Polynomial Chaos for random fractional order differential equations

González-Parra, Gilberto; Chen-Charpentier, Benito M.; Arenas, Abraham J.

doi:10.1016/j.amc.2013.10.051

Cited by 15 publications

(14 citation statements)

References 31 publications

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“…Finally, it is important to point out that there is a number of fractional derivatives such as Caputo, Riemann-Liouville, Grünwald-Letnikov [24,25]. In this paper we will only consider the Caputo derivative since we are interested in constructing a mean square solution to the general fractional linear first-order differential equation with random coefficients and random initial condition.…”

Section: Introductionmentioning

confidence: 99%

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

Burgos

López

Villafuerte

et al. 2017

Chaos, Solitons & Fractals

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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.

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Section: Introductionmentioning

confidence: 99%

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

Burgos

López

Villafuerte

et al. 2017

Chaos, Solitons & Fractals

View full text Add to dashboard Cite

show abstract

“…There are some different points of view for random differential systems, such as the Itô-Doob calculus approach and the sample calculus approach. The Itô-Doob calculus is used to study stochastic differential systems with the white noise (see e.g., Ren, Dai, & Sakthivel, 2013;Shen, Meng, & Shi, 2014;Shen, Shi, & Sun, 2010;Shen & Sun, 2012;Zamani & Abate, 2014;Zhao & Deng, 2014), while the sample calculus is used to study differential systems with random coefficients (see e.g., Balachandran & Kim, 2010;Charrier, 2012;Cortes, Jodar, Company, & Villafuerte, 2011;Dauer & Balachandran, 1997; Gonzalez-Parra, Chen-Charpentier, & Arenas, 2014;Park & Jeong, 2013;Zhang, Wang, Ding, & Shu, 2014b). In this paper, we focus ourselves on differential systems with random coefficients.…”

Section: Introductionmentioning

confidence: 99%

Sample controllability of impulsive differential systems with random coefficients

Zhang

2014

International Journal of Systems Science

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In this paper, we investigate the controllability of impulsive differential systems with random coefficients. Impulsive differential systems with random coefficients are a different stochastic model from stochastic differential equations. Sufficient conditions of sample controllability for impulsive differential systems with random coefficients are obtained by using random Sadovskii's fixed-point theorem. Finally, an example is given to illustrate our results.

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“…The analysis of RFDEs has been undertaken recently. Interesting results on the existence and uniqueness of solutions for initial value problems (IVPs) for RFDEs that extend their deterministic counterpart are exhibited in Lupulescu et al 13,14 In González-Parra, 15 one adapts the polynomial chaos method to solve some RFDEs. Recently, some of the authors have studied autonomous and nonautonomous linear RFDEs using the so-called mean-square calculus.…”

Section: Introductionmentioning

confidence: 99%

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

Burgos

Calatayud

López

et al. 2018

Math Methods in App Sciences

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This paper provides a full probabilistic solution of the randomized fractional linear nonhomogeneous differential equation with a random initial condition via the computation of the first probability density function of the solution stochastic process. To account for most generality in our analysis, we assume that uncertainty appears in all input parameters (diffusion coefficient, source term, and initial condition) and that a wide range of probabilistic distributions can be assigned to these parameters. Throughout our study, we will consider that the fractional order of Caputo derivative lies in ]0,1], that corresponds to the main standard case. To conduct our analysis, we take advantage of the random variable transformation technique to construct approximations of the first probability density function of the solution process from a suitable infinite series representation. We then prove these approximations do converge to the exact density assuming mild conditions on random input parameters. Our theoretical findings are illustrated through 2 numerical examples. KEYWORDSfirst probability density function, random fractional differential equations, random variable transformation technique

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Polynomial Chaos for random fractional order differential equations

Cited by 15 publications

References 31 publications

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

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

Sample controllability of impulsive differential systems with random coefficients

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

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