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

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. & COPY; 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).