Two-dimensional probability distribution of the solution to the random Burgers–Riemann problem

Abstract

We investigate Burgers’ partial differential equation with random Riemann initial states and random-field solution. Previous works obtained the generalized probability density function in the one-dimensional case, by using different methods. We explore a technique that has not been employed in this context before, which fully develops the potential of the Dirac delta function. We start with the one-dimensional scenario and afterward extend to two dimensions. The generalized probability density function is derived in those cases, both for non-independent and independent initial conditions, and general statistics are obtained as a consequence.