Symmetry and separability of the neutron diffusion equation

Abstract

Separation of variables is one of the oldest techniques for solving certain classes of partial differential equations (PDEs). As is the case with many other solution techniques for differential equations, separation of variables may be codified within the broader framework of symmetry analysis. Though the separation of variables technique is frequently used in the nuclear engineering context with various equations describing neutron transport, its connection to the symmetries of those equations has not yet been thoroughly established. It is thus the purpose of this work to establish that connection using neutron diffusion as both an initial step toward analysis of more generally applicable equations, and as a connection to previous results in related problems. Using Lie group analysis, it is found that the traditional space-time separable solution of the neutron diffusion equation (featuring a single αeigenvalue) corresponds to time translation and flux scaling symmetries. Additional solutions of this equation are also constructed using its broader symmetry set. IntroductionSeparation of variables is one of the oldest techniques for solving certain classes of partial differential equations (PDEs). Like many other such methods, it is often employed as a specific element out of a 'bag of tricks', so that given a certain equation the technique can either be expected to apply, or not. In this more general sense, it was not until the late 19th century that S. Lie began his investigation into a 'unification theory' of solution techniques for differential equations, inspired in part by the preceding successful application of group theory to algebraic equations. Lie was successful in his aim, and indeed it has since been found that a wide variety of solution techniques for differential equations can be integrated into the broader framework of symmetry analysis (as now set forth in rigorous detail by Ovsiannikov [1], Bluman and Anco [2], Ibragimov [3], Hydon [4], Olver [5], Cantwell [6], Stephani [7], and many others). Beginning in the 1970s, C Boyer, E Kalnins, and W Miller embarked upon a thorough program of analyzing the connection between Lie symmetries, separation of variables, and special functions in a variety of contexts, including Schrödinger equations, Helmholtz and Laplace equations, harmonic oscillator equations, Hamilton-Jacobi equations, wave equations, and multitudinous other structures in numerous coordinate systems and spaces. This effort is largely summarized in two books and a survey by Miller [8-10]. Following this work, a generalized attempt at demonstrating the connection in question was provided by Gegelia and Markovski [11]. Additional studies have been performed by Chou and Qu [12] for nonlinear diffusion-convection equations and Estevez et al [13] for a porous medium equation, and Polyanin and Zhurov [14] for the axisymmetric unsteady boundary-layer equations, to name a few. Modern developments intended to reconcile (or not) separation of variables within the symmetry analysis framework appear t...

Conditions for translation and scaling invariance of the neutron diffusion equation

“…Indeed, the outcomes of this work are closely related to those of Ramsey et al [27], who have noted that the usual space-time separable solutions of the monoenergetic neutron diffusion equation have well-defined ties to translational and scaling symmetry groups. These results are implicitly encoded in the current study, which presumably includes broader classes of space-time separable and other solutions with special physical properties.…”

“…• Various instantiations of separable time-dependent neutron diffusion equations by Ramsey, et al [27].…”