Abstract: Numerical methods are developed for solving the Cauchy problem for systems of ordinary differential equations with a single limiting singular point lying on the right boundary of the considered range of the argument. Such initial value problems arise in a variety of areas, such as inelastic deformation of metal structures at various temperatures and stresses under creep conditions, the computation of strength characteristics and the estimation of residual strain in the design of nuclear reactors, the construction and aerospace industries, and mechanical engineering. Since such initial value problems are ill-conditioned, their numerical solution faces considerable difficulties. Conventional explicit methods can be used for problems of this class, but only outside a neighborhood of limiting singular points, where the error of the numerical solution grows sharply. As a result, the step size has to be reduced significantly, which increases the computation time and makes explicit methods computationally expensive. For the numerical solution, it is possible to use the method of solution continuation, whereby the original argument of the problem is replaced by a new one such that the transformed problem is better conditioned. However, the best argument does not give desirable numerical advantages in this case, since the form of the original problem is complicated considerably. As a more suitable approach, we propose a specialized argument for solution continuation, which is called a modified best argument. For tubular samples of X18H10T steel subjected to tension to fracture, the advantages of the method of solution continuation with respect to a modified best argument are demonstrated by comparing it with traditional explicit methods for the Cauchy problem and with the best parametrization. The reliability of the results is confirmed by comparing them with experimental measurements and data of other authors. © 2020, Pleiades Publishing, Ltd.