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On the random fractional Bateman equations

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  • Jornet, Marc

Abstract

We study the random fractional Bateman equations for a radioactive decay chain, in the Caputo-derivative sense. On the one hand, a fractional order entails memory effects in the system, and on the other hand, randomness accounts for uncertainties on the physical quantities. Results on the deterministic fractional Bateman equations and on the random ordinary Bateman equations were recently published. The proposed stochastic fractional model extends these previous works. We construct the stochastic solution; first, in the pathwise sense by using the Laplace-transform method, and second, in the mean-square sense by applying Banach fixed-point theorem. The probability density function, that is associated to the solution of a chain of length three, is computed in the present work by a transformation method. Continuity of this density with respect to the fractional order, in the pointwise and in the total variation sense, is investigated. Numerical results are included for forward uncertainty quantification, with the purpose of illustrating the developed theory.

Suggested Citation

  • Jornet, Marc, 2023. "On the random fractional Bateman equations," Applied Mathematics and Computation, Elsevier, vol. 457(C).
    • Handle: RePEc:eee:apmaco:v:457:y:2023:i:c:s0096300323003661
    DOI: 10.1016/j.amc.2023.128197
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    Cited by:

    1. Jornet, Marc, 2024. "Generalized polynomial chaos expansions for the random fractional Bateman equations," Applied Mathematics and Computation, Elsevier, vol. 479(C).

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