The structure of gas-dynamic shock waves is of interest in hypersonic flow studies and also constitutes a straightforward test for competing kinetic theories. The description of the shock profiles may be obtained from a second-order theory in the Knudsen number. The BGK approximation to the Boltzmann equation introduces additional terms in the transport of momentum and energy. These relations, known as the Burnett equations, improve the agreement between calculated shock profiles and experiment. However, for some formulations of these equations, the solution breaks down at a critical Math number. In addition, certain terms in the Burnett equations allow unphysical effects in gas flow. A modified kinetic theory has been proposed by Woods (An Introduction to the Kinetic Theory of Gases and Magnetoplasmas, Oxford Univ. Press, Oxford, 1993) which eliminates the frame dependence of the standard kinetic theory and corrects some of the second-order terms. This article describes a novel method devised to solve the time-independent conservation equations, including the second-order terms. The method is used to solve the shock structure problem in one dimension. It is based on a finite difference global scheme (FDGS), in which a Newton procedure is applied to a discretized version of the governing equations and boundary conditions. The method is first applied to the Navier-Stokes formulation of the shock equations. It is then successfully used to integrate a modified version of the second-order equations derired by Woods for monatomic gases, up to a Mach number of 30. Results of the calculations are compared with experimental data for Argon gas flows characterized by up-stream Mach numbers up to 10. The agreement is good, well within the data point spread. The FDGS method converges rapidly and it may be used to study other problems of the same general nature.