady size distributions of growing cell populations”, J. Evol. Equ. 1 (2001) 405–409.CrossRefGoogle Scholar
[12]Derfel, G., “Probabilistic method for a class of functional differential equations”, Ukraïn. Mat. Zh. 41 (1989) 1322–1327; English translation: Ukrainian Math. J. 41 (1990) 1137–1141.Google Scholar
[13]Derfel, G. and Iserles, A., “The pantograph equation in the complex plane”, J. Math. Anal. Appl. 213 (1997) 117–132.CrossRefGoogle Scholar
[14]Flamant, P., “Sur une équation différentielle fonctionnelle linéaire”, Palermo Rend. 48 (1924) 135–208.CrossRefGoogle Scholar
[15]Fox, L., Mayers, D. F., Ockendon, J. R. and Tayler, A. B., “On a functional differential equation”, J. Inst. Math. Appl. 8 (1971) 271–307.CrossRefGoogle Scholar
[16]Fredrickson, P. O., “Dirichlet series solution for certain functional differential equations”, in: Japan–United States seminar on ordinary differential and functional equations, Volume 243 of Springer Lecture Notes in Mathematics (ed. Uradbe, M.), (Springer, New York, 1971) 247–254.Google Scholar
[17]Gaver, D. P., “An absorption probablity problem”, J. Math. Anal. Appl. 9 (1964) 384–393.CrossRefGoogle Scholar
[18]Hall, A. J. and Wake, G. C., “A functional differential equation arising in the modelling of cell-growth”, J. Aust. Math. Soc. Ser. B 30 (1989) 424–435.CrossRefGoogle Scholar
[19]Hall, A. J. and Wake, G. C., “Functional differential equations determining steady size distributions for populations of cells growing exponentially”, J. Aust. Math. Soc. Ser. B 31 (1990) 434–453.CrossRefGoogle Scholar
[20]Hall, A. J., Wake, G. C. and Gandar, P. W., “Steady size distributions for cells in one dimensional plant tissues”, J. Math. Biol. 30 (1991) 101–123.CrossRefGoogle Scholar
[21]Heard, M., “Asymptotic behaviour of solutions of the functional differential equation x ′(t)=ax(t)+bx(t α),α>1”, J. Math. Anal. Appl. 44 (1973) 745–757.CrossRefGoogle Scholar
[22]Iserles, A., “On the generalized pantograph functional–differential equation”, European J. Appl. Math. 4 (1993) 1–38.CrossRefGoogle Scholar
[23]Kato, T. and McLeod, J. B., “The functional differential equation y ′(x)=ay(λx)+by(x)”, Bull. Amer. Math. Soc. 77 (1971) 891–937.Google Scholar
[24]Kim, H. K., “Advanced second order functional differential equations”, Ph. D. Thesis, Massey University, 1998.Google Scholar
[25]Marshall, J. C., van Brunt, B. and Wake, G. C., “Natural boundaries for solutions to a certain class of functional differential equations”, J. Math. Anal. Appl. 268 (2002) 157–170.CrossRefGoogle Scholar
[26]Oberg, R. J., “Local theory of complex functional differential equations”, Trans. Amer. Math. Soc. 161 (1971) 302–327.CrossRefGoogle Scholar
[27]Ockendon, J. R. and Tayler, A. B., “The dynamics of a current collection system for an electric locomotive”, Proc. R. Soc. Lond. A 322 (1971) 447–468.Google Scholar
[28]van Brunt, B., Kim, H. O. and Derfel, G., “Holomorphic solutions to functional differential equations”, J. Math. Anal. Appl. 368 (2010) 350–357.CrossRefGoogle Scholar
[29]van Brunt, B., Marshall, J. C. and Wake, G. C., “An eigenvalue problem for holomorphic solutions to a certain class of functional differential equations”, European J. Appl. Math. 14 (2003) 571–585.CrossRefGoogle Scholar
[30]van Brunt, B., Marshall, J. C. and Wake, G. C., “Holomorphic solutions to pantograph type equations with neutral fixed points”, J. Math. Anal. Appl. 295 (2004) 557–569.CrossRefGoogle Scholar
[31]van Brunt, B. and Vlieg-Hulstman, M., “An eigenvalue problem involving a functional differential equation arising in a cell growth model”, ANZIAM J. 51 (2010) 383–393.CrossRefGoogle Scholar
[32]van Brunt, B., Wake, G. C. and Kim, H. K., “On a singular Sturm–Liouville problem involving an advanced functional differential equation”, European J. Appl. Math. 12 (2001) 625–644.CrossRefGoogle Scholar
[33]Wake, G. C., Cooper, S., Kim, H. K. and van Brunt, B., “Functional differential equations for cell-growth models with dispersion”, Commun. Appl. Anal. 4 (2000) 561–573.Google Scholar