AbstractA new method for the solution of the Boltzmann equation for time-dependent or stationary neutron transport is presented. The essentials of the method are a Fourier transform of the transport integral equation and the decomposition of the new kernel into a bilinear form. In the monoenergetic case, the transport problem is reduced to the solution of an (infinite) linear system of equations. It is important that the matrix elements in this linear system be determined only once for any geometry and that this be done in an analytical way. In the present paper, the matrix elements for plane, spherical, and cylindrical geometries are constructed explicitly and are presented in a form suitable for numerical applications. To illustrate the efficiency of the method, the critical dimensions for these geometries have been determined in various integral transformITNapproximations,Nbeing the order of the linear system after truncation. For samples of a thickness up to∼10 mean-free-paths, the results for the critical dimension in stationary problems, or for the fundamental time-decay constants in nonstationary problems, have in theIT4approximation, at least, the accuracy of the correspondingS16results.The convergence of theITNresults, with respect to the orderN, is very fast. The results forN= 4 can be considered as exact within one or two units in the fifth digit. The convergence behavior of theSNand thePNmethod is decidedly slower. In contrast to theSN, PNor Case’s method, the smaller the system, the better the accuracy of theITNcalculations for fixedNis. The increase in accuracy by increasingNrequires less computational effort for theITNapproximation than forSNorPNapproximation. Apart from providing an analytical standard for the check of numerical approximations, the present method is a simple and rigorous tool for the calculation of neutron distributions, particularly in small systems.