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Solving random fractional second-order linear equations via the mean square Laplace transform: Theory and statistical computing

Author

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

Abstract

This paper deals with random fractional differential equations of the form, CD0+αX(t)+AX˙(t)+BX(t)=0, t>0, with initial conditions, X(0)=C0 and X˙(0)=C1, where CD0+αX(t) stands for the Caputo fractional derivative of X(t). We consider the case that the fractional differentiation order is 1<α<2. For the sake of generality, we further assume that C0, C1, A and B are random variables satisfying certain mild hypotheses. Then, we first construct a solution stochastic process, via a generalized power series, which is mean square convergent for all t>0. Secondly, we provide explicit approximations of the expectation and variance functions of the solution. To complete the random analysis and from this latter key information, we take advantage of the Principle of Maximum Entropy to calculate approximations of the first probability density function of the solution. All the theoretical findings are illustrated via numerical experiments.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:418:y:2022:i:c:s0096300321009292
    DOI: 10.1016/j.amc.2021.126846
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    References listed on IDEAS

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    1. 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.
    2. Acedo, L. & Burgos, C. & Cortés, J.-C. & Villanueva, R.-J., 2017. "Probabilistic prediction of outbreaks of meningococcus W-135 infections over the next few years in Spain," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 486(C), pages 106-117.
    3. A. A. Hemeda, 2013. "Solution of Fractional Partial Differential Equations in Fluid Mechanics by Extension of Some Iterative Method," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-9, December.
    4. Corina D. Constantinescu & Jorge M. Ramirez & Wei R. Zhu, 2019. "An application of fractional differential equations to risk theory," Finance and Stochastics, Springer, vol. 23(4), pages 1001-1024, October.
    5. 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.
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    Cited by:

    1. 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.

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