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Applied Mathematics and Nonlinear Sciences

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Available online: July 28, 2017

Mean square calculus and random linear fractional differential equations: Theory and applications

Clara Burgos Simón, Juan Carlos Cortés López, Laura Villafuerte Altúzar, Rafael Jacinto Villanueva Micó
Volume 2 - Issue 2.     Year 2017.     Pages 317–328.

DOI: 10.21042/AMNS.2017.2.00026


The aim to this paper is to study, in the mean square sense, a class of random fractional linear differential equation where the initial condition and the forcing term are assumed to be second-order random variables. The solution stochastic process of its associated Cauchy problem is constructed combining the application of a mean square chain rule for differentiating second- order stochastic processes and the random Fröbenius method. To conduct our study, first the classical Caputo derivative is extended to the random framework, in mean square sense. Furthermore, a sufficient condition to guarantee the existence of this operator is provided. Afterwards, the solution of a random fractional initial value problem is built under mild conditions. The main statistical functions of the solution stochastic process are also computed. Finally, several examples illustrate our theoretical findings.

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Applied Mathematics, Nonlinear Sciences


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