Ortigueira, Manuel DuarteBengochea, Gabriel2023-07-112023-07-112023-01-082076-3417PURE: 65875499PURE UUID: a1225a6c-f0ad-41ba-8f67-c4f385a9b4f4Scopus: 85146730967WOS: 000916931200001ORCID: /0000-0003-4270-3284/work/151429123http://hdl.handle.net/10362/155105Funding Information: The second author was supported by the Autonomous University of Mexico City (UACM) under the project Ccyt-2021-11. Publisher Copyright: © 2023 by the authors.Fractionalisation and solution of the Ambartsumian equation is considered. The general approach to fractional calculus suitable for applications in physics and engineering is described. It is shown that Liouville-type derivatives are the necessary ones, because they fully preserve backward compatibility with classical results. Such derivatives are used to define and solve the fractional Ambartsumian equation. First, a solution in terms of a slowly convergent fractional Taylor series is obtained. Then, a simple solution expressed in terms of an infinite linear combination of Mittag–Leffler functions is deduced. A fast algorithm, based on a bilinear transformation and using the fast Fourier transform, is described and demonstrated for its approximate numerical realisation.15434671engAmbartsumian equationbilinear transformationfractional derivativeGrünwald–LetnikovMittag–Leffler functionGeneral Materials ScienceInstrumentationGeneral EngineeringProcess Chemistry and TechnologyComputer Science ApplicationsFluid Flow and Transfer Processesjournal article10.3390/app13020871https://www.scopus.com/pages/publications/85146730967