Methods and Applications of Analysis

Volume 27 (2020)

Number 2

On approximation of a hyper-singular transport operator and existence of solutions

Pages: 125 – 152



J. Tervo (Department of Applied Physics, University of Eastern Finland, Kuopio, Finland)

M. Herty (RWTH Aachen University, IGPM, Templergraben, Germany)


The paper considers a class of linear Boltzmann transport equations containing hyper-singular integrals in their collision terms. A partial integro-differential approximation is derived and the approximation error is analysed. Existence results of solutions for the approximative initial inflow boundary value problem are shown. The approximative operator of this type may be used as a refined model for the dose calculation of radiation therapy. The paper continues authors’ recent analytical results on the approximation of the linear transport equations.


linear Boltzmann transport equation, hyper-singular integral operators, variational formulation, charged particle transport

2010 Mathematics Subject Classification

35Q20, 35R09, 45E99

Received 6 March 2019

Accepted 2 April 2020

Published 20 August 2020