A system of differential equations with a small parameter: a numerical solution based on asymptotic representations

Background. There is a system of nonlinear differential equations with a small parameter on derivatives. In this paper we consider the initial value problem, which has a well-known form of asymptotic expansion in a small parameter. The numerical method is based on the approach of a numerical sum of initial asymptotic series. Materials and methods. For numerical solution it is necessary to calculate the first few functions in an asymptotic series. The authors set tasks for regular and boundary layer functions and constructed numerical algorithms for determining the required accuracy of regular functions on a segment and boundary layer functions on a semi-infinite interval. Results. The researchers have proved stability of the methods, allowing the use of the values of already computed functions in problems for subsequent functions. The artilce estimates the number of arithmetic operations required by this method, and compares it with conventional numerical methods. This estimate shows the computational efficiency of implementation of the method. Conclusions. The numerical method is easier to implement, especially when solving problems for series of different values of a small parameter. For the considered type of singularly perturbed system the method requires no additional restrictions on the coefficients and on the initial conditions.