Numerical Solution of a Degenerate Parabolic Equation from Boltzmann-Fokker-Planck Equation. Application to Radiotherapy

Abstract

The number of photons and electrons are needed for the dose calculation of radiotherapy of cancer treatment. The number of photons are calculated employing the Boltzmann Transport equation (BTE) by direct calculation. But it is very difficult to solve directly the BTE for electrons as the mean free path of electrons is much smaller than the mean free path of photons. Thus, alternatively, one can solve the Fokker- Planck equation instead of BTE, which is an approximation of the BTE based on the fact that the scattering process for electrons are very forward-peaked. In this paper, we propose a new numerical scheme for a degenerate parabolic partial differential equation using finite difference method of final value problem.

Keywords: Electron, Photon, Radiotherapy, forward-peaked, degenerate parabolic partial differential     equation.