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A full probabilistic solution of the random linear fractional differential equation via the random variable transformation technique
Author(s) -
Burgos C.,
Calatayud J.,
Cortés J.C.,
NavarroQuiles A.
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4881
Subject(s) - mathematics , probability density function , random variable , probabilistic logic , stochastic differential equation , transformation (genetics) , random function , stochastic process , range (aeronautics) , mathematical analysis , statistics , biochemistry , chemistry , materials science , composite material , gene
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.