A solution of the neutron diffusion equation in hemispherical symmetry using the homotopy perturbation method

The homotopy perturbation method is used to formulate a new analytic solution of the neutron diffusion equation both for a sphere and a hemisphere of fissile material. Different boundary conditions are investigated; including zero flux on boundary, zero flux on extrapolated boundary, and radiation boundary condition. The interaction between two hemispheres with opposite flat faces is also presented. Numerical results are provided for one-speed fast neutrons in 235U. A comparison with Bessel function based solutions demonstrates that the homotopy perturbation method can exactly reproduce the results. The computational implementation of the analytic solutions was found to improve the numeric results when compared to finite element calculations.