The Rothe method for the McKendrick-von Foerster equation

Henryk Leszczyński; Piotr Zwierkowski

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 3, page 589-602
  • ISSN: 0011-4642

Abstract

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We present the Rothe method for the McKendrick-von Foerster equation with initial and boundary conditions. This method is well known as an abstract Euler scheme in extensive literature, e.g. K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Reidel, Dordrecht, 1982. Various Banach spaces are exploited, the most popular being the space of bounded and continuous functions. We prove the boundedness of approximate solutions and stability of the Rothe method in L ∞ and L 1 norms. Proofs of these results are based on comparison inequalities. Our theory is illustrated by numerical experiments. Our research is motivated by certain models of mathematical biology.

How to cite

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Leszczyński, Henryk, and Zwierkowski, Piotr. "The Rothe method for the McKendrick-von Foerster equation." Czechoslovak Mathematical Journal 63.3 (2013): 589-602. <http://eudml.org/doc/260719>.

@article{Leszczyński2013,
abstract = {We present the Rothe method for the McKendrick-von Foerster equation with initial and boundary conditions. This method is well known as an abstract Euler scheme in extensive literature, e.g. K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Reidel, Dordrecht, 1982. Various Banach spaces are exploited, the most popular being the space of bounded and continuous functions. We prove the boundedness of approximate solutions and stability of the Rothe method in $L^\infty $ and $L^1$ norms. Proofs of these results are based on comparison inequalities. Our theory is illustrated by numerical experiments. Our research is motivated by certain models of mathematical biology.},
author = {Leszczyński, Henryk, Zwierkowski, Piotr},
journal = {Czechoslovak Mathematical Journal},
keywords = {Rothe method; stability; comparison; Rothe method; stability; comparison; semidiscretization; McKendrick-von Foerster partial differential equation; initial-boundary-value problem; convergence; consistency; numerical example},
language = {eng},
number = {3},
pages = {589-602},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Rothe method for the McKendrick-von Foerster equation},
url = {http://eudml.org/doc/260719},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Leszczyński, Henryk
AU - Zwierkowski, Piotr
TI - The Rothe method for the McKendrick-von Foerster equation
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 3
SP - 589
EP - 602
AB - We present the Rothe method for the McKendrick-von Foerster equation with initial and boundary conditions. This method is well known as an abstract Euler scheme in extensive literature, e.g. K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Reidel, Dordrecht, 1982. Various Banach spaces are exploited, the most popular being the space of bounded and continuous functions. We prove the boundedness of approximate solutions and stability of the Rothe method in $L^\infty $ and $L^1$ norms. Proofs of these results are based on comparison inequalities. Our theory is illustrated by numerical experiments. Our research is motivated by certain models of mathematical biology.
LA - eng
KW - Rothe method; stability; comparison; Rothe method; stability; comparison; semidiscretization; McKendrick-von Foerster partial differential equation; initial-boundary-value problem; convergence; consistency; numerical example
UR - http://eudml.org/doc/260719
ER -

References

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