STOCHASTIC GALERKIN METHOD FOR LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

We consider an initial value problem of a linear fractional differential equation (FDE) with a Caputo derivative. The right-hand side of the FDE includes a coefficient, which depends on a random variable to model a variability in uncertainty quantification. Consequently, the solution of the FDE becomes a random process. We expand the random process into the generalized polynomial chaos, where a series consists of unknown time-dependent coefficients and predetermined basis polynomials. A stochastic Galerkin approach yields a larger deterministic linear system of FDEs, whose solution represents an approximation of the coefficient functions. We show that the approximations of the stochastic Galerkin systems converge to the exact random process, provided that some common assumptions are satisfied. Furthermore, we present results of numerical computations, where initial value problems of the stochastic Galerkin systems are solved. These results verify the deduced convergence property.