| Abstract. The term singular perturbation problem applies to any problem whose solution has a near-singular behaviour relative to a perturbation parameter in the system. This usually represents the appearance of several time scales (or, for example, length scales) meaning that various components of the system change at different rates. The long term solution is of particular interest to engineers and scientists as it describes the long-term behaviour of the system. There are two general ways for finding solutions of singular perturbation problems problems: an analytical approach and numerical integration. The common analytical approach is to construct two asymptotic approximations: one in the neighbourhood of the singular point, another far away from it and then to connect them in the overlap region where both expansions are valid (method of matched asymptotic expansions). The main difficulties associated with this method are solving the outer problem in a closed form, finding the asymptotics at large times for the inner problem and matching the two. At the same time, despite a tremendous progress in numerical analysis, numerical integration of singular perturbation or stiff problems still remains time-consuming as fine meshes and small step sizes are required throughout the calculation to ensure it is stable. There are certainly advantages of combining the two approaches and developing hybrid symbolic-numerical algorithms for solving singular perturbation problems. Taking advantage of solution structure may help to avoid stiffness problems, greatly reduce computational time and resources. The so-called asymptotic-numerical methods for particular problems have been developed by several authors. However, in these papers asymptotic analysis has been performed on paper manually. It has then been manually included in the design of the numerical code for the specific problem. It is advantageous to implement asymptotic analysis in CAS and to automatically incorporate it into numerical code. This will yield higher precision, improve efficiency, and, in particular, result in more general numerical codes for stiff problems as is already the case for other types of odes (e.g. Taylor method). At least two approaches for applying the results of asymptotic analysis in the numerical computation can be envisaged. One approach starts from a robust domain decomposition method and adjusts the decomposition on the basis of asymptotic criteria. Another approach starts from a known asymptotic expansion and devises a numerical scheme to compute various terms in the expansion. I will discuss ideas of the second approach. A symbolic technique is based on the method of matched asymptotic expansions and in particular on the polyhedron algorithm for finding the appropriate scaling and local approximating systems near the singularity. I will show how application of the hybrid method can improve or facilitate the procedure of solving the singularly perturbed problems. A symbolic-numerical algorithm for tackling scalar semilinear boundary value problems will be discussed. The algorithm is based on the Newton quasilinearisation method and asymptotic solutions of linearised problems. |