Extending the fractional Legendre differential equation to the random setting: Computation of the solution, statistical moments and density

In this paper, we solve a class of fractional Legendre differential equations in Caputo sense where its coefficients and initial conditions are random variables. The analysis is performed using the mean square calculus. First, we apply the random Fröbenius method to construct a generalized power series solution for this equation. Under mild hypotheses on the coefficients and initial conditions, we prove that the solution is mean square convergent, with a radius of convergence equals 1. This result is optimum in the sense of retaining the same radius of convergence as in the deterministic scenario. Furthermore, we derive explicit approximations of the mean and variance functions of the solution. To complete the probabilistic analysis, we use the Random Variable Transformation method to construct approximations for the first density function of the solution. Two examples are given to illustrate the reliability of our theoretical findings using different probability distributions for coefficients and initial conditions.