Polynomial Chaos for random fractional order differential equations

Abstract

Fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models, where such effects are difficult to incorporate and often neglected. Also, many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations. Thus, to be able to deal with many real life problems, it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty. The aim of this paper is to study the application of the generalized Polynomial Chaos (gPC) to random fractional ordinary differential equations. The method of Polynomial Chaos has played an increasingly important role when dealing with uncertainties. The main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses. However, to apply Polynomial Chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations. In order to avoid more complex numerical computations and obtain accurate solutions we rely on Richardson extrapolation. It is shown that the application of generalized Polynomial Chaos method in conjunction with Richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations.