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

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.

A full probabilistic solution of the random linear fractional differential equation via the random variable transformation technique

“…Independence among random inputs is a usual assumption in the analysis of RFDEs (see, Burgos et al, 16,17 for instance); moreover, it is also natural from an applied standpoint since random inputs usually have not direct relationship. Hypothesis H2 is equivalent to assume that γ∈L 1 (Ω), being L 1 (Ω) the biggest of L p (Ω)-Banach spaces, 23 so this is a very general assumption since to this space belong most of important random variables (Gaussian, gamma, beta, etc).…”

“…Recently, some of the authors have studied autonomous and nonautonomous linear RFDEs using the so-called mean-square calculus. 16,17 In both contributions, we have taken advantage of random Fröbenius method to solve linear RFDEs through mean-square random generalized power series and then to obtain the statistical information of the solution stochastic process through the mean and the variance. However, if y(t,ω) denotes the solution stochastic process of a RFDE defined in a complete probability space ðΩ; F; PÞ, then a more ambitious target is the computation of the first probability density function (1-PDF), f 1 (y, t), of y(t, ω) because from it, one can compute all 1-dimensional statistical moments at every time instant t,…”

“…According to Burgos et al, 16 using a random Fröbenius method, the solution stochastic process of random fractional IVP (2) can be represented by yðt; ωÞ ¼ β 0 ðωÞS 1 ðt; α; λðωÞÞ þ γðωÞS 2 ðt; α; λðωÞÞ;…”

Advances on Uncertainty Quantification Techniques for Dynamical Systems: Theory and Modelling

“…The results of this chapter have been published in [24]. With regard to this paper, the PhD candidate has contributed by working in its complete development with more emphasis on the theoretical results (definition of random fractional operators, construction of a convergent solution) and preparing the numerical examples.…”

“…So far, we have constructed a formal solution SP to random IVP (3.1) and now, assuming that input RVs satisfy hypothesesĤ1 andĤ2, we need to check that conditions D1-D6 hold. As this can be done by taking the same steps shown in detail in [24], they will be skipped here. The analysis of m.s.…”