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.

Burgers的偏微分方程通常作为新的随机方法的基准。在本文中，我们使用微分变换方法对初始条件和粘度为随机的Burgers方程的不确定性传播进行了量化。通过使用由Cole-Hopf变换产生的精确随机场解作为参考，我们测试了逆差分变换的均方收敛。与对线性随机常微分方程的深入研究相反，我们表明，当输入随机参数的色散较小时，此处的收敛只能在时空的小范围内进行。但是，在趋同领域，