Nonlocal boundary value problem for the McKendrick – von Foerster loaded equation of fractional-order

F.M. Losanova, R.O. Kenetova

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Abstract: The paper considers McKendrick–von Foerster loaded equation of fractional-order.
Aim. The study aims to demonstrate the existence of a unique solution ‘loaded equation’ within Ω, contingent upon satisfaction of regularity conditions.
Research methods. The convergence towards a solution was achieved via a reduction to a Volterra integral equation system, specifically of the second order. Employed the fractional calculus operator.
Results. Given the McKendrick – von Foerster loaded equation of fractional-order, the existence and uniqueness of a solution to a nonlocal boundary value problem is proven. An explicit representation of the solution is derived, expressed as integral equations.
Conclusion. The derived results facilitate mathematical modeling, specifically applied to population dynamics. Consider age-structured populations and incorporate diffusion phenomena exhibiting memory effects, formally representable via fractional-order derivatives. The derived theorems augment the axiomatic foundation for analyzing said differential equations, enabling further investigation in mathematical biology and the theory of integro-differential equations.

Keywords: Gerasimov – Caputo derivative, loaded equation, McKendrick – von Foerster equations, Wright function, fractional order equations

For citation. Losanova F.M., Kenetova R.O. Nonlocal boundary value problem for the McKendrick – von Foerster loaded equation of fractional-order. News of the Kabardino-Balkarian Scientific Center of RAS. 2025. Vol. 27. No. 6. Pp. 24–29. DOI: 10.35330/1991-6639-2025-27-6-24-29

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Information about the authors

Fatima M. Losanova, Researcher, Laboratory of Synergetic Problems, Institute of Applied Mathematics and Automation – branch of the Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences;
89 A, Shortanov street, Nalchik, 360000, Russia;
losanovaf@gmail.com, ORCID: https://orcid.org/0000-0002-6342-7162, SPIN-code: 8328-6335
Raisa O. Kenetova, Candidate of Physics and Mathematics, Head of Laboratory of Synergetic Problems, Institute of Applied Mathematics and Automation – branch of the Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences;
89 A, Shortanov street, Nalchik, 360000, Russia;
kenetova_r@mail.ru, SPIN-code: 8888-9163