Uncertainty quantification for the random viscous Burgers’ partial differential equation by using the differential transform method

Abstract

Burgers’ partial differential equation has usually served as a benchmark for new stochastic methods. In this paper, we quantify the propagation of uncertainty for the Burgers’ equation when the initial condition and the viscosity are random, by using the differential transform method. By employing the exact random field solution that arises from the Cole–Hopf transformation as reference, we test the mean square convergence of the inverse differential transform. Contrary to the well-studied case of linear random ordinary differential equations, we show that convergence here can only be expected in a small neighborhood in space–time when the input random parameters have small dispersion. Nonetheless, in the region of convergence, rapid approximations of the main statistics and of the density function can be determined at virtually no computational cost.