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Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus

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  • Burgos, C.
  • Cortés, J.-C.
  • Debbouche, A.
  • Villafuerte, L.
  • Villanueva, R.-J.

Abstract

The aim of this paper is to study a generalization of fractional Airy differential equations whose input data (coefficient and initial conditions) are random variables. Under appropriate hypotheses assumed upon the input data, we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense. Afterwards, we provide reliable explicit approximations for the main statistical information of the solution process (mean, variance and covariance). Further, we show a set of numerical examples where our obtained theory is illustrated. More precisely, we show that our results for the random fractional Airy equation are in full agreement with the corresponding to classical random Airy differential equation available in the extant literature. Finally, we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional Airy differential equation by combining the knowledge of the mean and the variance and the Principle of Maximum Entropy.

Suggested Citation

  • Burgos, C. & Cortés, J.-C. & Debbouche, A. & Villafuerte, L. & Villanueva, R.-J., 2019. "Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 15-29.
    • Handle: RePEc:eee:apmaco:v:352:y:2019:i:c:p:15-29
    DOI: 10.1016/j.amc.2019.01.039
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    References listed on IDEAS

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    1. Wu, Guo-Cheng & Baleanu, Dumitru & Luo, Wei-Hua, 2017. "Lyapunov functions for Riemann–Liouville-like fractional difference equations," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 228-236.
    2. Wu, Chufen & Yang, Yong & Zhao, Qianyi & Tian, Yanling & Xu, Zhiting, 2017. "Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 122-143.
    3. Burgos, C. & Cortés, J.-C. & Villafuerte, L. & Villanueva, R.-J., 2017. "Extending the deterministic Riemann–Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 305-318.
    4. Li, Qiang & Kang, Ting & Zhang, Qimin, 2018. "Mean-square dissipative methods for stochastic age-dependent capital system with fractional Brownian motion and jumps," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 81-92.
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    Cited by:

    1. Dhama, Soniya & Abbas, Syed & Debbouche, Amar, 2020. "Doubly-weighted pseudo almost automorphic solutions for stochastic dynamic equations with Stepanov-like coefficients on time scales," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    2. Villafuerte, L., 2023. "Solution processes for second-order linear fractional differential equations with random inhomogeneous parts," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 17-48.
    3. Akinlar, M.A. & Inc, Mustafa & Gómez-Aguilar, J.F. & Boutarfa, B., 2020. "Solutions of a disease model with fractional white noise," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    4. Burgos, C. & Cortés, J.-C. & Villafuerte, L. & Villanueva, R.J., 2022. "Solving random fractional second-order linear equations via the mean square Laplace transform: Theory and statistical computing," Applied Mathematics and Computation, Elsevier, vol. 418(C).
    5. Jornet, Marc, 2021. "Beyond the hypothesis of boundedness for the random coefficient of the Legendre differential equation with uncertainties," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    6. Kim, Hyunsoo & Sakthivel, Rathinasamy & Debbouche, Amar & Torres, Delfim F.M., 2020. "Traveling wave solutions of some important Wick-type fractional stochastic nonlinear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).

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