A high order approximation, the SKN method-a mnemonic for synthetic kernel-is introduced for solving radiation transfer problems in one- and two-dimensional geometries. The method relies on approximating the integral transport kernel by a sum of exponential kernels. The integral equation is then reducible to a set of coupled second-order differential equations-SKN equations. In this study, two types of boundary conditions have been proposed and explored. Naive boundary condition assumes one-dimensional slab boundary conditions on each boundary. Corrected boundary conditions; tackles the error terms that result from the approximation and is based on the minimization of the error term. The solutions of a test problem-incident radiation, outgoing intensities and heat fluxes at the boundaries-are compared with those of obtained by direct numerical solution of the integral transfer equation. Solutions are obtained for N = 2, 3,4, and 5 and are in agreement with the direct numerical solutions even for N = 2 and 3. The corrected boundary condition gives better solutions than naive boundary condition in optically thin configurations; the relative errors for the intensities are below 1% while 2-4% errors are encountered in the radiation function solutions. (C) 2002 Elsevier Science Ltd. All rights reserved.