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Extending the deterministic Riemann–Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations

Author

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

Abstract

This paper extends both the deterministic fractional Riemann–Liouville integral and the Caputo fractional derivative to the random framework using the mean square random calculus. Characterizations and sufficient conditions to guarantee the existence of both fractional random operators are given. Assuming mild conditions on the random input parameters (initial condition, forcing term and diffusion coefficient), the solution of the general random fractional linear differential equation, whose fractional order of the derivative is α ∈ [0, 1], is constructed. The approach is based on a mean square chain rule, recently established, together with the random Fröbenius method. Closed formulae to construct reliable approximations for the mean and the covariance of the solution stochastic process are also given. Several examples illustrating the theoretical results are included.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:102:y:2017:i:c:p:305-318
    DOI: 10.1016/j.chaos.2017.02.008
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    Cited by:

    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. El-Beltagy, Mohamed & Etman, Ahmed & Maged, Sroor, 2022. "Development of a fractional Wiener-Hermite expansion for analyzing the fractional stochastic models," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    3. 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.
    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).

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