Extending the deterministic Riemann–Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations. (September 2017)

• Record Type: Journal Article
• Authors: Burgos, C.; Cortés, J.-C.; Villafuerte, L.; Villanueva, R.-J.
• Is Part Of: Chaos, solitons and fractals. Volume 102(2017)
• Page Start: 305
• Page End: 318
• Publication Date: 2017-09
• Subjects: Random mean square Riemann–Liouville integral -- Random mean square Caputo derivative -- Random fractional linear differential equation -- Random Fröbenius method; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2017.02.008
• Languages: English
• ISSNs: 0960-0779

Abstract

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.