Similarity reductions of the (1 + 3)-dimensional Burgers equation

We consider the (1 + 3)-dimensional Burgers equation ut=uxx+uyy+uzz+uuxut=uxx+uyy+uzz+uux which has considerable interest in mathematical physics. We complete the list of similarity reductions that are obtained from the Lie symmetries admitted by this equation. To achieve this goal we employ two- and three-dimensional subalgebras of the Lie symmetry algebra, in addition to the one-dimensional subalgebras that appears in the literature. We solve analytically, where it is possible, the reduced ordinary differential equations and hence, we obtain closed-form solutions for the original equation. Finally, we derive certain non-Lie solutions.