132
Views
0
CrossRef citations to date
0
Altmetric
Review

A convergent numerical scheme to a McKendrick–von Foerster equation with diffusion

&
Pages 1193-1211 | Received 04 Oct 2021, Accepted 26 Feb 2023, Published online: 10 Mar 2023

References

  • B. Abdellaoui and T.M. Touaoula, Decay solution for the renewal equation with diffusion, NoDEA Nonlinear Differential Equations Appl. 17 (2010), pp. 271–288.
  • L.M. Abia, O. Angulo, and J.C. Lopez-Marcos, Age-structured population dynamics models and their numerical solutions, Ecol. Modell. 188 (2005), pp. 112–136.
  • Q. Alfio, S. Riccardo, and S. Fausto, Numerical Mathematics, Springer-Verlag, New York, 2007.
  • O. Angulo, J.C. Lopez Marcos, M.A. Lopez Marcos, and F.A. Milner, A numerical method for nonlinear age-structured population models with finite maximum age, J. Math. Anal. Appl. 361 (2010), pp. 150–160.
  • O. Angulo, J.C. Lopez Marcos, and F.A. Milner, The application of an age-structured model with unbounded mortality to demography, Math. Biosci. 208 (2007), pp. 495–520.
  • B. Basse, B.C. Baguley, E.S. Marshall, W.R. Joseph, B. VanBrunt, G. Wake, and D.J.N. Wall, A mathematical model for analysis of the cell cycle in cell lines derived from human tumors, J. Math. Biol. 47(4) (2003), pp. 295–312.
  • M. Chipot, On the equations of age-dependent population dynamics, Arch. Ration. Mech. Anal. 82(1) (1983), pp. 13–25.
  • J.M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998.
  • B. Dimitri, K. Toshikazu, R. Jordi, and V. Rossana, Collocation of next-generation operators for computing the basic reproduction number of structured populations, J. Sci. Comput. 85(2) (2020), pp. 1–33.
  • J. Douglas Jr and T.F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM. J. Numer. Anal. 19(5) (1982), pp. 871–885.
  • F.Z. Farkas, Stability conditions for the nonlinear McKendrick equations, Appl. Math. Comput. 156 (2004), pp. 771–777.
  • J. Halder and S.K. Tumuluri, Numerical solution to a nonlinear McKendrick–Von Foerster equation with diffusion, Numer. Algorithms 92 (2023), pp. 1007–1039. https://doi.org/10.1007/s11075-022-01328-5
  • M. Iannelli, Mathematical Theory of Age-structured Population Dynamics, Applied Mathematics Monograph C.N.R. 7, Giardini editorie stampatori, Pisa, 1995.
  • M. Iannelli, M.Y. Kim, and E.J. Park, Splitting methods for the numerical approximation of some models of age structured population dynamics and epidemiology, Appl. Math. Comput. 87 (1997), pp. 69–93.
  • M. Iannelli and F.A. Milner, On the approximation of the Lotka–McKendrick equation with finite life-span, J. Comput. Appl. Math. 136 (2001), pp. 245–254.
  • B.K. Kakumani and S.K. Tumuluri, On a nonlinear renewal equation with diffusion, Math. Methods Appl. Sci. 39 (2016), pp. 697–708.
  • B.K. Kakumani and S.K. Tumuluri, A numerical scheme to the McKendrick–von Foerster equation with diffusion in age, Numer. Methods Partial Differ. Equ 34(6) (2018), pp. 2113–2128.
  • J.C. Lopez Marcos, An upwind scheme for a nonlinear hyperbolic integro-differential equation with integral boundary condition, Comput. Math. Appl. 22 (1991), pp. 15–28.
  • J.C. Lopez Marcos and J.M. Sanz-Serna, A definition of stability for nonlinear problems, in The Numerical Treatment of Differential Equations K. Strehmel, ed., Vol. 104, Teubner-Texte zur Mathematik, Leipzig, 1988, pp. 216–226.
  • J.C. Lopez Marcos and J.M. Sanz-Serna, Stability and convergence in numerical analysis, III: Linear investigation of nonlinear stability, IMA J. Numer. Anal. 8 (1988), pp. 71–84.
  • P. Michel, General relative entropy in a nonlinear mckendrick model, in Stochastic Analysis and Partial Differential Equations, Vol. 429, American Mathematical Society, Providence, RI, 2007, pp. 205–232.
  • P. Michel, S. Mischler, and B. Perthame, General relative entropy inequality: An illustrations on growth models, J. Math. Pures Appl. 84(9) (2005), pp. 1235–1260.
  • P. Michel and B.K. Kakumani, GRE methods for nonlinear model of evolution equation and limited resource environment, Discrete Contin. Dyn. Syst. Ser. B 24(12) (2019), pp. 6653–6673.
  • P. Michel and T.M. Touaoula, Asymptotic behavior for a class of the renewal nonlinear equation with diffusion, Math. Methods Appl. Sci. 36(3) (2012), pp. 323–335.
  • B. Perthame, Transport Equations in Biology, LN Series Frontiers in Mathematics, Birkhauser, 2007.
  • B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell division equation, J. Differ. Equ. 210 (2005), pp. 155–177.
  • B. Perthame and S.K. Tumuluri, Nonlinear renewal equations, in Selected Topics in Cancer Modeling, Modeling and Simulation in Science, Engineering and Technology, Birkhauser Boston, Boston, MA, 2008. pp. 65–96.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.