# 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 α-eigenvalue) corresponds to time translation and flux scaling symmetries. Additional solutions of this equation are also constructed using its broader symmetry set.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. of Illinois, Urbana, IL (United States). Dept. of Nuclear, Plasma, and Radiological Engineering
- Univ. of Wisconsin, Madison, WI (United States). Dept. of Engineering Physics

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1482936

- Report Number(s):
- LA-UR-18-21934

Journal ID: ISSN 2399-6528

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Physics Communications

- Additional Journal Information:
- Journal Volume: 2; Journal Issue: 10; Journal ID: ISSN 2399-6528

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; neutron diffusion; symmetry analysis; separation of variables

### Citation Formats

```
Ramsey, Scott D., Tellez, Jacob A., Riewski, Eric J., and Temple, Brian A. Symmetry and separability of the neutron diffusion equation. United States: N. p., 2018.
Web. doi:10.1088/2399-6528/aae2a4.
```

```
Ramsey, Scott D., Tellez, Jacob A., Riewski, Eric J., & Temple, Brian A. Symmetry and separability of the neutron diffusion equation. United States. doi:10.1088/2399-6528/aae2a4.
```

```
Ramsey, Scott D., Tellez, Jacob A., Riewski, Eric J., and Temple, Brian A. Wed .
"Symmetry and separability of the neutron diffusion equation". United States. doi:10.1088/2399-6528/aae2a4. https://www.osti.gov/servlets/purl/1482936.
```

```
@article{osti_1482936,
```

title = {Symmetry and separability of the neutron diffusion equation},

author = {Ramsey, Scott D. and Tellez, Jacob A. and Riewski, Eric J. and Temple, Brian A.},

abstractNote = {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 α-eigenvalue) corresponds to time translation and flux scaling symmetries. Additional solutions of this equation are also constructed using its broader symmetry set.},

doi = {10.1088/2399-6528/aae2a4},

journal = {Journal of Physics Communications},

number = 10,

volume = 2,

place = {United States},

year = {2018},

month = {9}

}

#### Figures / Tables:

_{∞}is constant, $γ$=ωr and $τ$=$α$t.

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*