A full probabilistic solution of the random linear fractional differential equation via the random variable transformation technique. (8th May 2018)

• Record Type: Journal Article
• Authors: Burgos, C.; Calatayud, J.; Cortés, J.‐C.; Navarro‐Quiles, A.
• Other Names: Anguelov Roumen guestEditor.; Banasiak Jacek guestEditor.; Debbouche Amar guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 41:Number 18(2018)
• Page Start: 9037
• Page End: 9047
• Publication Date: 2018-05-08
• Subjects: first probability density function -- random fractional differential equations -- random variable transformation technique; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.4881
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : 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.