# An Analytical Solution for the General Perturbed Diffusion Equation by an Integral Transform Technique

## Abstract

In the developments of nuclear energy, new reactor concepts are being proposed and explored, where innovative ideas need to be tested by means of simulations. Although the original neutron calculations start from a transport equation, many approaches reduce the calculation to diffusion equations, since the Boltzmann equation for neutron transport is still considered a challenge (see, for example, [Le05], [Se07], and the references therein). A detailed sequence, starting from a neutron transport equation (Boltzmann equation) until the reduction to a diffusion phenomenon using Fick’s hypothesis, is given, for instance, in [Se07]. Our principal concern here is an effective analytical method for the general perturbed neutron diffusion equation by an integral transform technique. To this end, we present a procedure that allows us to construct an analytical solution of the multi-group neutron diffusion equation in Cartesian geometry using well-established integral transform procedures [He05]. Once the general structure of the solution is determined, we may directly calculate the neutron flux (which is an analytical expression), and the only quantity which is determined numerically at the end of the calculation is criticality. In what follows we present the procedure, considering a generic multi-group calculation for an arbitrary number of energy intervals. Due to the fact that the geometric extension of the reactor core is typically very much larger in one dimension compared to the other two length scales, we may cast the calculation into a two-dimensional (2D) setting.

## Keywords

Boltzmann Equation Interface Condition Reactor Core Total Problem Liouville Problem## Preview

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## References

- [Co93]Cotta, R.M.:
*Integral Transforms in Computational Heat and Flow*, CRC Press, Boca Raton, FL (1993).Google Scholar - [CoMi97]Cotta, R.M., Mikhailov, M.D.:
*Heat Conduction: Lumped Analysis, Integral Transforms, Symbolic Computation*, McGraw-Hill, Chichester, UK (1997).Google Scholar - [He05]Heinen, I.R.: Master thesis. Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil (2005) (Portuguese).Google Scholar
- [Le05]Leppänen, J.: A new assembly-level Monte Carlo neutron transport code for reactor physics calculations, in
*Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications*, American Nuclear Society, LaGrange Park, IL (2005) (on CD-ROM).Google Scholar - [Se07]Sekimoto, H.:
*Nuclear Reactor Theory. Part II*, COE-INES Tokyo Institute of Technology (2007).Google Scholar